径向Kohn-Sham方程的谱有限元方法
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径向Kohn-Sham方程的谱有限元方法
本文以Kohn-Sham方程为例，介绍了现在已发展成型的几种近似交换相关泛函和密度泛函理论的计算步骤，主要以代码的形式呈现。
径向密度泛函理论给出了求解下面一个Kohn-Sham方程的一套思路： 
其中，
下面给出程序，后面再做解释：
- 能量求解及其子函数：
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % % Script file to run radialdft:%clc;clear all;close all%% Set timer%tstart = cputime;% % ==========================================% Problem data (Chemistry part of the input)% ==========================================% % For benchmarking, check:% https://www.nist.gov/pml/data/atomic-total-energies-and-eigenvalues-html% % Set up Problem Parameters:% fprintf('\n');fprintf('----> Setting up the Problem Parameter . . .\n');fprintf('\n');Zion = 30; % Carbonqtot = 0.0; % net charge; nonzero for ionic calculationnelec = Zion - qtot;problemdata = struct('Z',Zion, 'nelec',nelec);% % Domain:% rmin = 0.0;rmax = 10.0;%% Mesh:% domain = [rmin rmax];toplot = 'on';% % 1. Mesh for radial Schrodinger eq.:% meshtypeSch = 1; % type: 1 = uniform, 2 = logarithmic, 3 = exponentialaSch = 10.0;pSch = 8;numelSch = 50;% meshoptsSch = struct('element_order',pSch, 'numel',numelSch, 'domain',domain, ... 'verbose',toplot, 'element_type','spectral');% meshdatainputSch = struct('meshtype',meshtypeSch, 'param',aSch, ... 'meshopts',meshoptsSch);% % 2. Mesh for radial Poisson eq.:% meshtypePois = meshtypeSch; % type: 1 = uniform, 2 = logarithmic, 3 = exponentialaPois = aSch;pPois = pSch;numelPois = numelSch;% meshoptsPois = struct('element_order',pPois, 'numel',numelPois, 'domain',domain, ... 'verbose',toplot, 'element_type','spectral');% meshdatainputPois = struct('meshtype',meshtypePois, 'param',aPois, ... 'meshopts',meshoptsPois); % % Quadrature%global quadtype;quadtype = 1; % Type of quadrature: 1 = Gauss, 2 = Lobatto% % 1. Quadrature for radial Schrodinger eq.:% nspSch = pSch + 2;% nspSch = pSch + 1; % Number of quadrature points/weights; % nsp = p + 1 => diagonal overlap (S)-matrix. This % leads to exact integration of H-matrix but % "underintegrates" S-matrix. Nonetheless, % optimal convergence is preserved% quaddatainputSch = struct('quadtype',quadtype, 'nsp',nspSch);% % 2. Quadrature for radial Poisson eq.:% nspPois = nspSch;% nspPois = pPois + 1; % Number of quadrature points/weights; % nsp = p + 1 => diagonal overlap (S)-matrix. This % leads to exact integration of H-matrix but % "underintegrates" S-matrix. Nonetheless, % optimal convergence is preservedquaddatainputPois = struct('quadtype',quadtype, 'nsp',nspPois);% % SCF% maxscfiter = 200;scftol = 1.0E-6;scftolmax = 5.0E-5;mixingparam = 0.7;scfopts = struct('maxiter',maxscfiter, 'scftol',scftol, 'scftolmax',scftolmax, ... 'mixingparam',mixingparam);maxeigiter = 2000;eigtol = 1.0E-10;eigdisp = 0;eigopts = struct('maxit',maxeigiter, 'tol',eigtol, 'disp',eigdisp);[lam, u, scf_converged, Etot, Ekin, Ecoul, Eenuc, Exc, history, exitflag] = radialdft(problemdata, ... meshdatainputSch, meshdatainputPois, ... quaddatainputSch, quaddatainputPois, ... scfopts, eigopts); if exitflag ~= 0 fprintf('\n'); fprintf('Radial Kohn-Sham Equation could not be solved because \n'); fprintf('MATLAB''s eigensolver (eigs) failed! \n'); fprintf('\n'); timetot = cputime - tstart; fprintf('********************************************************* \n'); fprintf('\n'); fprintf(' Total Computing Time so far: %20.3e s \n', timetot); fprintf('\n'); fprintf('********************************************************* \n'); error('spectatom Aborted!');endif scf_converged == 1 fprintf('\n') fprintf(' ========================== \n') fprintf(' *** Converged Results *** \n') fprintf(' ========================== \n') fprintf('\n') fprintf(' Etot = %-11.8e \n', Etot); fprintf(' Ekin = %-11.8e \n', Ekin); fprintf(' Ecoul = %-11.8e \n', Ecoul); fprintf(' Eenuc = %-11.8e \n', Eenuc); fprintf(' Exc = %-11.8e \n', Exc); fprintf('\n'); % fprintf('The ground state energy is: %-11.8e Hartree \n', Etot(end));else fprintf('\n') fprintf('SCF did not converged. Maximum number of iterations (%-5d) reached \n', length(history)); fprintf('\n')endtimetot = cputime - tstart;fprintf('*************************************************************** \n');fprintf('\n');fprintf(' Time to complete the Simulation = %20.3e s\n', timetot);fprintf('\n');fprintf('*************************************************************** \n');function rho = thomasfermidensity(r, Z)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % Generalized Thomas-Fermi density (charge)% Inputs% ====== % r(:) : Radial grid% Z : Atomic number% % Output% ======% rho(:) : Thomas-Fermi charge density evaluated at the input grid pointsV = thomasfermipotential(r, Z, false);rho = -1.0./3.0/pi^2.*(-2.0*V).^(3.0/2.0); % Charge densityreturnendfunction V = thomasfermipotential(r, Z, cut)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % Generalized Thomas-Fermi atomic potential% % Inputs% ====== % r(:) : Radial grid% Z : Atomic number% cut : Cut the potential% % Output% ======% V(:) : Thomas-Fermi potential evaluated at the input grid points% if nargin < 3 cut = true;endx = r.*(128.0*Z/9.0/pi^2).^(1.0/3.0);% Z_eff(x) = Z * phi(x), where phi(x) satisfies the Thomas-Fermi equation:% phi'' = phi^(3/2) / sqrt(x)% with boundary conditions:% phi(0) = 1% phi(oo) = 0% There is no analytic solution, but one can solve this approximately. We use:% http://arxiv.org/abs/physics/0511017% Parameters:alpha = 0.7280642371;beta = -0.5430794693;gamma = 0.3612163121;sqrtx = sqrt(x);Zeff = Z.*(1 + alpha*sqrtx + beta*x.*exp(-gamma*sqrtx)).^2.* ... exp(-2.0*alpha*sqrtx);% This keeps all the eigenvalues of the radial problem negative:if cut == 1 Zeff(Zeff < 1.0) = 1.0;endV = -Zeff./r;returnendfunction r = sphericalintegral(xe, xq, wtq, fq)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % Spherical integral of function f(r) (no \theta, \phi dependence) % defined by values [fq] at the quadrature points on mesh xe. % i.e., \int_0^rmax 4 pi r^2 f(r) dr, where rmax = max r value in meshr = integrateonmesh(xe, wtq, 4.0.*pi.*xq.^2.*fq);returnendfunction [lambda, eigv, eigsflag] = solveradialschroed(xe, connect, numel, ... nodaldofs, dofinfo, numdofs, ... Veffq, l, xq, xiq, wtq, phiq, dphiq, nbeigs, eigopts)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % Inputs% ====== % Veffq(:,:) : quadrature-grid representatin of Veff - % Veffq(:,:) := VH[rhoq(:,:)] + Vxc[rhoq(:,:)] + Vext(:,:);% Veffq(:,:) = value at i-th quadrature point of j-th element% % phiq(:,:), dphiq(:,:) : parent basis fn values at % quadrature points xiq; % phihq(i,j) = value of j-th func% at i-th quadrature point% xq(:,:) : quadrature pts in physical space; % xq(i,j) = coordinate of i-th point in j-th element% Outputs% ======= %% global quadtype;m = 0; % size of index arraysfor e = 1:numel conn = connect(e,:); sctr = [ nodaldofs(conn).list ]; m = m + (length(sctr))^2;endI = zeros(m,1); J = I; XH = I; XS = I;%% compute and assemble Hamiltonian matrix H and overlap matrix S%% loop over all the elements%ind = 1; % indices for arrays I, J, XH, XSfor e = 1:numel conn = connect(e,:); sctr = [nodaldofs(conn).list]; dofs_to_nodes = dofinfo(sctr); % dofs = DOFs of the element % global node Ids detj = (xe(e+1) - xe(e))/2.0; % % compute element contributions to Stiffness matrix and load vector: % [he, se] = radialschroedingerelemeqns(conn, dofs_to_nodes, ... Veffq(:,e), l, phiq, dphiq, ... xq(:,e), wtq, detj); % % insert he, se into vector arrays % sctrlen = length(sctr); temp = meshgrid(sctr); I(ind:ind+sctrlen^2-1) = temp(:); temp = temp'; J(ind:ind+sctrlen^2-1) = temp(:); XH(ind:ind+sctrlen^2-1) = he(:); XS(ind:ind+sctrlen^2-1) = se(:); ind = ind + sctrlen^2;end%% Create H and S-matrices from the arrays%fprintf('Performing sparse assembly of system matrices (%d x %d) . . .\n\n',numdofs,numdofs);bigh = sparse(I, J, XH, numdofs, numdofs);bigs = sparse(I, J, XS, numdofs, numdofs);clear I; clear J; clear XH; clear XS;%% solve the eigen-problem (generalized/standard): H c = \lambda S c%% Symmetrize (in case bigs is nearly-symmetric due to round-off, etc.)bigs = 0.5*(bigs + bigs');% opt.TOLERANCE = 1.0E-10;% opt.MAXITERATION = 1000;% opt.iluThresh = 0.1; % % [lambda, eigv] = eigifp(bigh, bigs, nbeigs, opt);if (size(phiq,2) == length(xiq)) && (quadtype == 2) % nsp = p + 1 % Call eigs as a standard eigenproblem solver: fprintf('Solving standard eigenproblem: [H]{c} = e{c} . . .\n\n');% [eigv, D, ~] = eigs(bigs\bigh, nbeigs , 'SA', opts); [eigv, D, eigsflag] = eigs(bigs\bigh, nbeigs , 'SM', eigopts);else % Call eigs as a generalized eigenproblem solver: fprintf('Solving generalized eigenproblem: [H]{c} = e[S]{c} . . .\n\n'); [eigv, D, eigsflag] = eigs(bigh, bigs, nbeigs , 'SA', eigopts);endif eigsflag ~= 0 fprintf('\n') fprintf(' ************************** \n') fprintf(' FATAL ERROR! \n') fprintf(' ************************** \n') fprintf('\n') fprintf('All the required eigenvalues did not converge! \n'); fprintf('\n') fprintf('Consider changing eigopts.tol/eigopts.maxit and re-run. \n') condbigs = condest(bigs); condbigh = condest(bigh); fprintf('\n') fprintf('The Condition number of Overlap matrix at exit %-16.10e \n', condbigs) fprintf('\n') fprintf('and the Condition number of Hamiltonian matrix at exit %-16.10e \n', condbigh) fprintf('\n') fprintf('Aborting . . . \n'); fprintf('\n') lambda = NaN; returnelse fprintf('\n') fprintf('All the required eigenvalues converged within set tolerance. \n'); fprintf('\n') lambda = diag(D); returnendendfunction [Vh] = solveradialpoisson(xe, connect, numel, ... nodaldofs, dofinfo, numdofs, ... xq, wtq, phiq, dphiq, rhoq)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % Inputs% ====== % rhoq(:,) : quadrature-grid representatin of electron density; % rho(i,j) = rho value at i-th quadrature point % of j-th element% phiq(:,:), dphiq(:,:) : parent basis fn values at % quadrature points xiq; % phihq(i,j) = value of j-th func% at i-th quadrature point% % Outputs% ======= %m = 0; % size of index arraysfor e = 1:numel conn = connect(e,:); sctr = [ nodaldofs(conn).list ]; m = m + (length(sctr))^2;endI = zeros(m,1); J = I; XK = I; f = zeros(numdofs, 1);%% compute and assemble Stiffness matrix K and load vector f% ind = 1; % indices for arrays I, J, and XK%% loop over all the elements%for e = 1:numel conn = connect(e,:); sctr = [nodaldofs(conn).list]; dofs_to_nodes = dofinfo(sctr); % dofs = DOFs of the element % global node Ids detj = (xe(e+1) - xe(e))/2.0; % % compute element contributions to Stiffness matrix and load vector: % [ke, fe] = radialpoissonelemeqns(conn, dofs_to_nodes, phiq, dphiq, ... rhoq(:,e), xq(:,e), wtq, detj); % % insert ke, fe into vector arrays % sctrlen = length(sctr); temp = meshgrid(sctr); I(ind:ind+sctrlen^2-1) = temp(:); temp = temp'; J(ind:ind+sctrlen^2-1) = temp(:); XK(ind:ind+sctrlen^2-1) = ke(:); f(sctr) = f(sctr) + fe; ind = ind + sctrlen^2;end%% Create Stiffness matrix from the array%fprintf('Performing sparse assembly of Stiffness Matrix (%d x %d) . . .\n\n',numdofs,numdofs);bigk = sparse(I, J, XK, numdofs, numdofs);clear I; clear J; clear XK;%% solve the linear system Ku = f%fprintf('Solving the linear system [K]{u} = {b} . . .\n\n');Vh = bigk\f;returnend %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [NFE dNdxi] = shapefunctions(eorder,xi) % Purpose% =======% Compute the shape function and its local derivative at the Gauss point%function [NFE, dNdxi] = shapefunctions(eorder,xi) %% shape function%% nodes 1 to eorder+1%n = eorder+1;[NFE, dNdxi] = lagrange(xi,n);returnendfunction [he, se] = radialschroedingerelemeqns(nodes, dofs_to_nodes, ... Veffelem, l, phiq, dphiq, ... xqe, wtq, detj)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= %% Inputs% ====== % rhoq(:) : quadrature point values of the electron density.% rhoq(i) is the value of density at the i-th quadrature % point of the element under consideration % % Veff(:) : VH[rho(:)] + Vxc[rho(:)] + Vext(:);% A vector of length nsp. The i-th elements of Veff% vector contains the sum of the values of VH, % Vxc, and Vext evaluated at the i-th quadrature % point of the element under consideration %% Outputs% ======= %ndofs = length(dofs_to_nodes);dofs_to_locindices = zeros(1,ndofs);for i = 1:ndofs dofs_to_locindices(i) = find(nodes == dofs_to_nodes(i));end%% initialize element Hamiltonian (he) and overlap (se) matrices to zero%he = zeros(ndofs,ndofs);se = zeros(ndofs,ndofs);%% loop over Gauss points in xi-direction%for i = 1:length(xqe) r = xqe(i); % % FE shape functions and local derivatives % NFE = phiq(i,:); dNFEdxi = dphiq(i,:); N = NFE(dofs_to_locindices); dNdxi = dNFEdxi(dofs_to_locindices); % derivative of shape function with respect to global coordinate r dNdr = dNdxi/detj; % % The element Hamiltonian (he) and overlap matrices (se) % mass = N'*N; % Using u = P_{nl} = r * R_{nl} as the varibale to solve: he = he + detj*wtq(i)*(0.5*(dNdr'*dNdr) + ... (Veffelem(i) + l*(l + 1)/2.0/r/r)*mass); se = se + detj*wtq(i)*mass;endreturnend function [ke, fe] = radialpoissonelemeqns(nodes, dofs_to_nodes, ... phiq, dphiq, rhoq, xqe, wtq, detj)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= %% Inputs% ====== % rhoq(:) : quadrature point values of the electron density.% rhoq(i) is the value of density at the i-th quadrature % point of the element under consideration%% Outputs% ======= %ndofs = length(dofs_to_nodes);dofs_to_locindices = zeros(1,ndofs);for i = 1:ndofs dofs_to_locindices(i) = find(nodes == dofs_to_nodes(i));end%% initialize the stiffness matrix and load vector to zero%ke = zeros(ndofs,ndofs);fe = zeros(ndofs,1);%% loop over Gauss points in xi-direction%for i = 1:length(xqe) r = xqe(i); % % FE shape functions and local derivatives % NFE = phiq(i,:); dNFEdxi = dphiq(i,:); N = NFE(dofs_to_locindices); dNdxi = dNFEdxi(dofs_to_locindices); % derivative of shape function with respect to global coordinate r dNdr = dNdxi/detj; % % The element Stiffness matrix (ke) and load vector (fe): % % Considering Vp = r * Vh as the varibale to solve: ke = ke + detj*wtq(i)*(dNdr'*dNdr); fe = fe + detj*wtq(i)*N'*r*rhoq(i);endreturnendfunction [lam, u, scf_converged, Etot, Ekin, Ecoul, Eenuc, Exc, history, exitflag] = radialdft(problemdata, ... meshdatainputSch, meshdatainputPois, ... quaddatainputSch, quaddatainputPois, ... scfopts, eigopts)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Subhajit Banerjee and N. Sukumar% July 2017% UC Davis% % Purpose% ======= % Solves single-atom, all-electron, radial Kohn-Sham equation by % self-consistent field interation (SCF) % % Inputs% ====== % % % Outputs% ======= % % Set up Mesh for Spectral FEM% fprintf('\n');fprintf('----> Constructing finite element mesh . . . \n');fprintf('\n');meshtypeSch = meshdatainputSch.meshtype;paramSch = meshdatainputSch.param; meshoptsSch = meshdatainputSch.meshopts;% meshsetupSch = mesher(meshtypeSch, paramSch, meshoptsSch);% meshtypePois = meshdatainputPois.meshtype;paramPois = meshdatainputPois.param; meshoptsPois = meshdatainputPois.meshopts;% meshsetupPois = mesher(meshtypePois, paramPois, meshoptsPois);% % Set up Quadrature% fprintf('\n');fprintf('----> Setting up quadrature options . . . \n');fprintf('\n');% quadtypeSch = quaddatainputSch.quadtype;nspSch = quaddatainputSch.nsp;% quadsetupSch = getparentquadptswts(quadtypeSch, nspSch);% xiqSch = quadsetupSch.points;wtqSch = quadsetupSch.weights;% xeSch = meshsetupSch.elemendcoord;xqSch = getphysicalquadpts(xeSch, xiqSch);% quadtypePois = quaddatainputPois.quadtype;nspPois = quaddatainputPois.nsp;% quadsetupPois = getparentquadptswts(quadtypePois, nspPois);% xiqPois = quadsetupPois.points;wtqPois = quadsetupPois.weights;% xePois = meshsetupPois.elemendcoord;xqPois = getphysicalquadpts(xePois, xiqPois);% % Initialize density% fprintf('\n');fprintf('----> Constructing initial density . . . \n');fprintf('\n');nelec = problemdata.nelec;% rhoqSch = initialdensity(nelec, xeSch, xiqSch, wtqSch); % rhoqPois = initialdensity(nelec, xePois, xiqPois, wtqPois); rhoqSch = initialdensity(nelec, xeSch, xqSch, wtqSch); rhoqPois = initialdensity(nelec, xePois, xqPois, wtqPois); % % Commence SCF Iterations% [lam, u, scf_converged, Etot, Ekin, Ecoul, Eenuc, Exc, history, exitflag] = KSscfiteration(problemdata, rhoqSch, rhoqPois, ... meshsetupSch, meshsetupPois, ... meshoptsSch, meshoptsPois, ... quadsetupSch, quadsetupPois, ... xqSch, xqPois, scfopts, eigopts);if exitflag ~= 0 fprintf('\n'); fprintf('SCF Iteration Aborted . . . \n'); fprintf('\n'); fprintf('Hit any key to Abort the Radial Kohn-Sham Equation Solver . . . \n'); pause; returnelse fprintf('\n'); fprintf('SCF Iterations Ended successfully . . . \n'); fprintf('\n'); returnendendfunction [nodaldofs, dofinfo, numdofs] = Poissetupdofs(numnod)isboundarynode = zeros(1,numnod);% Dirichlet BCs on boundary nodesisboundarynode(1) = 1; isboundarynode(numnod) = 1;nodaldofs(numnod).list = [];numdofs = 0;for n = 1:numnod if (~isboundarynode(n)) numdofs = numdofs + 1; % classical DOF nodaldofs(n).list = [numdofs]; dofinfo(numdofs) = n; % node number for dof equal to numdofs endendreturnend%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Purpose% =======% Plot nodes%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function plotnodes(xe, coord) close all;x = coord(:,1);width = 6; % Width in inchesheight = 4; % Height in inchesalw = 5.0; % AxesLineWidthfsz = 18; % Fontsizelw = 2.0; % LineWidthfigureset(gcf,'InvertHardcopy','on');set(gcf,'PaperUnits', 'inches');pos = get(gcf, 'Position');set(gcf, 'Position', [pos(1) pos(2) width*1000, height*1000]);set(gca, 'FontSize', fsz, 'LineWidth', alw);set(get(gca,'xlabel'),'FontSize', fsz, 'FontWeight', 'Bold');set(get(gca,'ylabel'),'FontSize', fsz, 'FontWeight', 'Bold');set(get(gca,'title'),'FontSize', fsz, 'FontWeight', 'Bold');hold on;axis([min(xe) max(xe) -1.0/5.0 1.0/5.0 ]);xlabel('x','FontSize', fsz);% plot(xe, zeros(length(xe),1), 'mo', 'LineWidth', lw, ... 'MarkerEdgeColor', 'k', 'MarkerFaceColor', [.49 1 .63], ... 'MarkerSize', 15);plot(x, zeros(length(x),1),'x','Linewidth', lw, ... 'MarkerEdgeColor','k','MarkerFaceColor','r','MarkerSize',15);legend('Element Boundary Nodes', 'FE Nodes')returnend%----------------------------------------------------------------------------% Function Syntax% ===============% function [coord,connect] = mesh(meshtype, param, options)%% Inputs% ====== % mesh generated on [rin rout]% mesh type: 1 = uniform, 2 = logarithmic, 3 = exponential% % options : data structure with element information %% Outputs% ======= % coord(:) : 1D array which stores the global coordinates of% the nodes; coord(I,1) : node I coordinates% connect(:,:) : nodal connectivity; connect(i,j) = index of basis% node corresponding to local node j of element i% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function meshsetup = mesher(meshtype, param, meshopts)element_type = meshopts.element_type;switch element_type case 'spectral' otherwise fprintf('Mesh generator not coded for element type = %s \n',element_type); error('Aborting . .');endp = meshopts.element_order;numel = meshopts.numel;domain = meshopts.domain;plot = meshopts.verbose;rmin = domain(1);rmax = domain(2);xin = linspace(-1.0, 1.0, p+1);numnod = p*numel + 1;% Checks:if (rmax <= rmin) error('Error! rmax <= rmin. Aborting . .'); endif (numel < 1) error('Error! Number of elements < 1. Aborting . .');endswitch meshtype case 1 % uniform xe = linspace(rmin, rmax, numel+1); case 2 % logarithmic % using DOI: 10.1016/j.cpc.2013.02.014 % dx_Ne/dx_1 = mpars(1) = a; % dx_{i+1}/dx_i = q, where dx_i = length of i-th element % => r = mpars(1)^(1/(Ne - 1)) % x_i = (b - a)(q^i - 1)/(q^Ne - 1) + a, % where x_i = right boundary of i-th element % Some Checks: if (numel < 2), error('Error! Logarithmic mesh requires numel >= 2. Aborting . . .'); end a = param; if (a == 1), error('Error! Logarithmic mesh requires a != 1. Aborting . . .'); end % xe(i/i+1) = coord of left/right boundary of ith element xe = zeros(numel+1, 1); xe(1,1) = rmin; xe(end,1) = rmax; q = a^(1/(numel - 1)); for i = 1:numel xe(i+1,1) = (q^i - 1.0)/(q^numel - 1.0)*(rmax - rmin) + rmin; end disp(' ') case 3 % exponential mesh (Not tested) a = param; if (a < 0) error('Error! a > 0 required. Aborting . .'); elseif (a == 1) alpha = (rmax - rmin)/numel; for i = 1:numel+1 xe(i) = alpha*(i - 1.0) + rmin; end else if (numel > 1) xe = zeros(numel+1,1); beta = log(a)/(numel - 1); alpha = (rmax - rmin)/(exp(beta*numel) - 1.0); for i = 1:numel+1 xe(i) = alpha*(exp(beta*(i - 1)) - 1.0) + rmin; end elseif (numel == 1) xe(1) = rmin; xe(2) = rmax; else error('Error! numel >= 1 required. Aborting . .'); end end otherwise error('More coming soon!');end% % Generate nodal coordinates in equi-spaced settings% coord = zeros(numnod, 1);for e = 1:numel coord((e-1)*(p+1)+1:e*(p+1),1) = linspace(xe(e), xe(e+1), p+1);end%% connectivity%connect = zeros(numel,p+1);for e = 1:numel is = p*(e-1) + 1; connect(e,:) = is:is+p;end%% modify nodal coordinates (Lobatto points)%if (p >= 2) xin = lobatto_points(p+1); for e = 1:numel is = p*(e-1) + 1; a = xe(e); b = xe(e+1);% a = coord(is,:); b = coord(is+p,:); for n = is:is+p index = n-is+1; coord(n,:) = 0.5*(b + a) + 0.5*(b - a).*xin(index); end endendmeshsetup = struct('elemendcoord',xe, 'parentnodecoord',xin, 'coord',coord, 'connect',connect);%-------------------------------------------------------------------% Mesh plotting%-------------------------------------------------------------------if (strcmp(plot,'on')) if numel < 10 && p <= 5 plotnodes(xe, coord); hold off; endelse fprintf('\n'); fprintf('Disabling mesh-plotting because of too many elements/basis functions. \n'); fprintf('\n');endreturnendfunction w = lobatto_weights(n)w = ones(1,n);switch n case(2) w(1)=1.0000000000000000000000000000000; w(2)=1.0000000000000000000000000000000; case(3) w(1)=0.33333333333333333333333333333333; w(2)=1.3333333333333333333333333333333; w(3)=0.33333333333333333333333333333333; case(4) w(1)=0.16666666666666666666666666666667; w(2)=0.83333333333333333333333333333333; w(3)=0.83333333333333333333333333333333; w(4)=0.16666666666666666666666666666667; case(5) w(1)=0.10000000000000000000000000000000; w(2)=0.54444444444444444444444444444444; w(3)=0.71111111111111111111111111111111; w(4)=0.54444444444444444444444444444444; w(5)=0.10000000000000000000000000000000; case(6) w(1)=0.066666666666666666666666666666667; w(2)=0.37847495629784698031661280821202; w(3)=0.55485837703548635301672052512131; w(4)=0.55485837703548635301672052512131; w(5)=0.37847495629784698031661280821202; w(6)=0.066666666666666666666666666666667; case(7) w(1)=0.047619047619047619047619047619048; w(2)=0.27682604736156594801070040629007; w(3)=0.43174538120986262341787102228136; w(4)=0.48761904761904761904761904761905; w(5)=0.43174538120986262341787102228136; w(6)=0.27682604736156594801070040629007; w(7)=0.047619047619047619047619047619048; case(8) w(1)=0.035714285714285714285714285714286; w(2)=0.21070422714350603938299206577576; w(3)=0.34112269248350436476424067710775; w(4)=0.41245879465870388156705297140221; w(5)=0.41245879465870388156705297140221; w(6)=0.34112269248350436476424067710775; w(7)=0.21070422714350603938299206577576; w(8)=0.035714285714285714285714285714286; case(9) w(1)=0.027777777777777777777777777777778; w(2)=0.16549536156080552504633972002921; w(3)=0.27453871250016173528070561857937; w(4)=0.34642851097304634511513153213972; w(5)=0.37151927437641723356009070294785; w(6)=0.34642851097304634511513153213972; w(7)=0.27453871250016173528070561857937; w(8)=0.16549536156080552504633972002921; w(9)=0.027777777777777777777777777777778; case(10) w(1)=0.022222222222222222222222222222222; w(2)=0.13330599085107011112622717075539; w(3)=0.22488934206312645211945782173105; w(4)=0.29204268367968375787558225737444; w(5)=0.32753976118389745665651052791689; w(6)=0.32753976118389745665651052791689; w(7)=0.29204268367968375787558225737444; w(8)=0.22488934206312645211945782173105; w(9)=0.13330599085107011112622717075539; w(10)=0.022222222222222222222222222222222; case(11) w(1)=0.018181818181818181818181818181818; w(2)=0.10961227326699486446140344958035; w(3)=0.18716988178030520410814152189943; w(4)=0.24804810426402831404008486642187; w(5)=0.28687912477900808867922240333154; w(6)=0.30021759545569069378593188116998; w(7)=0.28687912477900808867922240333154; w(8)=0.24804810426402831404008486642187; w(9)=0.18716988178030520410814152189943; w(10)=0.10961227326699486446140344958035; w(11)=0.018181818181818181818181818181818; case(12) w(1)=0.015151515151515151515151515151515; w(2)=0.091684517413196130668342594134079; w(3)=0.15797470556437011516467106270034; w(4)=0.21250841776102114535830207736687; w(5)=0.25127560319920128029324441214760; w(6)=0.27140524091069617700028833849960; w(7)=0.27140524091069617700028833849960; w(8)=0.25127560319920128029324441214760; w(9)=0.21250841776102114535830207736687; w(10)=0.15797470556437011516467106270034; w(11)=0.091684517413196130668342594134079; w(12)=0.015151515151515151515151515151515; case(13) w(1)=0.012820512820512820512820512820513; w(2)=0.077801686746818927793588988333134; w(3)=0.13498192668960834911991476258937; w(4)=0.18364686520355009200749425874681; w(5)=0.22076779356611008608553400837940; w(6)=0.24401579030667635645857814836016; w(7)=0.25193084933344673604413864154124; w(8)=0.24401579030667635645857814836016; w(9)=0.22076779356611008608553400837940; w(10)=0.18364686520355009200749425874681; w(11)=0.13498192668960834911991476258937; w(12)=0.077801686746818927793588988333134; w(13)=0.012820512820512820512820512820513; case(14) w(1)=0.010989010989010989010989010989011; w(2)=0.066837284497681284634070660746053; w(3)=0.11658665589871165154099667065465; w(4)=0.16002185176295214241282099798759; w(5)=0.19482614937341611864033177837588; w(6)=0.21912625300977075487116252395417; w(7)=0.23161279446845705888962835729264; w(8)=0.23161279446845705888962835729264; w(9)=0.21912625300977075487116252395417; w(10)=0.19482614937341611864033177837588; w(11)=0.16002185176295214241282099798759; w(12)=0.11658665589871165154099667065465; w(13)=0.066837284497681284634070660746053; w(14)=0.010989010989010989010989010989011; case(15) w(1)=0.0095238095238095238095238095238095; w(2)=0.058029893028601249096880584025282; w(3)=0.10166007032571806760366617078880; w(4)=0.14051169980242810946044680564367; w(5)=0.17278964725360094905207709940835; w(6)=0.19698723596461335609250034650741; w(7)=0.21197358592682092012743007697722; w(8)=0.21704811634881564951495021425091; w(9)=0.21197358592682092012743007697722; w(10)=0.19698723596461335609250034650741; w(11)=0.17278964725360094905207709940835; w(12)=0.14051169980242810946044680564367; w(13)=0.10166007032571806760366617078880; w(14)=0.058029893028601249096880584025282; w(15)=0.0095238095238095238095238095238095; case(16) w(1)=0.0083333333333333333333333333333333; w(2)=0.050850361005919905403244919565455; w(3)=0.089393697325930800991052080166084; w(4)=0.12425538213251409834953633265731; w(5)=0.15402698080716428081564494048499; w(6)=0.17749191339170412530107566952836; w(7)=0.19369002382520358431691359885352; w(8)=0.20195830817822987148919912541094; w(9)=0.20195830817822987148919912541094; w(10)=0.19369002382520358431691359885352; w(11)=0.17749191339170412530107566952836; w(12)=0.15402698080716428081564494048499; w(13)=0.12425538213251409834953633265731; w(14)=0.089393697325930800991052080166084; w(15)=0.050850361005919905403244919565455; w(16)=0.0083333333333333333333333333333333; case(17) w(1)=0.0073529411764705882352941176470588; w(2)=0.044921940543254209647400954623212; w(3)=0.079198270503687119190264429952835; w(4)=0.11059290900702816137577270522008; w(5)=0.13798774620192655905620157495403; w(6)=0.16039466199762153951632836586475; w(7)=0.17700425351565787043694574536329; w(8)=0.18721633967761923589208848286062; w(9)=0.19066187475346943329940724702825; w(10)=0.18721633967761923589208848286062; w(11)=0.17700425351565787043694574536329; w(12)=0.16039466199762153951632836586475; w(13)=0.13798774620192655905620157495403; w(14)=0.11059290900702816137577270522008; w(15)=0.079198270503687119190264429952835; w(16)=0.044921940543254209647400954623212; w(17)=0.0073529411764705882352941176470588; case(18) w(1)=0.0065359477124183006535947712418301; w(2)=0.039970628810914066137599176410101; w(3)=0.070637166885633664999222960167786; w(4)=0.099016271717502802394423605318672; w(5)=0.12421053313296710026339635889675; w(6)=0.14541196157380226798300321049443; w(7)=0.16193951723760248926432670670023; w(8)=0.17326210948945622601061440382668; w(9)=0.17901586343970308229381880694353; w(10)=0.17901586343970308229381880694353; w(11)=0.17326210948945622601061440382668; w(12)=0.16193951723760248926432670670023; w(13)=0.14541196157380226798300321049443; w(14)=0.12421053313296710026339635889675; w(15)=0.099016271717502802394423605318672; w(16)=0.070637166885633664999222960167786; w(17)=0.039970628810914066137599176410101; w(18)=0.0065359477124183006535947712418301; case(19) w(1)=0.0058479532163742690058479532163743; w(2)=0.035793365186176477115425569035122; w(3)=0.063381891762629736851695690418317; w(4)=0.089131757099207084448008790556153; w(5)=0.11231534147730504407091001546378; w(6)=0.13226728044875077692604673390973; w(7)=0.14841394259593888500968064366841; w(8)=0.16029092404406124197991096818359; w(9)=0.16755658452714286727013727774026; w(10)=0.17000191928482723464467271561652; w(11)=0.16755658452714286727013727774026; w(12)=0.16029092404406124197991096818359; w(13)=0.14841394259593888500968064366841; w(14)=0.13226728044875077692604673390973; w(15)=0.11231534147730504407091001546378; w(16)=0.089131757099207084448008790556153; w(17)=0.063381891762629736851695690418317; w(18)=0.035793365186176477115425569035122; w(19)=0.0058479532163742690058479532163743; case(20) w(1)=0.0052631578947368421052631578947368; w(2)=0.032237123188488941491605028117294; w(3)=0.057181802127566826004753627173243; w(4)=0.080631763996119603144776846113721; w(5)=0.10199149969945081568378120573289; w(6)=0.12070922762867472509942970500239; w(7)=0.13630048235872418448978079298903; w(8)=0.14836155407091682581471301373397; w(9)=0.15658010264747548715816989679364; w(10)=0.16074328638784574900772672644908; w(11)=0.16074328638784574900772672644908; w(12)=0.15658010264747548715816989679364; w(13)=0.14836155407091682581471301373397; w(14)=0.13630048235872418448978079298903; w(15)=0.12070922762867472509942970500239; w(16)=0.10199149969945081568378120573289; w(17)=0.080631763996119603144776846113721; w(18)=0.057181802127566826004753627173243; w(19)=0.032237123188488941491605028117294; w(20)=0.0052631578947368421052631578947368; case(21) w(1)=0.0047619047619047619047619047619048; w(2)=0.029184840098505458609458543613171; w(3)=0.051843169000849625072722971852830; w(4)=0.073273918185074144252547861041894; w(5)=0.092985467957886065301137664149214; w(6)=0.11051708321912333526700048678439; w(7)=0.12545812119086894801515753570800; w(8)=0.13745846286004134358089961741515; w(9)=0.14623686244797745926727053063439; w(10)=0.15158757511168138445325068150529; w(11)=0.15338519033217494855158440506754; w(12)=0.15158757511168138445325068150529; w(13)=0.14623686244797745926727053063439; w(14)=0.13745846286004134358089961741515; w(15)=0.12545812119086894801515753570800; w(16)=0.11051708321912333526700048678439; w(17)=0.092985467957886065301137664149214; w(18)=0.073273918185074144252547861041894; w(19)=0.051843169000849625072722971852830; w(20)=0.029184840098505458609458543613171; w(21)=0.0047619047619047619047619047619048; case(22) w(1)=0.0043290043290043290043290043290043; w(2)=0.026545747682501757911627904520543; w(3)=0.047214465293740752123775734864792; w(4)=0.066865605864553076012404194157097; w(5)=0.085090060391838447815711236095748; w(6)=0.10150057480164767437243730374960; w(7)=0.11574764465393906659003636772146; w(8)=0.12752769665343027553084445930883; w(9)=0.13658968861374142668617736220617; w(10)=0.14274049227136140033623599356679; w(11)=0.14584901944424179361642043947997; w(12)=0.14584901944424179361642043947997; w(13)=0.14274049227136140033623599356679; w(14)=0.13658968861374142668617736220617; w(15)=0.12752769665343027553084445930883; w(16)=0.11574764465393906659003636772146; w(17)=0.10150057480164767437243730374960; w(18)=0.085090060391838447815711236095748; w(19)=0.066865605864553076012404194157097; w(20)=0.047214465293740752123775734864792; w(21)=0.026545747682501757911627904520543; w(22)=0.0043290043290043290043290043290043; case(23) w(1)=0.0039525691699604743083003952569170; w(2)=0.024248600771531736517399658937097; w(3)=0.043175871170241834748876465612042; w(4)=0.061252477129554206381382847440355; w(5)=0.078135449475569989741934255347965; w(6)=0.093497246163512341833500706906697; w(7)=0.10703910172433651153518362791547; w(8)=0.11849751066274913130212600472426; w(9)=0.12764947470175887663614855305567; w(10)=0.13431687263860381990156489770071; w(11)=0.13836993638580739452350273386294; w(12)=0.13972978001274736514015970647975; w(13)=0.13836993638580739452350273386294; w(14)=0.13431687263860381990156489770071; w(15)=0.12764947470175887663614855305567; w(16)=0.11849751066274913130212600472426; w(17)=0.10703910172433651153518362791547; w(18)=0.093497246163512341833500706906697; w(19)=0.078135449475569989741934255347965; w(20)=0.061252477129554206381382847440355; w(21)=0.043175871170241834748876465612042; w(22)=0.024248600771531736517399658937097; w(23)=0.0039525691699604743083003952569170; case(24) w(1)=0.0036231884057971014492753623188406; w(2)=0.022236853464711208992960434817299; w(3)=0.039631681333467809469262163363472; w(4)=0.056309848724646199020948339678118; w(5)=0.071981862055293982215639060668680; w(6)=0.086369029967929068216562819306583; w(7)=0.099214827684083587414149770995712; w(8)=0.11029008689296860411042412302760; w(9)=0.11939719370249131903229388835072; w(10)=0.12637364202802080012724809069137; w(11)=0.13109494187360394235445022581653; w(12)=0.13347684386698637759678572096508; w(13)=0.13347684386698637759678572096508; w(14)=0.13109494187360394235445022581653; w(15)=0.12637364202802080012724809069137; w(16)=0.11939719370249131903229388835072; w(17)=0.11029008689296860411042412302760; w(18)=0.099214827684083587414149770995712; w(19)=0.086369029967929068216562819306583; w(20)=0.071981862055293982215639060668680; w(21)=0.056309848724646199020948339678118; w(22)=0.039631681333467809469262163363472; w(23)=0.022236853464711208992960434817299; w(24)=0.0036231884057971014492753623188406; case(25) w(1)=0.0033333333333333333333333333333333; w(2)=0.020465168932974385308542471819163; w(3)=0.036504738794271372032382988755110; w(4)=0.051936228368491474643333889713489; w(5)=0.066513728675312784693869993313599; w(6)=0.079998774836292981801626436138836; w(7)=0.092170139910620421912689622714049; w(8)=0.10282803034795783082750364490713; w(9)=0.11179746626832088815624423243104; w(10)=0.11893117940681182540944424463419; w(11)=0.12411203893795029069521531243930; w(12)=0.12725497753833144701711441673876; w(13)=0.12830838929866192833739882612401; w(14)=0.12725497753833144701711441673876; w(15)=0.12411203893795029069521531243930; w(16)=0.11893117940681182540944424463419; w(17)=0.11179746626832088815624423243104; w(18)=0.10282803034795783082750364490713; w(19)=0.092170139910620421912689622714049; w(20)=0.079998774836292981801626436138836; w(21)=0.066513728675312784693869993313599; w(22)=0.051936228368491474643333889713489; w(23)=0.036504738794271372032382988755110; w(24)=0.020465168932974385308542471819163; w(25)=0.0033333333333333333333333333333333; case(26) w(1)=0.0030769230769230769230769230769231; w(2)=0.018896858024263465581352047676267; w(3)=0.033732303685955999377522748582756; w(4)=0.048048399081180627315979312780662; w(5)=0.061635025142547402782180054104279; w(6)=0.074287050122291137316084082986249; w(7)=0.085812863980004362187481252359068; w(8)=0.096037802353901310803697396370862; w(9)=0.10480688623073705298990583689136; w(10)=0.11198719411986033530021087987989; w(11)=0.11746988409380900707669562029648; w(12)=0.12117184628844334604428669570423; w(13)=0.12303696380008287630152714929097; w(14)=0.12303696380008287630152714929097; w(15)=0.12117184628844334604428669570423; w(16)=0.11746988409380900707669562029648; w(17)=0.11198719411986033530021087987989; w(18)=0.10480688623073705298990583689136; w(19)=0.096037802353901310803697396370862; w(20)=0.085812863980004362187481252359068; w(21)=0.074287050122291137316084082986249; w(22)=0.061635025142547402782180054104279; w(23)=0.048048399081180627315979312780662; w(24)=0.033732303685955999377522748582756; w(25)=0.018896858024263465581352047676267; w(26)=0.0030769230769230769230769230769231; case(27) w(1)=0.0028490028490028490028490028490028; w(2)=0.017501974876065579019369734362719; w(3)=0.031262951735202384324784990064795; w(4)=0.044577657933061698744074742241386; w(5)=0.057265569680162731739087228253941; w(6)=0.069149342360043276280634735030127; w(7)=0.080062321970538458168238070483950; w(8)=0.089851365259290559972136301946063; w(9)=0.098379074585952763179308931269989; w(10)=0.10552574782125301121838198947990; w(11)=0.11119106525743703292826195934215; w(12)=0.11529550025465198281375333337473; w(13)=0.11778143658595615906649907944836; w(14)=0.11861397766276302708523980370575; w(15)=0.11778143658595615906649907944836; w(16)=0.11529550025465198281375333337473; w(17)=0.11119106525743703292826195934215; w(18)=0.10552574782125301121838198947990; w(19)=0.098379074585952763179308931269989; w(20)=0.089851365259290559972136301946063; w(21)=0.080062321970538458168238070483950; w(22)=0.069149342360043276280634735030127; w(23)=0.057265569680162731739087228253941; w(24)=0.044577657933061698744074742241386; w(25)=0.031262951735202384324784990064795; w(26)=0.017501974876065579019369734362719; w(27)=0.0028490028490028490028490028490028; case(28) w(1)=0.0026455026455026455026455026455026; w(2)=0.016255883957504218198990152897110; w(3)=0.029054220677979144706852779823582; w(4)=0.041466915243006721037985848778627; w(5)=0.053338077047327488400325513471500; w(6)=0.064513658080354538390245693462478; w(7)=0.074848123509707702780567939143162; w(8)=0.084206795121510325739491187797920; w(9)=0.092467685997712175722035394699386; w(10)=0.099523110412495672875945354750915; w(11)=0.10528109376105589448237755179491; w(12)=0.10966657379597662508858586379121; w(13)=0.11262238007723901254063642411366; w(14)=0.11410997967262783453331479283004; w(15)=0.11410997967262783453331479283004; w(16)=0.11262238007723901254063642411366; w(17)=0.10966657379597662508858586379121; w(18)=0.10528109376105589448237755179491; w(19)=0.099523110412495672875945354750915; w(20)=0.092467685997712175722035394699386; w(21)=0.084206795121510325739491187797920; w(22)=0.074848123509707702780567939143162; w(23)=0.064513658080354538390245693462478; w(24)=0.053338077047327488400325513471500; w(25)=0.041466915243006721037985848778627; w(26)=0.029054220677979144706852779823582; w(27)=0.016255883957504218198990152897110; w(28)=0.0026455026455026455026455026455026; case(29) w(1)=0.0024630541871921182266009852216749; w(2)=0.015138169859967596790857926606856; w(3)=0.027070806296824827540851425445026; w(4)=0.038668439979712979747832574267124; w(5)=0.049795809093237560922668341175478; w(6)=0.060318503828522735028064463517634; w(7)=0.070108938000597999185266858424014; w(8)=0.079048313027885601784170232175109; w(9)=0.087028133441135605637626155215633; w(10)=0.093951542114796312497836506987570; w(11)=0.099734501637149860653569177534068; w(12)=0.10430681646367653742270711219124; w(13)=0.10761298583356853937358685404661; w(14)=0.10961287784589042013213241075660; w(15)=0.11028221677968261011245795287074; w(16)=0.10961287784589042013213241075660; w(17)=0.10761298583356853937358685404661; w(18)=0.10430681646367653742270711219124; w(19)=0.099734501637149860653569177534068; w(20)=0.093951542114796312497836506987570; w(21)=0.087028133441135605637626155215633; w(22)=0.079048313027885601784170232175109; w(23)=0.070108938000597999185266858424014; w(24)=0.060318503828522735028064463517634; w(25)=0.049795809093237560922668341175478; w(26)=0.038668439979712979747832574267124; w(27)=0.027070806296824827540851425445026; w(28)=0.015138169859967596790857926606856; w(29)=0.0024630541871921182266009852216749; case(30) w(1)=0.0022988505747126436781609195402299; w(2)=0.014131799327905387640732168668208; w(3)=0.025283166740551402204268253850042; w(4)=0.036142094199408535314732683123360; w(5)=0.046590694533142927401880491446022; w(6)=0.056511197923080383302193710472303; w(7)=0.065791336397790054944101374593132; w(8)=0.074326003324718253834067642000831; w(9)=0.082018512833406914799721545527160; w(10)=0.088781712319765210167256434455921; w(11)=0.094538975193860891781132715364036; w(12)=0.099225071004299830657675964954309; w(13)=0.10278690530723498947028080833538; w(14)=0.10518412159645464985621298085914; w(15)=0.10638955872366792494758230680993; w(16)=0.10638955872366792494758230680993; w(17)=0.10518412159645464985621298085914; w(18)=0.10278690530723498947028080833538; w(19)=0.099225071004299830657675964954309; w(20)=0.094538975193860891781132715364036; w(21)=0.088781712319765210167256434455921; w(22)=0.082018512833406914799721545527160; w(23)=0.074326003324718253834067642000831; w(24)=0.065791336397790054944101374593132; w(25)=0.056511197923080383302193710472303; w(26)=0.046590694533142927401880491446022; w(27)=0.036142094199408535314732683123360; w(28)=0.025283166740551402204268253850042; w(29)=0.014131799327905387640732168668208; w(30)=0.0022988505747126436781609195402299; case(31) w(1)=0.0021505376344086021505376344086022; w(2)=0.013222471025464670302635629869835; w(3)=0.023666433230270316730045782709426; w(4)=0.033853940405224057629810846931873; w(5)=0.043681818160066912829747288083268; w(6)=0.053046465493448782774280860067640; w(7)=0.061848741290454623390748584002232; w(8)=0.069995377594100570450215353373140; w(9)=0.077400032341475618634283878417423; w(10)=0.083984220517529730665258108226376; w(11)=0.089678151045260822248142932836209; w(12)=0.094421468377857957432190489816534; w(13)=0.098163893013712757119741879401421; w(14)=0.10086575479865051339556259610285; w(15)=0.10249841359547039767590565499043; w(16)=0.10304456295320733314178496152549; w(17)=0.10249841359547039767590565499043; w(18)=0.10086575479865051339556259610285; w(19)=0.098163893013712757119741879401421; w(20)=0.094421468377857957432190489816534; w(21)=0.089678151045260822248142932836209; w(22)=0.083984220517529730665258108226376; w(23)=0.077400032341475618634283878417423; w(24)=0.069995377594100570450215353373140; w(25)=0.061848741290454623390748584002232; w(26)=0.053046465493448782774280860067640; w(27)=0.043681818160066912829747288083268; w(28)=0.033853940405224057629810846931873; w(29)=0.023666433230270316730045782709426; w(30)=0.013222471025464670302635629869835; w(31)=0.0021505376344086021505376344086022 case(32) w(1)=0.0020161290322580645161290322580645; w(2)=0.012398106501373843788620349229117; w(3)=0.022199552889291964623832171161080; w(4)=0.031775135410915465781562278917906; w(5)=0.041034201586062723330403841719016; w(6)=0.049885271336221207011960153724443; w(7)=0.058240497248055869550798929919204; w(8)=0.066016877257154543932436331683494; w(9)=0.073137139602679032640370983569209; w(10)=0.079530525692106252292356728883049; w(11)=0.085133497949668230527527658506617; w(12)=0.089890372957357833072124789584664; w(13)=0.093753875546813813565908354145794; w(14)=0.096685608948002600560378147706395; w(15)=0.098656436540761777170651164243408; w(16)=0.099646771501276777634939084748541; w(17)=0.099646771501276777634939084748541; w(18)=0.098656436540761777170651164243408; w(19)=0.096685608948002600560378147706395; w(20)=0.093753875546813813565908354145794; w(21)=0.089890372957357833072124789584664; w(22)=0.085133497949668230527527658506617; w(23)=0.079530525692106252292356728883049; w(24)=0.073137139602679032640370983569209; w(25)=0.066016877257154543932436331683494; w(26)=0.058240497248055869550798929919204; w(27)=0.049885271336221207011960153724443; w(28)=0.041034201586062723330403841719016; w(29)=0.031775135410915465781562278917906; w(30)=0.022199552889291964623832171161080; w(31)=0.012398106501373843788620349229117; w(32)=0.0020161290322580645161290322580645; case(33) w(1)= 0.0018939393939393939393939393939394; w(2)=0.011648448392267734651222179870285; w(3)=0.020864609017603360095811664182613; w(4)=0.029881045916746477519971156454995; w(5)=0.038617814771813967563858875675217; w(6)=0.046993850461024170547973112041225; w(7)=0.054931059442626967951698841306657; w(8)=0.062355367852465305441060959167732; w(9)=0.069197469494016147559969796800326; w(10)=0.075393486923973828507129918402909; w(11)=0.080885572193455092179707696772196; w(12)=0.085622448531813132550191963745865; w(13)=0.089559889747077400661148462725561; w(14)=0.092661133442241463529935120285105; w(15)=0.094897224394591815824355414662604; w(16)=0.096247284972985461996554494277935; w(17)=0.096698710102716558960032808469672; w(18)=0.096247284972985461996554494277935; w(19)=0.094897224394591815824355414662604; w(20)=0.092661133442241463529935120285105; w(21)=0.089559889747077400661148462725561; w(22)=0.085622448531813132550191963745865; w(23)=0.080885572193455092179707696772196; w(24)=0.075393486923973828507129918402909; w(25)=0.069197469494016147559969796800326; w(26)=0.062355367852465305441060959167732; w(27)=0.054931059442626967951698841306657; w(28)=0.046993850461024170547973112041225; w(29)=0.038617814771813967563858875675217; w(30)=0.029881045916746477519971156454995; w(31)=0.020864609017603360095811664182613; w(32)=0.011648448392267734651222179870285; w(33)=0.0018939393939393939393939393939394; otherwise error('LOBATTO_WEIGHTS - Fatal error! Illegal value of n.');endreturnendfunction xi = lobatto_points(n)xi = ones(1,n);switch n case(2) xi(1)=-1.0000000000000000000000000000000; xi(2)=1.0000000000000000000000000000000; case(3) xi(1)=-1.0000000000000000000000000000000; xi(2)=0.0; xi(3)=1.0000000000000000000000000000000; case(4) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.44721359549995793928183473374626; xi(3)=0.44721359549995793928183473374626; xi(4)=1.0000000000000000000000000000000; case(5) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.65465367070797714379829245624686; xi(3)=0.0; xi(4)=0.65465367070797714379829245624686; xi(5)=1.0000000000000000000000000000000; case(6) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.76505532392946469285100297395934; xi(3)=-0.28523151648064509631415099404088; xi(4)=0.28523151648064509631415099404088; xi(5)=0.76505532392946469285100297395934; xi(6)=1.0000000000000000000000000000000; case(7) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.83022389627856692987203221396747; xi(3)=-0.46884879347071421380377188190877; xi(4)=0.0; xi(5)=0.46884879347071421380377188190877; xi(6)=0.83022389627856692987203221396747; xi(7)=1.0000000000000000000000000000000; case(8) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.87174014850960661533744576122066; xi(3)=-0.59170018143314230214451073139795; xi(4)=-0.20929921790247886876865726034535; xi(5)=0.20929921790247886876865726034535; xi(6)=0.59170018143314230214451073139795; xi(7)=0.87174014850960661533744576122066; xi(8)=1.0000000000000000000000000000000; case(9) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.89975799541146015731234524441834; xi(3)=-0.67718627951073775344588542709134; xi(4)=-0.36311746382617815871075206870866; xi(5)=0.0; xi(6)=0.36311746382617815871075206870866; xi(7)=0.67718627951073775344588542709134; xi(8)=0.89975799541146015731234524441834; xi(9)=1.0000000000000000000000000000000; case(10) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.91953390816645881382893266082234; xi(3)=-0.73877386510550507500310617485983; xi(4)=-0.47792494981044449566117509273126; xi(5)=-0.16527895766638702462621976595817; xi(6)=0.16527895766638702462621976595817; xi(7)=0.47792494981044449566117509273126; xi(8)=0.73877386510550507500310617485983; xi(9)=0.91953390816645881382893266082234; xi(10)=1.0000000000000000000000000000000; case(11) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.93400143040805913433227413609938; xi(3)=-0.78448347366314441862241781610846; xi(4)=-0.56523532699620500647096396947775; xi(5)=-0.29575813558693939143191151555906; xi(6)=0.0; xi(7)=0.29575813558693939143191151555906; xi(8)=0.56523532699620500647096396947775; xi(9)=0.78448347366314441862241781610846; xi(10)=0.93400143040805913433227413609938; xi(11)=1.0000000000000000000000000000000; case(12) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.94489927222288222340758013830322; xi(3)=-0.81927932164400667834864158171690; xi(4)=-0.63287615303186067766240485444366; xi(5)=-0.39953094096534893226434979156697; xi(6)=-0.13655293285492755486406185573969; xi(7)=0.13655293285492755486406185573969; xi(8)=0.39953094096534893226434979156697; xi(9)=0.63287615303186067766240485444366; xi(10)=0.81927932164400667834864158171690; xi(11)=0.94489927222288222340758013830322; xi(12)=1.0000000000000000000000000000000; case(13) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.95330984664216391189690546475545; xi(3)=-0.84634756465187231686592560709875; xi(4)=-0.68618846908175742607275903956636; xi(5)=-0.48290982109133620174693723363693; xi(6)=-0.24928693010623999256867370037423; xi(7)=0.0; xi(8)=0.24928693010623999256867370037423; xi(9)=0.48290982109133620174693723363693; xi(10)=0.68618846908175742607275903956636; xi(11)=0.84634756465187231686592560709875; xi(12)=0.95330984664216391189690546475545; xi(13)=1.0000000000000000000000000000000; case(14) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.95993504526726090135510016201542; xi(3)=-0.86780105383034725100022020290826; xi(4)=-0.72886859909132614058467240052088; xi(5)=-0.55063940292864705531662270585908; xi(6)=-0.34272401334271284504390340364167; xi(7)=-0.11633186888370386765877670973616; xi(8)=0.11633186888370386765877670973616; xi(9)=0.34272401334271284504390340364167; xi(10)=0.55063940292864705531662270585908; xi(11)=0.72886859909132614058467240052088; xi(12)=0.86780105383034725100022020290826; xi(13)=0.95993504526726090135510016201542; xi(14)=1.0000000000000000000000000000000; case(15) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.96524592650383857279585139206960; xi(3)=-0.88508204422297629882540163148223; xi(4)=-0.76351968995181520070411847597629; xi(5)=-0.60625320546984571112352993863673; xi(6)=-0.42063805471367248092189693873858; xi(7)=-0.21535395536379423822567944627292; xi(8)=0.0; xi(9)=0.21535395536379423822567944627292; xi(10)=0.42063805471367248092189693873858; xi(11)=0.60625320546984571112352993863673; xi(12)=0.76351968995181520070411847597629; xi(13)=0.88508204422297629882540163148223; xi(14)=0.96524592650383857279585139206960; xi(15)=1.0000000000000000000000000000000; case(16) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.96956804627021793295224273836746; xi(3)=-0.89920053309347209299462826151985; xi(4)=-0.79200829186181506393108827096315; xi(5)=-0.65238870288249308946788321964058; xi(6)=-0.48605942188713761178189078584687; xi(7)=-0.29983046890076320809835345472230; xi(8)=-0.10132627352194944784303300504592; xi(9)=0.10132627352194944784303300504592; xi(10)=0.29983046890076320809835345472230; xi(11)=0.48605942188713761178189078584687; xi(12)=0.65238870288249308946788321964058; xi(13)=0.79200829186181506393108827096315; xi(14)=0.89920053309347209299462826151985; xi(15)=0.96956804627021793295224273836746; xi(16)=1.0000000000000000000000000000000; case(17) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.97313217663141831415697950187372; xi(3)=-0.91087999591557359562380250639773; xi(4)=-0.81569625122177030710675055323753; xi(5)=-0.69102898062768470539491935737245; xi(6)=-0.54138539933010153912373340750406; xi(7)=-0.37217443356547704190723468073526; xi(8)=-0.18951197351831738830426301475311; xi(9)=0.0; xi(10)=0.18951197351831738830426301475311; xi(11)=0.37217443356547704190723468073526; xi(12)=0.54138539933010153912373340750406; xi(13)=0.69102898062768470539491935737245; xi(14)=0.81569625122177030710675055323753; xi(15)=0.91087999591557359562380250639773; xi(16)=0.97313217663141831415697950187372; xi(17)=1.0000000000000000000000000000000; case(18) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.97610555741219854286451892434170; xi(3)=-0.92064918534753387383785462543128; xi(4)=-0.83559353521809021371364636232794; xi(5)=-0.72367932928324268130621036530207; xi(6)=-0.58850483431866176117353589319356; xi(7)=-0.43441503691212397534228713674067; xi(8)=-0.26636265287828098416766533202560; xi(9)=-0.089749093484652111022645010088562; xi(10)=0.089749093484652111022645010088562; xi(11)=0.26636265287828098416766533202560; xi(12)=0.43441503691212397534228713674067; xi(13)=0.58850483431866176117353589319356; xi(14)=0.72367932928324268130621036530207; xi(15)=0.83559353521809021371364636232794; xi(16)=0.92064918534753387383785462543128; xi(17)=0.97610555741219854286451892434170; xi(18)=1.0000000000000000000000000000000; case(19) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.97861176622208009515263406311022; xi(3)=-0.92890152815258624371794025879655; xi(4)=-0.85246057779664609308595597004106; xi(5)=-0.75149420255261301416363748963394; xi(6)=-0.62890813726522049776683230622873; xi(7)=-0.48822928568071350277790963762492; xi(8)=-0.33350484782449861029850010384493; xi(9)=-0.16918602340928157137515415344488; xi(10)=0.0; xi(11)=0.16918602340928157137515415344488; xi(12)=0.33350484782449861029850010384493; xi(13)=0.48822928568071350277790963762492; xi(14)=0.62890813726522049776683230622873; xi(15)=0.75149420255261301416363748963394; xi(16)=0.85246057779664609308595597004106; xi(17)=0.92890152815258624371794025879655; xi(18)=0.97861176622208009515263406311022; xi(19)=1.0000000000000000000000000000000; case(20) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.98074370489391417192544643858423; xi(3)=-0.93593449881266543571618158493063; xi(4)=-0.86687797808995014130984721461629; xi(5)=-0.77536826095205587041431752759469; xi(6)=-0.66377640229031128984640332297116; xi(7)=-0.53499286403188626164813596182898; xi(8)=-0.39235318371390929938647470381582; xi(9)=-0.23955170592298649518240135692709; xi(10)=-0.080545937238821837975944518159554; xi(11)=0.080545937238821837975944518159554; xi(12)=0.23955170592298649518240135692709; xi(13)=0.39235318371390929938647470381582; xi(14)=0.53499286403188626164813596182898; xi(15)=0.66377640229031128984640332297116; xi(16)=0.77536826095205587041431752759469; xi(17)=0.86687797808995014130984721461629; xi(18)=0.93593449881266543571618158493063; xi(19)=0.98074370489391417192544643858423; xi(20)=1.0000000000000000000000000000000; case(21) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.98257229660454802823448127655541; xi(3)=-0.94197629695974553429610265066144; xi(4)=-0.87929475532359046445115359630494; xi(5)=-0.79600192607771240474431258966036; xi(6)=-0.69405102606222323262731639319467; xi(7)=-0.57583196026183068692702187033809; xi(8)=-0.44411578327900210119451634960735; xi(9)=-0.30198985650876488727535186785875; xi(10)=-0.15278551580218546600635832848567; xi(11)=0.0; xi(12)=0.15278551580218546600635832848567; xi(13)=0.30198985650876488727535186785875; xi(14)=0.44411578327900210119451634960735; xi(15)=0.57583196026183068692702187033809; xi(16)=0.69405102606222323262731639319467; xi(17)=0.79600192607771240474431258966036; xi(18)=0.87929475532359046445115359630494; xi(19)=0.94197629695974553429610265066144; xi(20)=0.98257229660454802823448127655541; xi(21)=1.0000000000000000000000000000000; case(22) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.98415243845764617655228962221207; xi(3)=-0.94720428399922868052421376661573; xi(4)=-0.89006229019090447052965782577909; xi(5)=-0.81394892761192113604544184805614; xi(6)=-0.72048723996120215811988189639847; xi(7)=-0.61166943828425897122621160586993; xi(8)=-0.48981487518990234980875123568327; xi(9)=-0.35752071013891953806095728024018; xi(10)=-0.21760658515928504178795509346539; xi(11)=-0.073054540010898334761088790464107; xi(12)=0.073054540010898334761088790464107; xi(13)=0.21760658515928504178795509346539; xi(14)=0.35752071013891953806095728024018; xi(15)=0.48981487518990234980875123568327; xi(16)=0.61166943828425897122621160586993; xi(17)=0.72048723996120215811988189639847; xi(18)=0.81394892761192113604544184805614; xi(19)=0.89006229019090447052965782577909; xi(20)=0.94720428399922868052421376661573; xi(21)=0.98415243845764617655228962221207; xi(22)=1.0000000000000000000000000000000; case(23) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.98552715587873257808146276673810; xi(3)=-0.95175795571071020413563967985143; xi(4)=-0.89945855804034501095016032034737; xi(5)=-0.82965109665128588622320061929000; xi(6)=-0.74369504117206068394516354306700; xi(7)=-0.64326364446013620847614553360277; xi(8)=-0.53031177113684416813011532015230; xi(9)=-0.40703793791447482919595048821510; xi(10)=-0.27584154894579306710687763267914; xi(11)=-0.13927620404066839859186261298277; xi(12)=0.0; xi(13)=0.13927620404066839859186261298277; xi(14)=0.27584154894579306710687763267914; xi(15)=0.40703793791447482919595048821510; xi(16)=0.53031177113684416813011532015230; xi(17)=0.64326364446013620847614553360277; xi(18)=0.74369504117206068394516354306700; xi(19)=0.82965109665128588622320061929000; xi(20)=0.89945855804034501095016032034737; xi(21)=0.95175795571071020413563967985143; xi(22)=0.98552715587873257808146276673810; xi(23)=1.0000000000000000000000000000000; case(24) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.98673055350516088355308673815447; xi(3)=-0.95574822092988635802697713055064; xi(4)=-0.90770567511350652199515299646621; xi(5)=-0.84346407015487204062330503742334; xi(6)=-0.76417048242049330778737528095229; xi(7)=-0.67124010526412869983566485818701; xi(8)=-0.56633135797929531218940954454228; xi(9)=-0.45131637321432261824821849156962; xi(10)=-0.32824761337551091203338917935961; xi(11)=-0.19932125339083266723657253912499; xi(12)=-0.066837993737228578113641808391677; xi(13)=0.066837993737228578113641808391677; xi(14)=0.19932125339083266723657253912499; xi(15)=0.32824761337551091203338917935961; xi(16)=0.45131637321432261824821849156962; xi(17)=0.56633135797929531218940954454228; xi(18)=0.67124010526412869983566485818701; xi(19)=0.76417048242049330778737528095229; xi(20)=0.84346407015487204062330503742334; xi(21)=0.90770567511350652199515299646621; xi(22)=0.95574822092988635802697713055064; xi(23)=0.98673055350516088355308673815447; xi(24)=1.0000000000000000000000000000000; case(25) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.98778994493149370927180407087316; xi(3)=-0.95926413825253447885984517883952; xi(4)=-0.91498277073462257832314993373685; xi(5)=-0.85567646583531657752381326017175; xi(6)=-0.78231965924071678039918795222875; xi(7)=-0.69611704881513436676036543378938; xi(8)=-0.59848414727999326809762099718107; xi(9)=-0.49102411481887838261895992256880; xi(10)=-0.37550145785922723322871461226072; xi(11)=-0.25381306416887658017988688168728; xi(12)=-0.12795705948310697270898462509465; xi(13)=0.0; xi(14)=0.12795705948310697270898462509465; xi(15)=0.25381306416887658017988688168728; xi(16)=0.37550145785922723322871461226072; xi(17)=0.49102411481887838261895992256880; xi(18)=0.59848414727999326809762099718107; xi(19)=0.69611704881513436676036543378938; xi(20)=0.78231965924071678039918795222875; xi(21)=0.85567646583531657752381326017175; xi(22)=0.91498277073462257832314993373685; xi(23)=0.95926413825253447885984517883952; xi(24)=0.98778994493149370927180407087316; xi(25)=1.0000000000000000000000000000000; case(26) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.98872741231147565442924669037130; xi(3)=-0.96237787476771732975066267128213; xi(4)=-0.92143554681755734112513052174869; xi(5)=-0.86652432395912357097445149780977; xi(6)=-0.79847718310743743903678763857034; xi(7)=-0.71832581636266508052540538790393; xi(8)=-0.62728529949231688780051890716122; xi(9)=-0.52673574202987854529824339438099; xi(10)=-0.41820138706624678556368388233385; xi(11)=-0.30332751285925272077404103790211; xi(12)=-0.18385549527005490132120567188576; xi(13)=-0.061596411781919728205487274781751; xi(14)=0.061596411781919728205487274781751; xi(15)=0.18385549527005490132120567188576; xi(16)=0.30332751285925272077404103790211; xi(17)=0.41820138706624678556368388233385; xi(18)=0.52673574202987854529824339438099; xi(19)=0.62728529949231688780051890716122; xi(20)=0.71832581636266508052540538790393; xi(21)=0.79847718310743743903678763857034; xi(22)=0.86652432395912357097445149780977; xi(23)=0.92143554681755734112513052174869; xi(24)=0.96237787476771732975066267128213; xi(25)=0.98872741231147565442924669037130; xi(26)=1.0000000000000000000000000000000; case(27) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.98956096372855062114887464005452; xi(3)=-0.96514840245081891926426680312113; xi(4)=-0.92718345872511577885974676407272; xi(5)=-0.87620208621452247866654304919414; xi(6)=-0.81292048689581228825809186475854; xi(7)=-0.73822714984645991083942722574841; xi(8)=-0.65317066369680951517870487518246; xi(9)=-0.55894506094256118006266807486812; xi(10)=-0.45687307561408240367861265083924; xi(11)=-0.34838758198902867043708657858717; xi(12)=-0.23501148310291813333680158051333; xi(13)=-0.11833633389852104843870826827532; xi(14)=0.0; xi(15)=0.11833633389852104843870826827532; xi(16)=0.23501148310291813333680158051333; xi(17)=0.34838758198902867043708657858717; xi(18)=0.45687307561408240367861265083924; xi(19)=0.55894506094256118006266807486812; xi(20)=0.65317066369680951517870487518246; xi(21)=0.73822714984645991083942722574841; xi(22)=0.81292048689581228825809186475854; xi(23)=0.87620208621452247866654304919414; xi(24)=0.92718345872511577885974676407272; xi(25)=0.96514840245081891926426680312113; xi(26)=0.98956096372855062114887464005452; xi(27)=1.0000000000000000000000000000000; case(28) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.99030540261845412551709291186460; xi(3)=-0.96762428585713130434377087914347; xi(4)=-0.93232516712155852492614241860899; xi(5)=-0.88487101721130283841230633825725; xi(6)=-0.82588097005633819604451442619660; xi(7)=-0.75612419400556976515209572526619; xi(8)=-0.67651012892957331771938124507339; xi(9)=-0.58807668983717560647831207976171; xi(10)=-0.49197675393157938042961668294448; xi(11)=-0.38946313757636280793588415520411; xi(12)=-0.28187226662160237195415250890933; xi(13)=-0.17060675530800436088270267774877; xi(14)=-0.057117121693512897626543651573747; xi(15)=0.057117121693512897626543651573747; xi(16)=0.17060675530800436088270267774877; xi(17)=0.28187226662160237195415250890933; xi(18)=0.38946313757636280793588415520411; xi(19)=0.49197675393157938042961668294448; xi(20)=0.58807668983717560647831207976171; xi(21)=0.67651012892957331771938124507339; xi(22)=0.75612419400556976515209572526619; xi(23)=0.82588097005633819604451442619660; xi(24)=0.88487101721130283841230633825725; xi(25)=0.93232516712155852492614241860899; xi(26)=0.96762428585713130434377087914347; xi(27)=0.99030540261845412551709291186460; xi(28)=1.0000000000000000000000000000000; case(29) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.99097298826856981627874511329957; xi(3)=-0.96984580728793629371240138788782; xi(4)=-0.93694271852098251154203684828336; xi(5)=-0.89266571997608827876422991405309; xi(6)=-0.83755273628178656524207331618682; xi(7)=-0.77227289720646874135417063413645; xi(8)=-0.69761866135636798839397442857709; xi(9)=-0.61449625220343294997090361742436; xi(10)=-0.52391467437196908357881657312361; xi(11)=-0.42697347171349438763757641096836; xi(12)=-0.32484938284191100626001402307884; xi(13)=-0.21878205828426101980905185232499; xi(14)=-0.11005901339559210970778760225568; xi(15)=0.0; xi(16)=0.11005901339559210970778760225568; xi(17)=0.21878205828426101980905185232499; xi(18)=0.32484938284191100626001402307884; xi(19)=0.42697347171349438763757641096836; xi(20)=0.52391467437196908357881657312361; xi(21)=0.61449625220343294997090361742436; xi(22)=0.69761866135636798839397442857709; xi(23)=0.77227289720646874135417063413645; xi(24)=0.83755273628178656524207331618682; xi(25)=0.89266571997608827876422991405309; xi(26)=0.93694271852098251154203684828336; xi(27)=0.96984580728793629371240138788782; xi(28)=0.99097298826856981627874511329957; xi(29)=1.0000000000000000000000000000000; case(30) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.99157394284050029333881822590865; xi(3)=-0.97184660316626924167659295220919; xi(4)=-0.94110478095105708230718954941552; xi(5)=-0.89969921819927685955334282314945; xi(6)=-0.84809948718019810955142383218307; xi(7)=-0.78689035723754708044995952059240; xi(8)=-0.71676539863708513163379859612990; xi(9)=-0.63851917580755840737109348270974; xi(10)=-0.55303826009505285238455023004066; xi(11)=-0.46129119016824068522655722110715; xi(12)=-0.36431750042244899775598510555179; xi(13)=-0.26321594371957379126709946900007; xi(14)=-0.15913204262585046782494857709885; xi(15)=-0.053245110485486669363007834451354; xi(16)=0.053245110485486669363007834451354; xi(17)=0.15913204262585046782494857709885; xi(18)=0.26321594371957379126709946900007; xi(19)=0.36431750042244899775598510555179; xi(20)=0.46129119016824068522655722110715; xi(21)=0.55303826009505285238455023004066; xi(22)=0.63851917580755840737109348270974; xi(23)=0.71676539863708513163379859612990; xi(24)=0.78689035723754708044995952059240; xi(25)=0.84809948718019810955142383218307; xi(26)=0.89969921819927685955334282314945; xi(27)=0.94110478095105708230718954941552; xi(28)=0.97184660316626924167659295220919; xi(29)=0.99157394284050029333881822590865; xi(30)=1.0000000000000000000000000000000; case(31) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.99211684434648108858932461564871; xi(3)=-0.97365493581573647482606954847523; xi(4)=-0.94486917020803922980640553764986; xi(5)=-0.90606695144126981314451152324746; xi(6)=-0.85765999529745561036700538239647; xi(7)=-0.80016154319246298927233133398318; xi(8)=-0.73418113630907524213414083483764; xi(9)=-0.66041820261152552333704127459677; xi(10)=-0.57965465720800191300329686219773; xi(11)=-0.49274661909883240570380023307021; xi(12)=-0.40061533828056180630266441939609; xi(13)=-0.30423743127287262095945514762626; xi(14)=-0.20463452924752450114109358572771; xi(15)=-0.10286244876068227708911472330805; xi(16)=0.0; xi(17)=0.10286244876068227708911472330805; xi(18)=0.20463452924752450114109358572771; xi(19)=0.30423743127287262095945514762626; xi(20)=0.40061533828056180630266441939609; xi(21)=0.49274661909883240570380023307021; xi(22)=0.57965465720800191300329686219773; xi(23)=0.66041820261152552333704127459677; xi(24)=0.73418113630907524213414083483764; xi(25)=0.80016154319246298927233133398318; xi(26)=0.85765999529745561036700538239647; xi(27)=0.90606695144126981314451152324746; xi(28)=0.94486917020803922980640553764986; xi(29)=0.97365493581573647482606954847523; xi(30)=0.99211684434648108858932461564871; xi(31)=1.0000000000000000000000000000000; case(32) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.99260893397276135937154131527745; xi(3)=-0.97529469048270922806245595038815; xi(4)=-0.94828483841723237808332824158843; xi(5)=-0.91184993906373190407453145237407; xi(6)=-0.86635247601267551983081476942454; xi(7)=-0.81224473177744234454682178807258; xi(8)=-0.75006449393667479771727596467830; xi(9)=-0.68042975561555081594239575769885; xi(10)=-0.60403258714842112613717291261189; xi(11)=-0.52163226288156529060659257339202; xi(12)=-0.43404771720184693960331935700639; xi(13)=-0.34214940653888148625381222069872; xi(14)=-0.24685065885020530441624143115887; xi(15)=-0.14909859681364749491438187617004; xi(16)=-0.049864725046593252306297744728426; xi(17)=0.049864725046593252306297744728426; xi(18)=0.14909859681364749491438187617004; xi(19)=0.24685065885020530441624143115887; xi(20)=0.34214940653888148625381222069872; xi(21)=0.43404771720184693960331935700639; xi(22)=0.52163226288156529060659257339202; xi(23)=0.60403258714842112613717291261189; xi(24)=0.68042975561555081594239575769885; xi(25)=0.75006449393667479771727596467830; xi(26)=0.81224473177744234454682178807258; xi(27)=0.86635247601267551983081476942454; xi(28)=0.91184993906373190407453145237407; xi(29)=0.94828483841723237808332824158843; xi(30)=0.97529469048270922806245595038815; xi(31)=0.99260893397276135937154131527745; xi(32)=1.0000000000000000000000000000000; case(33) xi(1)=-1.0000000000000000000000000000000; xi(2)=-0.99305635843365834366733608675624; xi(3)=-0.97678616331690630148562760838132; xi(4)=-0.95139345139699574337552441691356; xi(5)=-0.91711730345094124082567906975682; xi(6)=-0.87427810075056222064656062933034; xi(7)=-0.82327592300406746955907131326671; xi(8)=-0.76458700179352862800665848042309; xi(9)=-0.69875931661816259569967724754442; xi(10)=-0.62640749128126825726383134241071; xi(11)=-0.54820705991911162311040471176063; xi(12)=-0.46488816163210675597300372098526; xi(13)=-0.37722872425339363504193466071800; xi(14)=-0.2860472014876740416053445029630; xi(15)=-0.19219493146747722574130699544116; xi(16)=-0.096548188176107006316957788091625; xi(17)=0.0; xi(18)=0.096548188176107006316957788091625; xi(19)=0.19219493146747722574130699544116; xi(20)=0.28604720148767404160534450296304; xi(21)=0.37722872425339363504193466071800; xi(22)=0.46488816163210675597300372098526; xi(23)=0.54820705991911162311040471176063; xi(24)=0.62640749128126825726383134241071; xi(25)=0.69875931661816259569967724754442; xi(26)=0.76458700179352862800665848042309; xi(27)=0.82327592300406746955907131326671; xi(28)=0.87427810075056222064656062933034; xi(29)=0.91711730345094124082567906975682; xi(30)=0.95139345139699574337552441691356; xi(31)=0.97678616331690630148562760838132; xi(32)=0.99305635843365834366733608675624; xi(33)=1.0000000000000000000000000000000; otherwise error('LOBATTO_POINTS - Fatal error! Illegal value of n.');endreturnendfunction [cdin] = linearmix(cdin, cdout, linearmixing)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % % Inputs% ====== % cdin(:) : input cd to current iteration% overwritten by next input% cdout(:) : output cd from current iteration% linearmixing : mixing parameter - 0.1 means% mix 10% or output with input% Check that linear mixing parameter is between 0 and 1 % and that input and output charge densities have the % same dimensionif(linearmixing <= 0.0 || linearmixing >= 1.0) error(' Error in linearMix: linearmixing out of range');endif (size(cdin) ~= size(cdout) ) error(' error in linearMix: mixmatch in cd array dimensions');end% Now just mix them and we're donecdin = cdin + linearmixing*(cdout - cdin);end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [phi dphi] = lagrange(xi,n)%% Purpose% =======% Compute the Lagrange shape functions and its derivatives for xi \in [-1,1]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [phi, dphi] = lagrange(xi,n)% n = number of shape functions (n = order+1)xip = lobatto_points(n);xivec = xi*ones(1,n);vec = xivec - xip;phi = zeros(1,n);dphi = zeros(1,n);denom = zeros(1,n);for i = 1:n num = 1; den = 1.; for j = 1:n if (j ~= i) num = num*vec(j); den = den*(xip(i) - xip(j)); end end denom(i) = den; phi(i) = num/den;endfor i = 1:n avec = vec; avec(i) = 1.; num = 0.; for j = 1:n if (j ~= i) bvec = avec; bvec(j) = 1.; num = num + prod(bvec); end end dphi(i) = num/denom(i);endreturnendfunction [nodaldofs, dofinfo, numdofs] = KSsetupdofs(numnod)isboundarynode = zeros(1,numnod);% Dirichlet BCs on boundary nodesisboundarynode(1) = 1; isboundarynode(numnod) = 1;nodaldofs(numnod).list = [];numdofs = 0;for n = 1:numnod if (~isboundarynode(n)) numdofs = numdofs + 1; % classical DOF nodaldofs(n).list = [numdofs]; dofinfo(numdofs) = n; % node number for dof equal to numdofs endendreturnendfunction [lam, u, scf_converged, Etot, Ekin, Ecoul, Eenuc, Exc, history, eigsflag] = KSscfiteration(problemdata, rhoqSch, rhoqPois, ... meshsetupSch, meshsetupPois, ... meshoptsSch, meshoptsPois, ... quadsetupSch, quadsetupPois, ... xqSch, xqPois, scfopts, eigopts) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Purpose% ======= % Perform SCF iterations:% Inputs% ====== % % % Outputs% ======= global numdofsSch;% % Unpack:% Zion = problemdata.Z;nelec = problemdata.nelec;[~, ~, lo, fo] = getatomicstates(Zion); % no(:), lo(:) : quantum numbers "n" and "l" % fo(:) : occupancy of the (n, l) stateLmax = max(lo);xeSch = meshsetupSch.elemendcoord;xinSch = meshsetupSch.parentnodecoord;connectSch = meshsetupSch.connect;pSch = meshoptsSch.element_order;numelSch = meshoptsSch.numel;numnodSch = pSch*numelSch + 1;% xePois = meshsetupPois.elemendcoord;xinPois = meshsetupPois.parentnodecoord;connectPois = meshsetupPois.connect;pPois = meshoptsPois.element_order;numelPois = meshoptsPois.numel;numnodPois = pPois*numelPois + 1;% % Set up arrays for DOF related info% fprintf('\n');fprintf(' ----> Constructing arrays for DOF info \n');fprintf('\n');[nodaldofsSch, dofinfoSch, numdofsSch] = KSsetupdofs(numnodSch);[nodaldofsPois, dofinfoPois, numdofsPois] = Poissetupdofs(numnodPois);xiqSch = quadsetupSch.points;wtqSch = quadsetupSch.weights;xiqPois = quadsetupPois.points;wtqPois = quadsetupPois.weights;% % Set up basis functions in parent domain% fprintf('\n');fprintf('----> Constructing basis functions . . . \n');fprintf('\n');[phiqSch, dphiqSch] = getshapefunc(pSch, xiqSch);[phiqPois, dphiqPois] = getshapefunc(pPois, xiqPois);maxiter = scfopts.maxiter;scftol = scfopts.scftol;scftolmax = scfopts.scftolmax;mixingparam = scfopts.mixingparam;% Initialization% lam = zeros(neigs,Lmax+1);% u = zeros(numdofs,neigs,Lmax+1); lam = cell(Lmax+1,1); % to store eigenvalues of H u = lam S u: % lam(i,j) = i-th eigenvalue of j-th % L-value, j=0..Lmaxu = cell(Lmax+1,1); % to store eigenvectors of H u = lam S u; % u(i,j,k) = i-th component of j-th % vector of k-th L-value, k=0..Lmax% Computing the number of eigenvalues required as each l = 0, ..., Lmaxneigs = zeros(Lmax + 1,1);for l = 0:Lmax neigs(l+1) = length(find(lo == l));endscf_converged = false;% SCF iterations beginsfprintf('\n');fprintf('----> SCF iterations begin . . . \n');fprintf('\n');for iter = 1:maxiter fprintf('\n'); fprintf(' ----> Starting SCF Iteration # %-5d . . .\n', iter); fprintf('\n'); % Normalize density on Poisson and K-S mesh: rhointSch = sphericalintegral(xeSch, xqSch, wtqSch, rhoqSch); rhointPois = sphericalintegral(xePois, xqPois, wtqPois, rhoqPois); fprintf('\n'); fprintf(' Integral of input density before normalization (on K-S Mesh): %-11.8e \n', rhointSch); fprintf('\n'); fprintf(' Integral of input density before normalization (on Poisson Mesh): %-11.8e \n', rhointPois); fprintf('\n'); rhoqSch = -rhoqSch./rhointSch*nelec; % charge-density on K-S Mesh rhoqPois = -rhoqPois./rhointPois*nelec; % charge-density on Poisson Mesh % Construct Potential: fprintf('\n'); fprintf('----> Constructing Hartree potential . . . \n'); fprintf('\n'); % First get nodal solution for Vp(r) = r * Vh(r) % -ve sign as Vp" = -4 * pi * n(r) * r VpPois = solveradialpoisson(xePois, connectPois, numelPois, ... nodaldofsPois, dofinfoPois, numdofsPois, ... xqPois, wtqPois, phiqPois, dphiqPois, -4.0*pi*rhoqPois); % Transform to Poisson quadrature grid VpqSch = fetoquadgen(xinPois, xiqSch, xePois, xeSch, connectPois, ... [0.0; VpPois; 0.0]); % zero Dirichlet on the left % and zero Dirichlet on the right % Construct solution satisfying the BC Vh(r=rmax) = Z/rmax, and we % divide by "r" to obtain the Hartree potential. % This is because we were solving for Vp(r) = r * Vh(r): VhqSch = VpqSch./xqSch + Zion/xeSch(end); % Vnq(:,:) : quadrature-grid representatin of Vn(x) = -Z/(|x - R|); % Vn(i,j) = value at i-th quadrature point of j-th element %Vnq = computenuclearpotential(Zion, xqSch); %Veffq = Vhq + Vnq; VeffqSch = VhqSch - Zion./xqSch; % Add Nuclear potential [excqSch, VxcqSch] = exhangecorrelation('lda', -rhoqSch); VeffqSch = VeffqSch + VxcqSch; % Find eigenstates fprintf('\n'); fprintf('----> Computing eigenstates . . . \n'); fprintf('\n'); for l = 0:Lmax % u{l+1,1} = zeros(numdofs, neigs(l+1)); % lam{l+1,1} = zeros(neigs(l+1),1); [lam{l+1,1}, u{l+1,1}, eigsflag] = solveradialschroed(xeSch, connectSch, numelSch, ... nodaldofsSch, dofinfoSch, numdofsSch, ... VeffqSch, l, xqSch, xiqSch, wtqSch, phiqSch, ... dphiqSch, neigs(l+1), eigopts); if eigsflag ~= 0 break else continue end end if eigsflag ~= 0 scf_converged = false; Etot = NaN; Ekin = NaN; Ecoul = NaN; Eenuc = NaN; Exc = NaN; history = NaN; return end % Construct density: fprintf('\n'); fprintf('----> Constructing output density . . . \n'); fprintf('\n'); rhooutqSch = 0.0; rhooutqPois = 0.0; Eband = 0.0; for l = 0:Lmax nbands = neigs(l+1); U = reshape([u{l+1}(:,:)], [numdofsSch,nbands]); LAM = reshape([lam{l+1,1}(:)], [nbands, 1]);% occ = fo((fo(l+1) == fo)); occ = fo(lo == l);% occ = fo(l+1); for n = 1:nbands % transform to quadrature grid fullc = zeros(numdofsSch+2,1); fullc(2:numdofsSch+1) = U(:,n); % Generate quadrature point values of P_{nl} on Schrodinger mesh: PqSch = fetoquad(xinSch, xiqSch, connectSch, fullc); % Generate quadrature point values of P_{nl} on Poisson mesh: PqPois = fetoquadgen(xinSch, xiqPois, xeSch, xePois, connectSch, fullc); % % Form density and accumulate % % On Schrodinger mesh: rhooutqSch = rhooutqSch - occ(n)*PqSch.^2./(xqSch.^2)/(4.0*pi); % On Poisson mesh: rhooutqPois = rhooutqPois - occ(n)*PqPois.^2./(xqPois.^2)/(4.0*pi); Eband = Eband + occ(n)*LAM(n); end end [T_s(iter), E_ee(iter), E_en(iter), E_xc(iter), E_tot(iter)] = ... computetotalenergy(xeSch, xqSch, wtqSch, rhooutqSch, VeffqSch, VhqSch, ... excqSch, Zion, Eband); history(iter) = iter; rhointSch = sphericalintegral(xeSch, xqSch, wtqSch, rhooutqSch); rhointPois = sphericalintegral(xePois, xqPois, wtqPois, rhooutqPois); fprintf('\n'); fprintf(' Integral of input density before normalization: %-11.8e \n', rhointSch); fprintf('\n'); rhooutqSch = -rhooutqSch./rhointSch*nelec; rhooutqPois = -rhooutqPois./rhointPois*nelec; % Assess convergence, exit if achieved fprintf('\n'); fprintf('----> Assessing convergence . . . \n'); fprintf('\n'); drhoq = norm((rhooutqSch - rhoqSch),'fro')/norm(rhoqSch,'fro'); fprintf('\n'); fprintf('Fractional difference of input and output densities = %-11.8e \n', drhoq); fprintf('\n'); if (iter > 3) Emax = max(E_tot(iter-3:iter)); Emin = min(E_tot(iter-3:iter)); if (drhoq < scftol && Emax-Emin < scftolmax) fprintf('\n'); fprintf('Self-consistent convergence criterion achieved \n'); fprintf('\n'); fprintf('SCF Iteration: %-4d \n', iter); fprintf('tolscf: %-11.8e \n', scftol); fprintf('drhoq: %-11.8e \n', drhoq); fprintf('tolscfmax: %-11.8e \n', scftolmax); fprintf('Emax - Emin: %-11.8e \n', Emax-Emin); fprintf('\n'); scf_converged = true; break end end fprintf('\n'); fprintf('----> SCF iterations end . . . \n'); fprintf('\n'); % if not converged, form new input density scf_converged = false; fprintf('\n'); fprintf('----> Forming new input density on Poisson mesh. . . \n'); fprintf('\n'); rhoqPois = linearmix(rhoqPois, rhooutqPois, mixingparam); rhoqSch = linearmix(rhoqSch, rhooutqSch, mixingparam);endEtot = E_tot(end);Ekin = T_s(end);Ecoul = E_ee(end);Eenuc = E_en(end);Exc = E_xc(end);endfunction sm = integrateonmesh(xe, wtq, fq)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % Integral of function f(x) defined by values fq at quadrature points % on mesh xe, without additional factors; i.e., \int_xmin^xmax f(x) dx% % Inputs% ====== % xe(:) : xe(i/i+1) = coord of left/right boundary of i-th element% wtq(:) : quadrature weights% fq(:,:) : integrand values at quadrature points; % fq(i,j) = func. value at i-th quad pt in j-th element% % Output% ======% numel = length(xe) - 1;sm = 0;for e = 1:numel % sum over elements % Jacobian of affine transformation from parent coords % xi in [-1,1] to coords x in element [xa,xb]: % x = al xi + be, al=(xb-xa)/2, be=(xb+xa)/2 jac = 0.5*abs(xe(e+1) - xe(e)); sm = sm + sum(wtq'.*fq(:,e))*jac; % dot(wtq, fq(:,e)*jac);endreturnendfunction rhoq = initialdensity(Q, xe, xq, wtq)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % Constructs localized charge density of total charge -Q at quadrature points% % Inputs% ====== % Q : total charge; density of total charge -Q is constructed% xe(:) : xe(i/i+1) = coord of left/right boundary of ith element% xiq(:) : quadrature points% wtq(:) : quadrature weights% xq(:,:) : quadrature pts in physical space; % xq(i,j) = coordinate of i-th point in j-th element% % Output% ======% rhoq(:,:) : radial charge density at quadrature points; % rhoq(i,j) = value at i-th quadrature point of j-th element% numel = length(xe) - 1;rhoq = zeros(size(xq));% rhoq = zeros(length(xi),numel);for e = 1:numel % tabulate over elements xqe = xq(:,e); % rhoq(:,e) = exp(-xqe.^2./4.0) % Gaussian % rhoq(:,e) = 1.0./cosh(xqe).^2 % 1/cosh^2 rhoq(:,e) = -thomasfermidensity(xqe, Q); % Thomas-Fermi end% % The foloowing two operations are already being done outside at the % first SCF iteration step% integ = sphericalintegral(xe, xq, wtq, rhoq);rhoq = -rhoq./integ*Q;returnendfunction [phiq, dphiq] = getshapefunc(p, xiq)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % Tabulate parent basis at quadrature points:% % Inputs% ====== % % % Outputs% ======= nsp = length(xiq);phiq = zeros(nsp,p+1); % parent basis fn values at % quadrature points xiq; % phihq(i,j) = value of j-th func% at i-th quadrature pointdphiq = zeros(nsp,p+1); % derivative of the parent basis fn values at % quadrature points xiq; % dphihq(i,j) = value of j-th func% at i-th quadrature pointfor iq = 1:nsp [phiq(iq,:), dphiq(iq,:)] = shapefunctions(p, xiq(iq)); % all the shapefunctionsendfunction xq = getphysicalquadpts(xe, xiq)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % Generates quadrature points xq in physical coordinate system via % affine mapping of parent quad points xiq to elements xe% % Inputs% ====== % xe(:) : xe(i/i+1) = coord of left/right boundary of ith element% xiq(:) : quadrature points in [-1, 1] (parent domain)% % Output% ======% xq(:,:) : quadrature pts in physical space; % xq(i,j) = coordinate of i-th point in j-th elementnumel = length(xe) - 1; nsp = length(xiq);xq = zeros(nsp,numel); % assumes same nsp for all the elementsfor e = 1:numel xl = xe(e); xr = xe(e+1); % affine transformation: [-1 1] --> [xl xr] xq(:,e) = (xr - xl)/2.0*xiq + (xr + xl)/2.0;endfunction quadsetup = getparentquadptswts(quadtype, nsp)% Generate the quadrature points and weights in the biunit parent domainswitch quadtype case 1 % Use Gauss quadrature xiq = gausspoints(nsp); wtq = gaussweights(nsp); case 2 % Use Lobatto quadrature xiq = lobatto_points(nsp); wtq = lobatto_weights(nsp);endquadsetup = struct('points',xiq, 'weights',wtq);returnend function [n, no, lo, fo] = getatomicstates(Z)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% =======% Returns all electron states of the form (n, l, f) for a given Z% Note: sum(fo) == Z%% Inputs% ======% Z : atomic number%% Output% ======% no(:), lo(:) : quantum numbers "n" and "l"% fo(:) : occupancy of the (n, l) stateswitch Z case (1) n = 1; no = 1; lo = 0; fo = 1; case (2) n = 1; no = 1; lo = 0; fo = 2; case (3) n = 2; no = [ 1, 2 ]; lo = [ 0, 0 ]; fo = [ 2, 1 ]; case (4) n = 2; no = [ 1, 2 ]; lo = [ 0, 0 ]; fo = [ 2, 2 ]; case (5) n = 3; no = [ 1, 2, 2 ]; lo = [ 0, 0, 1 ]; fo = [ 2, 2, 1 ]; case (6) n = 3; no = [ 1, 2, 2 ]; lo = [ 0, 0, 1 ]; fo = [ 2, 2, 2 ]; case (7) n = 3; no = [ 1, 2, 2 ]; lo = [ 0, 0, 1 ]; fo = [ 2, 2, 3 ]; case (8) n = 3; no = [ 1, 2, 2 ]; lo = [ 0, 0, 1 ]; fo = [ 2, 2, 4 ]; case (9) n = 3; no = [ 1, 2, 2 ]; lo = [ 0, 0, 1 ]; fo = [ 2, 2, 5 ]; case (10) n = 3; no = [ 1, 2, 2 ]; lo = [ 0, 0, 1 ]; fo = [ 2, 2, 6 ]; case (11) n = 4; no = [ 1, 2, 2, 3 ]; lo = [ 0, 0, 1, 0 ]; fo = [ 2, 2, 6, 1 ]; case (12) n = 4; no = [ 1, 2, 2, 3 ]; lo = [ 0, 0, 1, 0 ]; fo = [ 2, 2, 6, 2 ]; case (13) n = 5; no = [ 1, 2, 2, 3, 3 ]; lo = [ 0, 0, 1, 0, 1 ]; fo = [ 2, 2, 6, 2, 1 ]; case (14) n = 5; no = [ 1, 2, 2, 3, 3 ]; lo = [ 0, 0, 1, 0, 1 ]; fo = [ 2, 2, 6, 2, 2 ]; case (15) n = 5; no = [ 1, 2, 2, 3, 3 ]; lo = [ 0, 0, 1, 0, 1 ]; fo = [ 2, 2, 6, 2, 3 ]; case (16) n = 5; no = [ 1, 2, 2, 3, 3 ]; lo = [ 0, 0, 1, 0, 1 ]; fo = [ 2, 2, 6, 2, 4 ]; case (17) n = 5; no = [ 1, 2, 2, 3, 3 ]; lo = [ 0, 0, 1, 0, 1 ]; fo = [ 2, 2, 6, 2, 5 ]; case (18) n = 5; no = [ 1, 2, 2, 3, 3 ]; lo = [ 0, 0, 1, 0, 1 ]; fo = [ 2, 2, 6, 2, 6 ]; case (19) n = 6; no = [ 1, 2, 2, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 1 ]; case (20) n = 6; no = [ 1, 2, 2, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 2 ]; case (21) n = 7; no = [ 1, 2, 2, 3, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 1, 2 ]; case (22) n = 7; no = [ 1, 2, 2, 3, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 2, 2 ]; case (23) n = 7; no = [ 1, 2, 2, 3, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 3, 2 ]; case (24) n = 7; no = [ 1, 2, 2, 3, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 5, 1 ]; case (25) n = 7; no = [ 1, 2, 2, 3, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 5, 2 ]; case (26) n = 7; no = [ 1, 2, 2, 3, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 6, 2 ]; case (27) n = 7; no = [ 1, 2, 2, 3, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 7, 2 ]; case (28) n = 7; no = [ 1, 2, 2, 3, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 8, 2 ]; case (29) n = 7; no = [ 1, 2, 2, 3, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 1 ]; case (30) n = 7; no = [ 1, 2, 2, 3, 3, 3, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2 ]; case (31) n = 8; no = [ 1, 2, 2, 3, 3, 3, 4, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 1 ]; case (32) n = 8; no = [ 1, 2, 2, 3, 3, 3, 4, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 2 ]; case (33) n = 8; no = [ 1, 2, 2, 3, 3, 3, 4, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 3 ]; case (34) n = 8; no = [ 1, 2, 2, 3, 3, 3, 4, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 4 ]; case (35) n = 8; no = [ 1, 2, 2, 3, 3, 3, 4, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 5 ]; case (36) n = 8; no = [ 1, 2, 2, 3, 3, 3, 4, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6 ]; case (37) n = 9; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 1 ]; case (38) n = 9; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 2 ]; case (39) n = 10; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 1, 2 ]; case (40) n = 10; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 2, 2 ]; case (41) n = 10; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 4, 1 ]; case (42) n = 10; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 5, 1 ]; case (43) n = 10; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 5, 2 ]; case (44) n = 10; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 7, 1 ]; case (45) n = 10; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 8, 1 ]; case (46) n = 9; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10 ]; case (47) n = 10; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 1 ]; case (48) n = 10; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 2 ]; case (49) n = 11; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 2, 1 ]; case (50) n = 11; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 2, 2 ]; case (51) n = 11; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 2, 3 ]; case (52) n = 11; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 2, 4 ]; case (53) n = 11; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 2, 5 ]; case (54) n = 11; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 2, 6 ]; case (55) n = 12; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 2, 6, 1 ]; case (56) n = 12; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 2, 6, 2 ]; case (57) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 2, 6, 1, 2 ]; case (58) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 1, 2, 6, 1, 2 ]; case (59) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 3, 2, 6, 2 ]; case (60) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 4, 2, 6, 2 ]; case (61) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 5, 2, 6, 2 ]; case (62) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 6, 2, 6, 2 ]; case (63) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 7, 2, 6, 2 ]; case (64) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 7, 2, 6, 1, 2 ]; case (65) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 9, 2, 6, 2 ]; case (66) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 10, 2, 6, 2 ]; case (67) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 11, 2, 6, 2 ]; case (68) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 12, 2, 6, 2 ]; case (69) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 13, 2, 6, 2 ]; case (70) n = 13; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 2 ]; case (71) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 1, 2 ]; case (72) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 2, 2 ]; case (73) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 3, 2 ]; case (74) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 4, 2 ]; case (75) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 5, 2 ]; case (76) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 6, 2 ]; case (77) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 7, 2 ]; case (78) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 9, 1 ]; case (79) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 1 ]; case (80) n = 14; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2 ]; case (81) n = 15; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2, 1 ]; case (82) n = 15; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2, 2 ]; case (83) n = 15; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2, 3 ]; case (84) n = 15; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2, 4 ]; case (85) n = 15; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2, 5 ]; case (86) n = 15; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2, 6 ]; case (87) n = 16; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2, 6, 1 ]; case (88) n = 16; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2, 6, 2 ]; case (89) n = 17; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2, 6, 1, 2 ]; case (90) n = 17; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2, 6, 2, 2 ]; case (91) n = 18; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 2, 2, 6, 1, 2 ]; case (92) n = 18; no = [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7 ]; lo = [ 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 0 ]; fo = [ 2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 3, 2, 6, 1, 2 ]; otherwise fprintf('Z = %-3d is an invalid Z-value\n', Z); error('Z not supported. Aborting . . .')endend function w = gaussweights(n)%% function w = gauss_weights_QUAD(n);%% For 1 <= n <= 64, returns the weights w of an% n point Gauss-Legendre quadrature rule over the interval [-1,1].%w = ones(1,n); switch n case(1) w(1)=2.0000000000000000; case(2) w(1)=1.0000000000000000; w(2)=1.0000000000000000; case(3) w(1)=0.55555555555555556; w(2)=0.88888888888888889; w(3)=0.55555555555555556; case(4) w(1)=0.34785484513745386; w(2)=0.65214515486254614; w(3)=0.65214515486254614; w(4)=0.34785484513745386; case(5) w(1)=0.23692688505618909; w(2)=0.47862867049936647; w(3)=0.56888888888888889; w(4)=0.47862867049936647; w(5)=0.23692688505618909; case(6) w(1)=0.17132449237917035; w(2)=0.36076157304813861; w(3)=0.46791393457269105; w(4)=0.46791393457269105; w(5)=0.36076157304813861; w(6)=0.17132449237917035; case(7) w(1)=0.12948496616886969; w(2)=0.27970539148927667; w(3)=0.38183005050511894; w(4)=0.41795918367346939; w(5)=0.38183005050511894; w(6)=0.27970539148927667; w(7)=0.12948496616886969; case(8) w(1)=0.10122853629037626; w(2)=0.22238103445337447; w(3)=0.31370664587788729; w(4)=0.36268378337836198; w(5)=0.36268378337836198; w(6)=0.31370664587788729; w(7)=0.22238103445337447; w(8)=0.10122853629037626; case(9) w(1)=0.081274388361574412; w(2)=0.18064816069485740; w(3)=0.26061069640293546; w(4)=0.31234707704000284; w(5)=0.33023935500125976; w(6)=0.31234707704000284; w(7)=0.26061069640293546; w(8)=0.18064816069485740; w(9)=0.081274388361574412; case(10) w(1)=0.066671344308688138; w(2)=0.14945134915058059; w(3)=0.21908636251598204; w(4)=0.26926671930999636; w(5)=0.29552422471475287; w(6)=0.29552422471475287; w(7)=0.26926671930999636; w(8)=0.21908636251598204; w(9)=0.14945134915058059; w(10)=0.066671344308688138; case(11) w(1)=0.055668567116173666; w(2)=0.12558036946490462; w(3)=0.18629021092773425; w(4)=0.23319376459199048; w(5)=0.26280454451024666; w(6)=0.27292508677790063; w(7)=0.26280454451024666; w(8)=0.23319376459199048; w(9)=0.18629021092773425; w(10)=0.12558036946490462; w(11)=0.055668567116173666; case(12) w(1)=0.047175336386511827; w(2)=0.10693932599531843; w(3)=0.16007832854334623; w(4)=0.20316742672306592; w(5)=0.23349253653835481; w(6)=0.24914704581340279; w(7)=0.24914704581340279; w(8)=0.23349253653835481; w(9)=0.20316742672306592; w(10)=0.16007832854334623; w(11)=0.10693932599531843; w(12)=0.047175336386511827; case(13) w(1)=0.040484004765315880; w(2)=0.092121499837728448; w(3)=0.13887351021978724; w(4)=0.17814598076194574; w(5)=0.20781604753688850; w(6)=0.22628318026289724; w(7)=0.23255155323087391; w(8)=0.22628318026289724; w(9)=0.20781604753688850; w(10)=0.17814598076194574; w(11)=0.13887351021978724; w(12)=0.092121499837728448; w(13)=0.040484004765315880; case(14) w(1)=0.035119460331751863; w(2)=0.080158087159760210; w(3)=0.12151857068790318; w(4)=0.15720316715819353; w(5)=0.18553839747793781; w(6)=0.20519846372129560; w(7)=0.21526385346315779; w(8)=0.21526385346315779; w(9)=0.20519846372129560; w(10)=0.18553839747793781; w(11)=0.15720316715819353; w(12)=0.12151857068790318; w(13)=0.080158087159760210; w(14)=0.035119460331751863; case(15) w(1)=0.030753241996117268; w(2)=0.070366047488108125; w(3)=0.10715922046717194; w(4)=0.13957067792615431; w(5)=0.16626920581699393; w(6)=0.18616100001556221; w(7)=0.19843148532711158; w(8)=0.20257824192556127; w(9)=0.19843148532711158; w(10)=0.18616100001556221; w(11)=0.16626920581699393; w(12)=0.13957067792615431; w(13)=0.10715922046717194; w(14)=0.070366047488108125; w(15)=0.030753241996117268; case(16) w(1)=0.027152459411754095; w(2)=0.062253523938647893; w(3)=0.095158511682492785; w(4)=0.12462897125553387; w(5)=0.14959598881657673; w(6)=0.16915651939500254; w(7)=0.18260341504492359; w(8)=0.18945061045506850; w(9)=0.18945061045506850; w(10)=0.18260341504492359; w(11)=0.16915651939500254; w(12)=0.14959598881657673; w(13)=0.12462897125553387; w(14)=0.095158511682492785; w(15)=0.062253523938647893; w(16)=0.027152459411754095; case(17) w(1)=0.024148302868547932; w(2)=0.055459529373987201; w(3)=0.085036148317179181; w(4)=0.11188384719340397; w(5)=0.13513636846852547; w(6)=0.15404576107681029; w(7)=0.16800410215645004; w(8)=0.17656270536699265; w(9)=0.17944647035620653; w(10)=0.17656270536699265; w(11)=0.16800410215645004; w(12)=0.15404576107681029; w(13)=0.13513636846852547; w(14)=0.11188384719340397; w(15)=0.085036148317179181; w(16)=0.055459529373987201; w(17)=0.024148302868547932; case(18) w(1)=0.021616013526483310; w(2)=0.049714548894969796; w(3)=0.076425730254889057; w(4)=0.10094204410628717; w(5)=0.12255520671147846; w(6)=0.14064291467065065; w(7)=0.15468467512626524; w(8)=0.16427648374583272; w(9)=0.16914238296314359; w(10)=0.16914238296314359; w(11)=0.16427648374583272; w(12)=0.15468467512626524; w(13)=0.14064291467065065; w(14)=0.12255520671147846; w(15)=0.10094204410628717; w(16)=0.076425730254889057; w(17)=0.049714548894969796; w(18)=0.021616013526483310; case(19) w(1)=0.019461788229726477; w(2)=0.044814226765699600; w(3)=0.069044542737641227; w(4)=0.091490021622449999; w(5)=0.11156664554733399; w(6)=0.12875396253933623; w(7)=0.14260670217360661; w(8)=0.15276604206585967; w(9)=0.15896884339395435; w(10)=0.16105444984878370; w(11)=0.15896884339395435; w(12)=0.15276604206585967; w(13)=0.14260670217360661; w(14)=0.12875396253933623; w(15)=0.11156664554733399; w(16)=0.091490021622449999; w(17)=0.069044542737641227; w(18)=0.044814226765699600; w(19)=0.019461788229726477; case(20) w(1)=0.017614007139152118; w(2)=0.040601429800386941; w(3)=0.062672048334109064; w(4)=0.083276741576704749; w(5)=0.10193011981724044; w(6)=0.11819453196151842; w(7)=0.13168863844917663; w(8)=0.14209610931838205; w(9)=0.14917298647260375; w(10)=0.15275338713072585; w(11)=0.15275338713072585; w(12)=0.14917298647260375; w(13)=0.14209610931838205; w(14)=0.13168863844917663; w(15)=0.11819453196151842; w(16)=0.10193011981724044; w(17)=0.083276741576704749; w(18)=0.062672048334109064; w(19)=0.040601429800386941; w(20)=0.017614007139152118; case(21) w(1)=0.016017228257774333; w(2)=0.036953789770852494; w(3)=0.057134425426857208; w(4)=0.076100113628379302; w(5)=0.093444423456033862; w(6)=0.10879729916714838; w(7)=0.12183141605372853; w(8)=0.13226893863333746; w(9)=0.13988739479107315; w(10)=0.14452440398997006; w(11)=0.14608113364969043; w(12)=0.14452440398997006; w(13)=0.13988739479107315; w(14)=0.13226893863333746; w(15)=0.12183141605372853; w(16)=0.10879729916714838; w(17)=0.093444423456033862; w(18)=0.076100113628379302; w(19)=0.057134425426857208; w(20)=0.036953789770852494; w(21)=0.016017228257774333; case(22) w(1)=0.014627995298272201; w(2)=0.033774901584814155; w(3)=0.052293335152683286; w(4)=0.069796468424520488; w(5)=0.085941606217067727; w(6)=0.10041414444288096; w(7)=0.11293229608053922; w(8)=0.12325237681051242; w(9)=0.13117350478706237; w(10)=0.13654149834601517; w(11)=0.13925187285563199; w(12)=0.13925187285563199; w(13)=0.13654149834601517; w(14)=0.13117350478706237; w(15)=0.12325237681051242; w(16)=0.11293229608053922; w(17)=0.10041414444288096; w(18)=0.085941606217067727; w(19)=0.069796468424520488; w(20)=0.052293335152683286; w(21)=0.033774901584814155; w(22)=0.014627995298272201; case(23) w(1)=0.013411859487141772; w(2)=0.030988005856979444; w(3)=0.048037671731084669; w(4)=0.064232421408525852; w(5)=0.079281411776718955; w(6)=0.092915766060035147; w(7)=0.10489209146454141; w(8)=0.11499664022241136; w(9)=0.12304908430672953; w(10)=0.12890572218808215; w(11)=0.13246203940469662; w(12)=0.13365457218610618; w(13)=0.13246203940469662; w(14)=0.12890572218808215; w(15)=0.12304908430672953; w(16)=0.11499664022241136; w(17)=0.10489209146454141; w(18)=0.092915766060035147; w(19)=0.079281411776718955; w(20)=0.064232421408525852; w(21)=0.048037671731084669; w(22)=0.030988005856979444; w(23)=0.013411859487141772; case(24) w(1)=0.012341229799987200; w(2)=0.028531388628933663; w(3)=0.044277438817419806; w(4)=0.059298584915436781; w(5)=0.073346481411080306; w(6)=0.086190161531953276; w(7)=0.097618652104113888; w(8)=0.10744427011596563; w(9)=0.11550566805372560; w(10)=0.12167047292780339; w(11)=0.12583745634682830; w(12)=0.12793819534675216; w(13)=0.12793819534675216; w(14)=0.12583745634682830; w(15)=0.12167047292780339; w(16)=0.11550566805372560; w(17)=0.10744427011596563; w(18)=0.097618652104113888; w(19)=0.086190161531953276; w(20)=0.073346481411080306; w(21)=0.059298584915436781; w(22)=0.044277438817419806; w(23)=0.028531388628933663; w(24)=0.012341229799987200; case(25) w(1)=0.011393798501026288; w(2)=0.026354986615032137; w(3)=0.040939156701306313; w(4)=0.054904695975835192; w(5)=0.068038333812356917; w(6)=0.080140700335001018; w(7)=0.091028261982963650; w(8)=0.10053594906705064; w(9)=0.10851962447426365; w(10)=0.11485825914571165; w(11)=0.11945576353578477; w(12)=0.12224244299031004; w(13)=0.12317605372671545; w(14)=0.12224244299031004; w(15)=0.11945576353578477; w(16)=0.11485825914571165; w(17)=0.10851962447426365; w(18)=0.10053594906705064; w(19)=0.091028261982963650; w(20)=0.080140700335001018; w(21)=0.068038333812356917; w(22)=0.054904695975835192; w(23)=0.040939156701306313; w(24)=0.026354986615032137; w(25)=0.011393798501026288; case(26) w(1)=0.010551372617343007; w(2)=0.024417851092631909; w(3)=0.037962383294362764; w(4)=0.050975825297147812; w(5)=0.063274046329574836; w(6)=0.074684149765659746; w(7)=0.085045894313485239; w(8)=0.094213800355914148; w(9)=0.10205916109442542; w(10)=0.10847184052857659; w(11)=0.11336181654631967; w(12)=0.11666044348529658; w(13)=0.11832141527926228; w(14)=0.11832141527926228; w(15)=0.11666044348529658; w(16)=0.11336181654631967; w(17)=0.10847184052857659; w(18)=0.10205916109442542; w(19)=0.094213800355914148; w(20)=0.085045894313485239; w(21)=0.074684149765659746; w(22)=0.063274046329574836; w(23)=0.050975825297147812; w(24)=0.037962383294362764; w(25)=0.024417851092631909; w(26)=0.010551372617343007; case(27) w(1)=0.0097989960512943603; w(2)=0.022686231596180623; w(3)=0.035297053757419711; w(4)=0.047449412520615063; w(5)=0.058983536859833599; w(6)=0.069748823766245593; w(7)=0.079604867773057771; w(8)=0.088423158543756950; w(9)=0.096088727370028508; w(10)=0.10250163781774580; w(11)=0.10757828578853319; w(12)=0.11125248835684519; w(13)=0.11347634610896515; w(14)=0.11422086737895699; w(15)=0.11347634610896515; w(16)=0.11125248835684519; w(17)=0.10757828578853319; w(18)=0.10250163781774580; w(19)=0.096088727370028508; w(20)=0.088423158543756950; w(21)=0.079604867773057771; w(22)=0.069748823766245593; w(23)=0.058983536859833599; w(24)=0.047449412520615063; w(25)=0.035297053757419711; w(26)=0.022686231596180623; w(27)=0.0097989960512943603; case(28) w(1)=0.0091242825930945177; w(2)=0.021132112592771260; w(3)=0.032901427782304380; w(4)=0.044272934759004228; w(5)=0.055107345675716745; w(6)=0.065272923966999596; w(7)=0.074646214234568779; w(8)=0.083113417228901218; w(9)=0.090571744393032841; w(10)=0.096930657997929916; w(11)=0.10211296757806077; w(12)=0.10605576592284642; w(13)=0.10871119225829414; w(14)=0.11004701301647520; w(15)=0.11004701301647520; w(16)=0.10871119225829414; w(17)=0.10605576592284642; w(18)=0.10211296757806077; w(19)=0.096930657997929916; w(20)=0.090571744393032841; w(21)=0.083113417228901218; w(22)=0.074646214234568779; w(23)=0.065272923966999596; w(24)=0.055107345675716745; w(25)=0.044272934759004228; w(26)=0.032901427782304380; w(27)=0.021132112592771260; w(28)=0.0091242825930945177; case(29) w(1)=0.0085169038787464097; w(2)=0.019732085056122706; w(3)=0.030740492202093623; w(4)=0.041402062518682836; w(5)=0.051594826902497924; w(6)=0.061203090657079139; w(7)=0.070117933255051279; w(8)=0.078238327135763784; w(9)=0.085472257366172528; w(10)=0.091737757139258763; w(11)=0.096963834094408606; w(12)=0.10109127375991497; w(13)=0.10407331007772937; w(14)=0.10587615509732094; w(15)=0.10647938171831424; w(16)=0.10587615509732094; w(17)=0.10407331007772937; w(18)=0.10109127375991497; w(19)=0.096963834094408606; w(20)=0.091737757139258763; w(21)=0.085472257366172528; w(22)=0.078238327135763784; w(23)=0.070117933255051279; w(24)=0.061203090657079139; w(25)=0.051594826902497924; w(26)=0.041402062518682836; w(27)=0.030740492202093623; w(28)=0.019732085056122706; w(29)=0.0085169038787464097; case(30) w(1)=0.0079681924961666056; w(2)=0.018466468311090959; w(3)=0.028784707883323369; w(4)=0.038799192569627050; w(5)=0.048402672830594053; w(6)=0.057493156217619066; w(7)=0.065974229882180495; w(8)=0.073755974737705206; w(9)=0.080755895229420215; w(10)=0.086899787201082980; w(11)=0.092122522237786129; w(12)=0.096368737174644260; w(13)=0.099593420586795267; w(14)=0.10176238974840550; w(15)=0.10285265289355884; w(16)=0.10285265289355884; w(17)=0.10176238974840550; w(18)=0.099593420586795267; w(19)=0.096368737174644260; w(20)=0.092122522237786129; w(21)=0.086899787201082980; w(22)=0.080755895229420215; w(23)=0.073755974737705206; w(24)=0.065974229882180495; w(25)=0.057493156217619066; w(26)=0.048402672830594053; w(27)=0.038799192569627050; w(28)=0.028784707883323369; w(29)=0.018466468311090959; w(30)=0.0079681924961666056; case(31) w(1)=0.0074708315792487759; w(2)=0.017318620790310582; w(3)=0.027009019184979422; w(4)=0.036432273912385464; w(5)=0.045493707527201103; w(6)=0.054103082424916854; w(7)=0.062174786561028427; w(8)=0.069628583235410366; w(9)=0.076390386598776616; w(10)=0.082392991761589264; w(11)=0.087576740608477876; w(12)=0.091890113893641478; w(13)=0.095290242912319513; w(14)=0.097743335386328725; w(15)=0.099225011226672308; w(16)=0.099720544793426451; w(17)=0.099225011226672308; w(18)=0.097743335386328725; w(19)=0.095290242912319513; w(20)=0.091890113893641478; w(21)=0.087576740608477876; w(22)=0.082392991761589264; w(23)=0.076390386598776616; w(24)=0.069628583235410366; w(25)=0.062174786561028427; w(26)=0.054103082424916854; w(27)=0.045493707527201103; w(28)=0.036432273912385464; w(29)=0.027009019184979422; w(30)=0.017318620790310582; w(31)=0.0074708315792487759; case(32) w(1)=0.0070186100094700966; w(2)=0.016274394730905671; w(3)=0.025392065309262059; w(4)=0.034273862913021433; w(5)=0.042835898022226681; w(6)=0.050998059262376176; w(7)=0.058684093478535547; w(8)=0.065822222776361847; w(9)=0.072345794108848506; w(10)=0.078193895787070306; w(11)=0.083311924226946755; w(12)=0.087652093004403811; w(13)=0.091173878695763885; w(14)=0.093844399080804566; w(15)=0.095638720079274859; w(16)=0.096540088514727801; w(17)=0.096540088514727801; w(18)=0.095638720079274859; w(19)=0.093844399080804566; w(20)=0.091173878695763885; w(21)=0.087652093004403811; w(22)=0.083311924226946755; w(23)=0.078193895787070306; w(24)=0.072345794108848506; w(25)=0.065822222776361847; w(26)=0.058684093478535547; w(27)=0.050998059262376176; w(28)=0.042835898022226681; w(29)=0.034273862913021433; w(30)=0.025392065309262059; w(31)=0.016274394730905671; w(32)=0.0070186100094700966; case(33) w(1)=0.0066062278475873781; w(2)=0.015321701512934676; w(3)=0.023915548101749480; w(4)=0.032300358632328953; w(5)=0.040401541331669592; w(6)=0.048147742818711696; w(7)=0.055470846631663561; w(8)=0.062306482530317480; w(9)=0.068594572818656713; w(10)=0.074279854843954149; w(11)=0.079312364794886738; w(12)=0.083647876067038708; w(13)=0.087248287618844338; w(14)=0.090081958660638577; w(15)=0.092123986643316846; w(16)=0.093356426065596116; w(17)=0.093768446160209997; w(18)=0.093356426065596116; w(19)=0.092123986643316846; w(20)=0.090081958660638577; w(21)=0.087248287618844338; w(22)=0.083647876067038708; w(23)=0.079312364794886738; w(24)=0.074279854843954149; w(25)=0.068594572818656713; w(26)=0.062306482530317480; w(27)=0.055470846631663561; w(28)=0.048147742818711696; w(29)=0.040401541331669592; w(30)=0.032300358632328953; w(31)=0.023915548101749480; w(32)=0.015321701512934676; w(33)=0.0066062278475873781; case(34) w(1)=0.0062291405559086847; w(2)=0.014450162748595035; w(3)=0.022563721985494970; w(4)=0.030491380638446132; w(5)=0.038166593796387516; w(6)=0.045525611523353272; w(7)=0.052507414572678106; w(8)=0.059054135827524493; w(9)=0.065111521554076411; w(10)=0.070629375814255725; w(11)=0.075561974660031931; w(12)=0.079868444339771845; w(13)=0.083513099699845655; w(14)=0.086465739747035750; w(15)=0.088701897835693869; w(16)=0.090203044370640730; w(17)=0.090956740330259874; w(18)=0.090956740330259874; w(19)=0.090203044370640730; w(20)=0.088701897835693869; w(21)=0.086465739747035750; w(22)=0.083513099699845655; w(23)=0.079868444339771845; w(24)=0.075561974660031931; w(25)=0.070629375814255725; w(26)=0.065111521554076411; w(27)=0.059054135827524493; w(28)=0.052507414572678106; w(29)=0.045525611523353272; w(30)=0.038166593796387516; w(31)=0.030491380638446132; w(32)=0.022563721985494970; w(33)=0.014450162748595035; w(34)=0.0062291405559086847; case(35) w(1)=0.0058834334204430850; w(2)=0.013650828348361492; w(3)=0.021322979911483581; w(4)=0.028829260108894254; w(5)=0.036110115863463381; w(6)=0.043108422326170219; w(7)=0.049769370401353530; w(8)=0.056040816212370129; w(9)=0.061873671966080189; w(10)=0.067222285269086904; w(11)=0.072044794772560065; w(12)=0.076303457155442054; w(13)=0.079964942242324263; w(14)=0.083000593728856588; w(15)=0.085386653392099125; w(16)=0.087104446997183534; w(17)=0.088140530430275463; w(18)=0.088486794907104291; w(19)=0.088140530430275463; w(20)=0.087104446997183534; w(21)=0.085386653392099125; w(22)=0.083000593728856588; w(23)=0.079964942242324263; w(24)=0.076303457155442054; w(25)=0.072044794772560065; w(26)=0.067222285269086904; w(27)=0.061873671966080189; w(28)=0.056040816212370129; w(29)=0.049769370401353530; w(30)=0.043108422326170219; w(31)=0.036110115863463381; w(32)=0.028829260108894254; w(33)=0.021322979911483581; w(34)=0.013650828348361492; w(35)=0.0058834334204430850; case(36) w(1)=0.0055657196642450454; w(2)=0.012915947284065574; w(3)=0.020181515297735472; w(4)=0.027298621498568779; w(5)=0.034213810770307230; w(6)=0.040875750923644895; w(7)=0.047235083490265978; w(8)=0.053244713977759919; w(9)=0.058860144245324817; w(10)=0.064039797355015490; w(11)=0.068745323835736443; w(12)=0.072941885005653061; w(13)=0.076598410645870675; w(14)=0.079687828912071602; w(15)=0.082187266704339710; w(16)=0.084078218979661935; w(17)=0.085346685739338627; w(18)=0.085983275670394747; w(19)=0.085983275670394747; w(20)=0.085346685739338627; w(21)=0.084078218979661935; w(22)=0.082187266704339710; w(23)=0.079687828912071602; w(24)=0.076598410645870675; w(25)=0.072941885005653061; w(26)=0.068745323835736443; w(27)=0.064039797355015490; w(28)=0.058860144245324817; w(29)=0.053244713977759919; w(30)=0.047235083490265978; w(31)=0.040875750923644895; w(32)=0.034213810770307230; w(33)=0.027298621498568779; w(34)=0.020181515297735472; w(35)=0.012915947284065574; w(36)=0.0055657196642450454; case(37) w(1)=0.0052730572794979394; w(2)=0.012238780100307557; w(3)=0.019129044489083966; w(4)=0.025886036990558934; w(5)=0.032461639847521481; w(6)=0.038809602501934544; w(7)=0.044885364662437167; w(8)=0.050646297654824602; w(9)=0.056051987998274918; w(10)=0.061064516523225986; w(11)=0.065648722872751249; w(12)=0.069772451555700345; w(13)=0.073406777248488173; w(14)=0.076526207570529238; w(15)=0.079108861837529381; w(16)=0.081136624508465031; w(17)=0.082595272236437251; w(18)=0.083474573625862787; w(19)=0.083768360993138905; w(20)=0.083474573625862787; w(21)=0.082595272236437251; w(22)=0.081136624508465031; w(23)=0.079108861837529381; w(24)=0.076526207570529238; w(25)=0.073406777248488173; w(26)=0.069772451555700345; w(27)=0.065648722872751249; w(28)=0.061064516523225986; w(29)=0.056051987998274918; w(30)=0.050646297654824602; w(31)=0.044885364662437167; w(32)=0.038809602501934544; w(33)=0.032461639847521481; w(34)=0.025886036990558934; w(35)=0.019129044489083966; w(36)=0.012238780100307557; w(37)=0.0052730572794979394; case(38) w(1)=0.0050028807496393457; w(2)=0.011613444716468674; w(3)=0.018156577709613237; w(4)=0.024579739738232376; w(5)=0.030839500545175055; w(6)=0.036894081594024738; w(7)=0.042703158504674434; w(8)=0.048228061860758683; w(9)=0.053432019910332320; w(10)=0.058280399146997206; w(11)=0.062740933392133054; w(12)=0.066783937979140412; w(13)=0.070382507066898955; w(14)=0.073512692584743457; w(15)=0.076153663548446396; w(16)=0.078287844658210948; w(17)=0.079901033243527822; w(18)=0.080982493770597101; w(19)=0.081525029280385787; w(20)=0.081525029280385787; w(21)=0.080982493770597101; w(22)=0.079901033243527822; w(23)=0.078287844658210948; w(24)=0.076153663548446396; w(25)=0.073512692584743457; w(26)=0.070382507066898955; w(27)=0.066783937979140412; w(28)=0.062740933392133054; w(29)=0.058280399146997206; w(30)=0.053432019910332320; w(31)=0.048228061860758683; w(32)=0.042703158504674434; w(33)=0.036894081594024738; w(34)=0.030839500545175055; w(35)=0.024579739738232376; w(36)=0.018156577709613237; w(37)=0.011613444716468674; w(38)=0.0050028807496393457; case(39) w(1)=0.0047529446916351014; w(2)=0.011034788939164594; w(3)=0.017256229093724919; w(4)=0.023369384832178165; w(5)=0.029334955983903379; w(6)=0.035115111498131331; w(7)=0.040673276847933844; w(8)=0.045974301108916632; w(9)=0.050984665292129405; w(10)=0.055672690340916300; w(11)=0.060008736088596150; w(12)=0.063965388138682389; w(13)=0.067517630966231265; w(14)=0.070643005970608761; w(15)=0.073321753414268617; w(16)=0.075536937322836058; w(17)=0.077274552544682017; w(18)=0.078523613287371177; w(19)=0.079276222568368471; w(20)=0.079527622139442852; w(21)=0.079276222568368471; w(22)=0.078523613287371177; w(23)=0.077274552544682017; w(24)=0.075536937322836058; w(25)=0.073321753414268617; w(26)=0.070643005970608761; w(27)=0.067517630966231265; w(28)=0.063965388138682389; w(29)=0.060008736088596150; w(30)=0.055672690340916300; w(31)=0.050984665292129405; w(32)=0.045974301108916632; w(33)=0.040673276847933844; w(34)=0.035115111498131331; w(35)=0.029334955983903379; w(36)=0.023369384832178165; w(37)=0.017256229093724919; w(38)=0.011034788939164594; w(39)=0.0047529446916351014; case(40) w(1)=0.0045212770985331913; w(2)=0.010498284531152814; w(3)=0.016421058381907889; w(4)=0.022245849194166957; w(5)=0.027937006980023401; w(6)=0.033460195282547847; w(7)=0.038782167974472018; w(8)=0.043870908185673272; w(9)=0.048695807635072232; w(10)=0.053227846983936824; w(11)=0.057439769099391551; w(12)=0.061306242492928939; w(13)=0.064804013456601038; w(14)=0.067912045815233904; w(15)=0.070611647391286780; w(16)=0.072886582395804059; w(17)=0.074723169057968264; w(18)=0.076110361900626242; w(19)=0.077039818164247966; w(20)=0.077505947978424811; w(21)=0.077505947978424811; w(22)=0.077039818164247966; w(23)=0.076110361900626242; w(24)=0.074723169057968264; w(25)=0.072886582395804059; w(26)=0.070611647391286780; w(27)=0.067912045815233904; w(28)=0.064804013456601038; w(29)=0.061306242492928939; w(30)=0.057439769099391551; w(31)=0.053227846983936824; w(32)=0.048695807635072232; w(33)=0.043870908185673272; w(34)=0.038782167974472018; w(35)=0.033460195282547847; w(36)=0.027937006980023401; w(37)=0.022245849194166957; w(38)=0.016421058381907889; w(39)=0.010498284531152814; w(40)=0.0045212770985331913; case(41) w(1)=0.0043061403581648877; w(2)=0.0099999387739059453; w(3)=0.015644938407818589; w(4)=0.021201063368779553; w(5)=0.026635899207110445; w(6)=0.031918211731699282; w(7)=0.037017716703507988; w(8)=0.041905195195909689; w(9)=0.046552648369014342; w(10)=0.050933454294617495; w(11)=0.055022519242578742; w(12)=0.058796420949871945; w(13)=0.062233542580966316; w(14)=0.065314196453527410; w(15)=0.068020736760876767; w(16)=0.070337660620817497; w(17)=0.072251696861023073; w(18)=0.073751882027223470; w(19)=0.074829623176221552; w(20)=0.075478747092715824; w(21)=0.075695535647298372; w(22)=0.075478747092715824; w(23)=0.074829623176221552; w(24)=0.073751882027223470; w(25)=0.072251696861023073; w(26)=0.070337660620817497; w(27)=0.068020736760876767; w(28)=0.065314196453527410; w(29)=0.062233542580966316; w(30)=0.058796420949871945; w(31)=0.055022519242578742; w(32)=0.050933454294617495; w(33)=0.046552648369014342; w(34)=0.041905195195909689; w(35)=0.037017716703507988; w(36)=0.031918211731699282; w(37)=0.026635899207110445; w(38)=0.021201063368779553; w(39)=0.015644938407818589; w(40)=0.0099999387739059453; w(41)=0.0043061403581648877; case(42) w(1)=0.0041059986046490846; w(2)=0.0095362203017485024; w(3)=0.014922443697357494; w(4)=0.020227869569052645; w(5)=0.025422959526113048; w(6)=0.030479240699603468; w(7)=0.035369071097592111; w(8)=0.040065735180692262; w(9)=0.044543577771965878; w(10)=0.048778140792803245; w(11)=0.052746295699174070; w(12)=0.056426369358018382; w(13)=0.059798262227586654; w(14)=0.062843558045002576; w(15)=0.065545624364908979; w(16)=0.067889703376521945; w(17)=0.069862992492594160; w(18)=0.071454714265170983; w(19)=0.072656175243804105; w(20)=0.073460813453467528; w(21)=0.073864234232172880; w(22)=0.073864234232172880; w(23)=0.073460813453467528; w(24)=0.072656175243804105; w(25)=0.071454714265170983; w(26)=0.069862992492594160; w(27)=0.067889703376521945; w(28)=0.065545624364908979; w(29)=0.062843558045002576; w(30)=0.059798262227586654; w(31)=0.056426369358018382; w(32)=0.052746295699174070; w(33)=0.048778140792803245; w(34)=0.044543577771965878; w(35)=0.040065735180692262; w(36)=0.035369071097592111; w(37)=0.030479240699603468; w(38)=0.025422959526113048; w(39)=0.020227869569052645; w(40)=0.014922443697357494; w(41)=0.0095362203017485024; w(42)=0.0041059986046490846; case(43) w(1)=0.0039194902538441273; w(2)=0.0091039966374014033; w(3)=0.014248756431576486; w(4)=0.019319901423683900; w(5)=0.024290456613838816; w(6)=0.029134413261498495; w(7)=0.033826492086860292; w(8)=0.038342222194132658; w(9)=0.042658057197982084; w(10)=0.046751494754346580; w(11)=0.050601192784390157; w(12)=0.054187080318881787; w(13)=0.057490461956910519; w(14)=0.060494115249991295; w(15)=0.063182380449396112; w(16)=0.065541242126322797; w(17)=0.067558402229365169; w(18)=0.069223344193656684; w(19)=0.070527387765085028; w(20)=0.071463734252514141; w(21)=0.072027501971421974; w(22)=0.072215751693798988; w(23)=0.072027501971421974; w(24)=0.071463734252514141; w(25)=0.070527387765085028; w(26)=0.069223344193656684; w(27)=0.067558402229365169; w(28)=0.065541242126322797; w(29)=0.063182380449396112; w(30)=0.060494115249991295; w(31)=0.057490461956910519; w(32)=0.054187080318881787; w(33)=0.050601192784390157; w(34)=0.046751494754346580; w(35)=0.042658057197982084; w(36)=0.038342222194132658; w(37)=0.033826492086860292; w(38)=0.029134413261498495; w(39)=0.024290456613838816; w(40)=0.019319901423683900; w(41)=0.014248756431576486; w(42)=0.0091039966374014033; w(43)=0.0039194902538441273; case(44) w(1)=0.0037454048031127775; w(2)=0.0087004813675248441; w(3)=0.013619586755579986; w(4)=0.018471481736814749; w(5)=0.023231481902019211; w(6)=0.027875782821281010; w(7)=0.032381222812069821; w(8)=0.036725347813808874; w(9)=0.040886512310346219; w(10)=0.044843984081970031; w(11)=0.048578046448352038; w(12)=0.052070096091704462; w(13)=0.055302735563728053; w(14)=0.058259859877595495; w(15)=0.060926736701561968; w(16)=0.063290079733203855; w(17)=0.065338114879181435; w(18)=0.067060638906293652; w(19)=0.068449070269366661; w(20)=0.069496491861572578; w(21)=0.070197685473558213; w(22)=0.070549157789354069; w(23)=0.070549157789354069; w(24)=0.070197685473558213; w(25)=0.069496491861572578; w(26)=0.068449070269366661; w(27)=0.067060638906293652; w(28)=0.065338114879181435; w(29)=0.063290079733203855; w(30)=0.060926736701561968; w(31)=0.058259859877595495; w(32)=0.055302735563728053; w(33)=0.052070096091704462; w(34)=0.048578046448352038; w(35)=0.044843984081970031; w(36)=0.040886512310346219; w(37)=0.036725347813808874; w(38)=0.032381222812069821; w(39)=0.027875782821281010; w(40)=0.023231481902019211; w(41)=0.018471481736814749; w(42)=0.013619586755579986; w(43)=0.0087004813675248441; w(44)=0.0037454048031127775; case(45) w(1)=0.0035826631552835589; w(2)=0.0083231892962182416; w(3)=0.013031104991582784; w(4)=0.017677535257937591; w(5)=0.022239847550578732; w(6)=0.026696213967577665; w(7)=0.031025374934515467; w(8)=0.035206692201609016; w(9)=0.039220236729302448; w(10)=0.043046880709164971; w(11)=0.046668387718373365; w(12)=0.050067499237952030; w(13)=0.053228016731268952; w(14)=0.056134878759786477; w(15)=0.058774232718841739; w(16)=0.061133500831066523; w(17)=0.063201440073819938; w(18)=0.064968195750723431; w(19)=0.066425348449842528; w(20)=0.067565954163607536; w(21)=0.068384577378669675; w(22)=0.068877316977661323; w(23)=0.069041824829232020; w(24)=0.068877316977661323; w(25)=0.068384577378669675; w(26)=0.067565954163607536; w(27)=0.066425348449842528; w(28)=0.064968195750723431; w(29)=0.063201440073819938; w(30)=0.061133500831066523; w(31)=0.058774232718841739; w(32)=0.056134878759786477; w(33)=0.053228016731268952; w(34)=0.050067499237952030; w(35)=0.046668387718373365; w(36)=0.043046880709164971; w(37)=0.039220236729302448; w(38)=0.035206692201609016; w(39)=0.031025374934515467; w(40)=0.026696213967577665; w(41)=0.022239847550578732; w(42)=0.017677535257937591; w(43)=0.013031104991582784; w(44)=0.0083231892962182416; w(45)=0.0035826631552835589; case(46) w(1)=0.0034303008681070483; w(2)=0.0079698982297246225; w(3)=0.012479883770988684; w(4)=0.016933514007836238; w(5)=0.021309998754136501; w(6)=0.025589286397130011; w(7)=0.029751829552202756; w(8)=0.033778627999106897; w(9)=0.037651305357386071; w(10)=0.041352190109678730; w(11)=0.044864395277318127; w(12)=0.048171895101712201; w(13)=0.051259598007143021; w(14)=0.054113415385856754; w(15)=0.056720325843991236; w(16)=0.059068434595546315; w(17)=0.061147027724650481; w(18)=0.062946621064394508; w(19)=0.064459003467139070; w(20)=0.065677274267781207; w(21)=0.066595874768454887; w(22)=0.067210613600678176; w(23)=0.067518685849036459; w(24)=0.067518685849036459; w(25)=0.067210613600678176; w(26)=0.066595874768454887; w(27)=0.065677274267781207; w(28)=0.064459003467139070; w(29)=0.062946621064394508; w(30)=0.061147027724650481; w(31)=0.059068434595546315; w(32)=0.056720325843991236; w(33)=0.054113415385856754; w(34)=0.051259598007143021; w(35)=0.048171895101712201; w(36)=0.044864395277318127; w(37)=0.041352190109678730; w(38)=0.037651305357386071; w(39)=0.033778627999106897; w(40)=0.029751829552202756; w(41)=0.025589286397130011; w(42)=0.021309998754136501; w(43)=0.016933514007836238; w(44)=0.012479883770988684; w(45)=0.0079698982297246225; w(46)=0.0034303008681070483; case(47) w(1)=0.0032874538425280149; w(2)=0.0076386162958488336; w(3)=0.011962848464312321; w(4)=0.016235333146433060; w(5)=0.020436938147668428; w(6)=0.024549211659658819; w(7)=0.028554150700643387; w(8)=0.032434235515184757; w(9)=0.036172496584174952; w(10)=0.039752586122531004; w(11)=0.043158848648479538; w(12)=0.046376389086505911; w(13)=0.049391137747361170; w(14)=0.052189911780057145; w(15)=0.054760472781530226; w(16)=0.057091580293231540; w(17)=0.059173040942338876; w(18)=0.060995753008739645; w(19)=0.062551746220921663; w(20)=0.063834216605717031; w(21)=0.064837556238945727; w(22)=0.065557377766549740; w(23)=0.065990533588810475; w(24)=0.066135129623655480; w(25)=0.065990533588810475; w(26)=0.065557377766549740; w(27)=0.064837556238945727; w(28)=0.063834216605717031; w(29)=0.062551746220921663; w(30)=0.060995753008739645; w(31)=0.059173040942338876; w(32)=0.057091580293231540; w(33)=0.054760472781530226; w(34)=0.052189911780057145; w(35)=0.049391137747361170; w(36)=0.046376389086505911; w(37)=0.043158848648479538; w(38)=0.039752586122531004; w(39)=0.036172496584174952; w(40)=0.032434235515184757; w(41)=0.028554150700643387; w(42)=0.024549211659658819; w(43)=0.020436938147668428; w(44)=0.016235333146433060; w(45)=0.011962848464312321; w(46)=0.0076386162958488336; w(47)=0.0032874538425280149; case(48) w(1)=0.0031533460523058386; w(2)=0.0073275539012762621; w(3)=0.011477234579234539; w(4)=0.015579315722943849; w(5)=0.019616160457355528; w(6)=0.023570760839324379; w(7)=0.027426509708356948; w(8)=0.031167227832798089; w(9)=0.034777222564770439; w(10)=0.038241351065830706; w(11)=0.041545082943464749; w(12)=0.044674560856694280; w(13)=0.047616658492490475; w(14)=0.050359035553854475; w(15)=0.052890189485193667; w(16)=0.055199503699984163; w(17)=0.057277292100403216; w(18)=0.059114839698395636; w(19)=0.060704439165893880; w(20)=0.062039423159892664; w(21)=0.063114192286254026; w(22)=0.063924238584648187; w(23)=0.064466164435950082; w(24)=0.064737696812683923; w(25)=0.064737696812683923; w(26)=0.064466164435950082; w(27)=0.063924238584648187; w(28)=0.063114192286254026; w(29)=0.062039423159892664; w(30)=0.060704439165893880; w(31)=0.059114839698395636; w(32)=0.057277292100403216; w(33)=0.055199503699984163; w(34)=0.052890189485193667; w(35)=0.050359035553854475; w(36)=0.047616658492490475; w(37)=0.044674560856694280; w(38)=0.041545082943464749; w(39)=0.038241351065830706; w(40)=0.034777222564770439; w(41)=0.031167227832798089; w(42)=0.027426509708356948; w(43)=0.023570760839324379; w(44)=0.019616160457355528; w(45)=0.015579315722943849; w(46)=0.011477234579234539; w(47)=0.0073275539012762621; w(48)=0.0031533460523058386; case(49) w(1)=0.0030272789889229051; w(2)=0.0070350995900864515; w(3)=0.011020551031593580; w(4)=0.014962144935624651; w(5)=0.018843595853089458; w(6)=0.022649201587446676; w(7)=0.026363618927066017; w(8)=0.029971884620583825; w(9)=0.033459466791622174; w(10)=0.036812320963000690; w(11)=0.040016945766373021; w(12)=0.043060436981259598; w(13)=0.045930539355595854; w(14)=0.048615695887828240; w(15)=0.051105094330144591; w(16)=0.053388710708258969; w(17)=0.055457349674803589; w(18)=0.057302681530187475; w(19)=0.058917275760027266; w(20)=0.060294630953152017; w(21)=0.061429200979192936; w(22)=0.062316417320057267; w(23)=0.062952707465195699; w(24)=0.063335509296491749; w(25)=0.063463281404790598; w(26)=0.063335509296491749; w(27)=0.062952707465195699; w(28)=0.062316417320057267; w(29)=0.061429200979192936; w(30)=0.060294630953152017; w(31)=0.058917275760027266; w(32)=0.057302681530187475; w(33)=0.055457349674803589; w(34)=0.053388710708258969; w(35)=0.051105094330144591; w(36)=0.048615695887828240; w(37)=0.045930539355595854; w(38)=0.043060436981259598; w(39)=0.040016945766373021; w(40)=0.036812320963000690; w(41)=0.033459466791622174; w(42)=0.029971884620583825; w(43)=0.026363618927066017; w(44)=0.022649201587446676; w(45)=0.018843595853089458; w(46)=0.014962144935624651; w(47)=0.011020551031593580; w(48)=0.0070350995900864515; w(49)=0.0030272789889229051; case(50) w(1)=0.0029086225531551410; w(2)=0.0067597991957454015; w(3)=0.010590548383650969; w(4)=0.014380822761485574; w(5)=0.018115560713489390; w(6)=0.021780243170124793; w(7)=0.025360673570012390; w(8)=0.028842993580535198; w(9)=0.032213728223578017; w(10)=0.035459835615146154; w(11)=0.038568756612587675; w(12)=0.041528463090147697; w(13)=0.044327504338803275; w(14)=0.046955051303948433; w(15)=0.049400938449466315; w(16)=0.051655703069581138; w(17)=0.053710621888996247; w(18)=0.055557744806212518; w(19)=0.057189925647728384; w(20)=0.058600849813222446; w(21)=0.059785058704265458; w(22)=0.060737970841770216; w(23)=0.061455899590316664; w(24)=0.061936067420683243; w(25)=0.062176616655347262; w(26)=0.062176616655347262; w(27)=0.061936067420683243; w(28)=0.061455899590316664; w(29)=0.060737970841770216; w(30)=0.059785058704265458; w(31)=0.058600849813222446; w(32)=0.057189925647728384; w(33)=0.055557744806212518; w(34)=0.053710621888996247; w(35)=0.051655703069581138; w(36)=0.049400938449466315; w(37)=0.046955051303948433; w(38)=0.044327504338803275; w(39)=0.041528463090147697; w(40)=0.038568756612587675; w(41)=0.035459835615146154; w(42)=0.032213728223578017; w(43)=0.028842993580535198; w(44)=0.025360673570012390; w(45)=0.021780243170124793; w(46)=0.018115560713489390; w(47)=0.014380822761485574; w(48)=0.010590548383650969; w(49)=0.0067597991957454015; w(50)=0.0029086225531551410; case(51) w(1)=0.0027968071710898956; w(2)=0.0065003377832526003; w(3)=0.010185191297821730; w(4)=0.013832634006477822; w(5)=0.017428714723401052; w(6)=0.020959988401703211; w(7)=0.024413300573781434; w(8)=0.027775798594162477; w(9)=0.031034971290160008; w(10)=0.034178693204188336; w(11)=0.037195268923260293; w(12)=0.040073476285496453; w(13)=0.042802607997880087; w(14)=0.045372511407650069; w(15)=0.047773626240623102; w(16)=0.049997020150057410; w(17)=0.052034421936697088; w(18)=0.053878252313045561; w(19)=0.055521652095738693; w(20)=0.056958507720258662; w(21)=0.058183473982592141; w(22)=0.059191993922961544; w(23)=0.059980315777503252; w(24)=0.060545506934737795; w(25)=0.060885464844856344; w(26)=0.060998924841205880; w(27)=0.060885464844856344; w(28)=0.060545506934737795; w(29)=0.059980315777503252; w(30)=0.059191993922961544; w(31)=0.058183473982592141; w(32)=0.056958507720258662; w(33)=0.055521652095738693; w(34)=0.053878252313045561; w(35)=0.052034421936697088; w(36)=0.049997020150057410; w(37)=0.047773626240623102; w(38)=0.045372511407650069; w(39)=0.042802607997880087; w(40)=0.040073476285496453; w(41)=0.037195268923260293; w(42)=0.034178693204188336; w(43)=0.031034971290160008; w(44)=0.027775798594162477; w(45)=0.024413300573781434; w(46)=0.020959988401703211; w(47)=0.017428714723401052; w(48)=0.013832634006477822; w(49)=0.010185191297821730; w(50)=0.0065003377832526003; w(51)=0.0027968071710898956; case(52) w(1)=0.0026913169500471111; w(2)=0.0062555239629732769; w(3)=0.0098026345794627521; w(4)=0.013315114982340961; w(5)=0.016780023396300736; w(6)=0.020184891507980792; w(7)=0.023517513553984462; w(8)=0.026765953746504013; w(9)=0.029918581147143947; w(10)=0.032964109089718798; w(11)=0.035891634835097233; w(12)=0.038690678310423979; w(13)=0.041351219500560272; w(14)=0.043863734259000408; w(15)=0.046219228372784794; w(16)=0.048409269744074897; w(17)=0.050426018566342377; w(18)=0.052262255383906993; w(19)=0.053911406932757265; w(20)=0.055367569669302653; w(21)=0.056625530902368597; w(22)=0.057680787452526828; w(23)=0.058529561771813869; w(24)=0.059168815466042970; w(25)=0.059596260171248158; w(26)=0.059810365745291860; w(27)=0.059810365745291860; w(28)=0.059596260171248158; w(29)=0.059168815466042970; w(30)=0.058529561771813869; w(31)=0.057680787452526828; w(32)=0.056625530902368597; w(33)=0.055367569669302653; w(34)=0.053911406932757265; w(35)=0.052262255383906993; w(36)=0.050426018566342377; w(37)=0.048409269744074897; w(38)=0.046219228372784794; w(39)=0.043863734259000408; w(40)=0.041351219500560272; w(41)=0.038690678310423979; w(42)=0.035891634835097233; w(43)=0.032964109089718798; w(44)=0.029918581147143947; w(45)=0.026765953746504013; w(46)=0.023517513553984462; w(47)=0.020184891507980792; w(48)=0.016780023396300736; w(49)=0.013315114982340961; w(50)=0.0098026345794627521; w(51)=0.0062555239629732769; w(52)=0.0026913169500471111; case(53) w(1)=0.0025916837205670318; w(2)=0.0060242762269486733; w(3)=0.0094412022849403444; w(4)=0.012826026144240379; w(5)=0.016166725256687464; w(6)=0.019451721107636895; w(7)=0.022669673057070208; w(8)=0.025809482510757518; w(9)=0.028860323617823736; w(10)=0.031811678459019323; w(11)=0.034653372583534238; w(12)=0.037375609803482916; w(13)=0.039969005843540382; w(14)=0.042424620634520014; w(15)=0.044733989103672810; w(16)=0.046889150340750314; w(17)=0.048882675032699140; w(18)=0.050707691069292715; w(19)=0.052357907229872718; w(20)=0.053827634868731029; w(21)=0.055111807523933599; w(22)=0.056205998381739710; w(23)=0.057106435536267192; w(24)=0.057810014991713196; w(25)=0.058314311362256008; w(26)=0.058617586232720263; w(27)=0.058718794151164365; w(28)=0.058617586232720263; w(29)=0.058314311362256008; w(30)=0.057810014991713196; w(31)=0.057106435536267192; w(32)=0.056205998381739710; w(33)=0.055111807523933599; w(34)=0.053827634868731029; w(35)=0.052357907229872718; w(36)=0.050707691069292715; w(37)=0.048882675032699140; w(38)=0.046889150340750314; w(39)=0.044733989103672810; w(40)=0.042424620634520014; w(41)=0.039969005843540382; w(42)=0.037375609803482916; w(43)=0.034653372583534238; w(44)=0.031811678459019323; w(45)=0.028860323617823736; w(46)=0.025809482510757518; w(47)=0.022669673057070208; w(48)=0.019451721107636895; w(49)=0.016166725256687464; w(50)=0.012826026144240379; w(51)=0.0094412022849403444; w(52)=0.0060242762269486733; w(53)=0.0025916837205670318; case(54) w(1)=0.0024974818357615858; w(2)=0.0058056110152399849; w(3)=0.0090993694555093969; w(4)=0.012363328128847644; w(5)=0.015586303035924132; w(6)=0.018757527621469378; w(7)=0.021866451422853086; w(8)=0.024902741467208773; w(9)=0.027856309310595870; w(10)=0.030717342497870676; w(11)=0.033476336464372646; w(12)=0.036124125840383553; w(13)=0.038651914782102517; w(14)=0.041051306136644974; w(15)=0.043314329309597015; w(16)=0.045433466728276714; w(17)=0.047401678806444991; w(18)=0.049212427324528886; w(19)=0.050859697146188144; w(20)=0.052338016198298745; w(21)=0.053642473647553611; w(22)=0.054768736213057986; w(23)=0.055713062560589988; w(24)=0.056472315730625965; w(25)=0.057043973558794599; w(26)=0.057426137054112115; w(27)=0.057617536707147025; w(28)=0.057617536707147025; w(29)=0.057426137054112115; w(30)=0.057043973558794599; w(31)=0.056472315730625965; w(32)=0.055713062560589988; w(33)=0.054768736213057986; w(34)=0.053642473647553611; w(35)=0.052338016198298745; w(36)=0.050859697146188144; w(37)=0.049212427324528886; w(38)=0.047401678806444991; w(39)=0.045433466728276714; w(40)=0.043314329309597015; w(41)=0.041051306136644974; w(42)=0.038651914782102517; w(43)=0.036124125840383553; w(44)=0.033476336464372646; w(45)=0.030717342497870676; w(46)=0.027856309310595870; w(47)=0.024902741467208773; w(48)=0.021866451422853086; w(49)=0.018757527621469378; w(50)=0.015586303035924132; w(51)=0.012363328128847644; w(52)=0.0090993694555093969; w(53)=0.0058056110152399849; w(54)=0.0024974818357615858; case(55) w(1)=0.0024083236199797888; w(2)=0.0055986322665607674; w(3)=0.0087757461070585282; w(4)=0.011925160719848612; w(5)=0.015036458333511788; w(6)=0.018099614520729062; w(7)=0.021104801668016454; w(8)=0.024042388009725622; w(9)=0.026902961456396271; w(10)=0.029677357765161041; w(11)=0.032356689226185832; w(12)=0.034932372873589887; w(13)=0.037396157867965545; w(14)=0.039740151874337180; w(15)=0.041956846317718762; w(16)=0.044039140421606590; w(17)=0.045980363946283838; w(18)=0.047774298551200696; w(19)=0.049415197711551739; w(20)=0.050897805124493979; w(21)=0.052217371545632085; w(22)=0.053369670001605473; w(23)=0.054351009329911102; w(24)=0.055158246002508688; w(25)=0.055788794195284087; w(26)=0.056240634071084368; w(27)=0.056512318249772001; w(28)=0.056602976444560425; w(29)=0.056512318249772001; w(30)=0.056240634071084368; w(31)=0.055788794195284087; w(32)=0.055158246002508688; w(33)=0.054351009329911102; w(34)=0.053369670001605473; w(35)=0.052217371545632085; w(36)=0.050897805124493979; w(37)=0.049415197711551739; w(38)=0.047774298551200696; w(39)=0.045980363946283838; w(40)=0.044039140421606590; w(41)=0.041956846317718762; w(42)=0.039740151874337180; w(43)=0.037396157867965545; w(44)=0.034932372873589887; w(45)=0.032356689226185832; w(46)=0.029677357765161041; w(47)=0.026902961456396271; w(48)=0.024042388009725622; w(49)=0.021104801668016454; w(50)=0.018099614520729062; w(51)=0.015036458333511788; w(52)=0.011925160719848612; w(53)=0.0087757461070585282; w(54)=0.0055986322665607674; w(55)=0.0024083236199797888; case(56) w(1)=0.0023238553757732155; w(2)=0.0054025222460153378; w(3)=0.0084690631633078877; w(4)=0.011509824340383382; w(5)=0.014515089278021472; w(6)=0.017475512911400947; w(7)=0.020381929882402573; w(8)=0.023225351562565317; w(9)=0.025996987058391952; w(10)=0.028688268473822742; w(11)=0.031290876747310448; w(12)=0.033796767115611761; w(13)=0.036198193872315186; w(14)=0.038487734259247662; w(15)=0.040658311384744518; w(16)=0.042703216084667087; w(17)=0.044616127652692283; w(18)=0.046391133373001897; w(19)=0.048022746793600258; w(20)=0.049505924683047579; w(21)=0.050836082617798481; w(22)=0.052009109151741400; w(23)=0.053021378524010764; w(24)=0.053869761865714486; w(25)=0.054551636870889421; w(26)=0.055064895901762426; w(27)=0.055407952503245123; w(28)=0.055579746306514396; w(29)=0.055579746306514396; w(30)=0.055407952503245123; w(31)=0.055064895901762426; w(32)=0.054551636870889421; w(33)=0.053869761865714486; w(34)=0.053021378524010764; w(35)=0.052009109151741400; w(36)=0.050836082617798481; w(37)=0.049505924683047579; w(38)=0.048022746793600258; w(39)=0.046391133373001897; w(40)=0.044616127652692283; w(41)=0.042703216084667087; w(42)=0.040658311384744518; w(43)=0.038487734259247662; w(44)=0.036198193872315186; w(45)=0.033796767115611761; w(46)=0.031290876747310448; w(47)=0.028688268473822742; w(48)=0.025996987058391952; w(49)=0.023225351562565317; w(50)=0.020381929882402573; w(51)=0.017475512911400947; w(52)=0.014515089278021472; w(53)=0.011509824340383382; w(54)=0.0084690631633078877; w(55)=0.0054025222460153378; w(56)=0.0023238553757732155; case(57) w(1)=0.0022437538722506629; w(2)=0.0052165334747187794; w(3)=0.0081781600678212326; w(4)=0.011115763732335990; w(5)=0.014020270790753556; w(6)=0.016882959023441549; w(7)=0.019695270699488520; w(8)=0.022448807890776438; w(9)=0.025135350990918123; w(10)=0.027746881402180192; w(11)=0.030275604842693999; w(12)=0.032713974366371569; w(13)=0.035054712782312618; w(14)=0.037290834324417317; w(15)=0.039415665475480114; w(16)=0.041422864870801110; w(17)=0.043306442216215197; w(18)=0.045060776161381158; w(19)=0.046680631073641504; w(20)=0.048161172661687751; w(21)=0.049497982402019679; w(22)=0.050687070724927409; w(23)=0.051724888920517825; w(24)=0.052608339729177432; w(25)=0.053334786584819158; w(26)=0.053902061483298575; w(27)=0.054308471452498643; w(28)=0.054552803604761886; w(29)=0.054634328756584024; w(30)=0.054552803604761886; w(31)=0.054308471452498643; w(32)=0.053902061483298575; w(33)=0.053334786584819158; w(34)=0.052608339729177432; w(35)=0.051724888920517825; w(36)=0.050687070724927409; w(37)=0.049497982402019679; w(38)=0.048161172661687751; w(39)=0.046680631073641504; w(40)=0.045060776161381158; w(41)=0.043306442216215197; w(42)=0.041422864870801110; w(43)=0.039415665475480114; w(44)=0.037290834324417317; w(45)=0.035054712782312618; w(46)=0.032713974366371569; w(47)=0.030275604842693999; w(48)=0.027746881402180192; w(49)=0.025135350990918123; w(50)=0.022448807890776438; w(51)=0.019695270699488520; w(52)=0.016882959023441549; w(53)=0.014020270790753556; w(54)=0.011115763732335990; w(55)=0.0081781600678212326; w(56)=0.0052165334747187794; w(57)=0.0022437538722506629; case(58) w(1)=0.0021677232496274499; w(2)=0.0050399816126502431; w(3)=0.0079019738499986748; w(4)=0.010741553532878774; w(5)=0.013550237112988812; w(6)=0.016319874234970965; w(7)=0.019042465461893409; w(8)=0.021710156140146236; w(9)=0.024315252724963953; w(10)=0.026850243181981868; w(11)=0.029307818044160491; w(12)=0.031680891253809327; w(13)=0.033962620493416011; w(14)=0.036146426867087271; w(15)=0.038226013845858433; w(16)=0.040195385409867797; w(17)=0.042048863329582126; w(18)=0.043781103533640251; w(19)=0.045387111514819803; w(20)=0.046862256729026347; w(21)=0.048202285945417748; w(22)=0.049403335508962393; w(23)=0.050461942479953125; w(24)=0.051375054618285725; w(25)=0.052140039183669819; w(26)=0.052754690526370833; w(27)=0.053217236446579014; w(28)=0.053526343304058252; w(29)=0.053681119863334849; w(30)=0.053681119863334849; w(31)=0.053526343304058252; w(32)=0.053217236446579014; w(33)=0.052754690526370833; w(34)=0.052140039183669819; w(35)=0.051375054618285725; w(36)=0.050461942479953125; w(37)=0.049403335508962393; w(38)=0.048202285945417748; w(39)=0.046862256729026347; w(40)=0.045387111514819803; w(41)=0.043781103533640251; w(42)=0.042048863329582126; w(43)=0.040195385409867797; w(44)=0.038226013845858433; w(45)=0.036146426867087271; w(46)=0.033962620493416011; w(47)=0.031680891253809327; w(48)=0.029307818044160491; w(49)=0.026850243181981868; w(50)=0.024315252724963953; w(51)=0.021710156140146236; w(52)=0.019042465461893409; w(53)=0.016319874234970965; w(54)=0.013550237112988812; w(55)=0.010741553532878774; w(56)=0.0079019738499986748; w(57)=0.0050399816126502431; w(58)=0.0021677232496274499; case(59) w(1)=0.0020954922845412234; w(2)=0.0048722391682652848; w(3)=0.0076395294534875751; w(4)=0.010385885500995862; w(5)=0.013103366306345191; w(6)=0.015784347313081466; w(7)=0.018421342753610029; w(8)=0.021006998288437187; w(9)=0.023534105393713363; w(10)=0.025995619731298500; w(11)=0.028384680200534798; w(12)=0.030694627836111683; w(13)=0.032919024271045278; w(14)=0.035051669636400109; w(15)=0.037086619818870923; w(16)=0.039018203016160010; w(17)=0.040841035538686708; w(18)=0.042550036811067639; w(19)=0.044140443530297381; w(20)=0.045607822940509770; w(21)=0.046948085186962019; w(22)=0.048157494714606440; w(23)=0.049232680679361986; w(24)=0.050170646342996903; w(25)=0.050968777425393917; w(26)=0.051624849390891482; w(27)=0.052137033648375391; w(28)=0.052503902647828739; w(29)=0.052724433859127932; w(30)=0.052798012621990421; w(31)=0.052724433859127932; w(32)=0.052503902647828739; w(33)=0.052137033648375391; w(34)=0.051624849390891482; w(35)=0.050968777425393917; w(36)=0.050170646342996903; w(37)=0.049232680679361986; w(38)=0.048157494714606440; w(39)=0.046948085186962019; w(40)=0.045607822940509770; w(41)=0.044140443530297381; w(42)=0.042550036811067639; w(43)=0.040841035538686708; w(44)=0.039018203016160010; w(45)=0.037086619818870923; w(46)=0.035051669636400109; w(47)=0.032919024271045278; w(48)=0.030694627836111683; w(49)=0.028384680200534798; w(50)=0.025995619731298500; w(51)=0.023534105393713363; w(52)=0.021006998288437187; w(53)=0.018421342753610029; w(54)=0.015784347313081466; w(55)=0.013103366306345191; w(56)=0.010385885500995862; w(57)=0.0076395294534875751; w(58)=0.0048722391682652848; w(59)=0.0020954922845412234; case(60) w(1)=0.0020268119688737585; w(2)=0.0047127299269535686; w(3)=0.0073899311633454555; w(4)=0.010047557182287984; w(5)=0.012678166476815960; w(6)=0.015274618596784799; w(7)=0.017829901014207720; w(8)=0.020337120729457287; w(9)=0.022789516943997820; w(10)=0.025180477621521248; w(11)=0.027503556749924792; w(12)=0.029752491500788945; w(13)=0.031921219019296329; w(14)=0.034003892724946423; w(15)=0.035994898051084503; w(16)=0.037888867569243444; w(17)=0.039680695452380799; w(18)=0.041365551235584756; w(19)=0.042938892835935642; w(20)=0.044396478795787113; w(21)=0.045734379716114487; w(22)=0.046948988848912205; w(23)=0.048037031819971181; w(24)=0.048995575455756835; w(25)=0.049822035690550181; w(26)=0.050514184532509375; w(27)=0.051070156069855627; w(28)=0.051488451500980934; w(29)=0.051767943174910188; w(30)=0.051907877631220640; w(31)=0.051907877631220640; w(32)=0.051767943174910188; w(33)=0.051488451500980934; w(34)=0.051070156069855627; w(35)=0.050514184532509375; w(36)=0.049822035690550181; w(37)=0.048995575455756835; w(38)=0.048037031819971181; w(39)=0.046948988848912205; w(40)=0.045734379716114487; w(41)=0.044396478795787113; w(42)=0.042938892835935642; w(43)=0.041365551235584756; w(44)=0.039680695452380799; w(45)=0.037888867569243444; w(46)=0.035994898051084503; w(47)=0.034003892724946423; w(48)=0.031921219019296329; w(49)=0.029752491500788945; w(50)=0.027503556749924792; w(51)=0.025180477621521248; w(52)=0.022789516943997820; w(53)=0.020337120729457287; w(54)=0.017829901014207720; w(55)=0.015274618596784799; w(56)=0.012678166476815960; w(57)=0.010047557182287984; w(58)=0.0073899311633454555; w(59)=0.0047127299269535686; w(60)=0.0020268119688737585; case(61) w(1)=0.0019614533616702827; w(2)=0.0045609240060124172; w(3)=0.0071523549917490896; w(4)=0.0097254618303561337; w(5)=0.012273263507812105; w(6)=0.014789065884937915; w(7)=0.017266292987613744; w(8)=0.019698477746101181; w(9)=0.022079273148319044; w(10)=0.024402467187544203; w(11)=0.026661998524150890; w(12)=0.028851972088183402; w(13)=0.030966674368397395; w(14)=0.033000588275907411; w(15)=0.034948407516533351; w(16)=0.036805050423154817; w(17)=0.038565673207008173; w(18)=0.040225682590998247; w(19)=0.041780747790888492; w(20)=0.043226811812496098; w(21)=0.044560102035083488; w(22)=0.045777140053145959; w(23)=0.046874750750809066; w(24)=0.047850070585095607; w(25)=0.048700555056411526; w(26)=0.049423985346735590; w(27)=0.050018474108178253; w(28)=0.050482470386797405; w(29)=0.050814763668818343; w(30)=0.051014487038697264; w(31)=0.051081119440786218; w(32)=0.051014487038697264; w(33)=0.050814763668818343; w(34)=0.050482470386797405; w(35)=0.050018474108178253; w(36)=0.049423985346735590; w(37)=0.048700555056411526; w(38)=0.047850070585095607; w(39)=0.046874750750809066; w(40)=0.045777140053145959; w(41)=0.044560102035083488; w(42)=0.043226811812496098; w(43)=0.041780747790888492; w(44)=0.040225682590998247; w(45)=0.038565673207008173; w(46)=0.036805050423154817; w(47)=0.034948407516533351; w(48)=0.033000588275907411; w(49)=0.030966674368397395; w(50)=0.028851972088183402; w(51)=0.026661998524150890; w(52)=0.024402467187544203; w(53)=0.022079273148319044; w(54)=0.019698477746101181; w(55)=0.017266292987613744; w(56)=0.014789065884937915; w(57)=0.012273263507812105; w(58)=0.0097254618303561337; w(59)=0.0071523549917490896; w(60)=0.0045609240060124172; w(61)=0.0019614533616702827; case(62) w(1)=0.0018992056795136905; w(2)=0.0044163334569309048; w(3)=0.0069260419018309609; w(4)=0.0094185794284203876; w(5)=0.011887390117010502; w(6)=0.014326191823806518; w(7)=0.016728811790177316; w(8)=0.019089176658573199; w(9)=0.021401322277669969; w(10)=0.023659407208682793; w(11)=0.025857726954024698; w(12)=0.027990728163314638; w(13)=0.030053022573989870; w(14)=0.032039400581624678; w(15)=0.033944844379410545; w(16)=0.035764540622768141; w(17)=0.037493892582280030; w(18)=0.039128531751963084; w(19)=0.040664328882417441; w(20)=0.042097404410385097; w(21)=0.043424138258047420; w(22)=0.044641178977124414; w(23)=0.045745452214570181; w(24)=0.046734168478415525; w(25)=0.047604830184101232; w(26)=0.048355237963477673; w(27)=0.048983496220517837; w(28)=0.049488017919699293; w(29)=0.049867528594952394; w(30)=0.050121069569043288; w(31)=0.050248000375256282; w(32)=0.050248000375256282; w(33)=0.050121069569043288; w(34)=0.049867528594952394; w(35)=0.049488017919699293; w(36)=0.048983496220517837; w(37)=0.048355237963477673; w(38)=0.047604830184101232; w(39)=0.046734168478415525; w(40)=0.045745452214570181; w(41)=0.044641178977124414; w(42)=0.043424138258047420; w(43)=0.042097404410385097; w(44)=0.040664328882417441; w(45)=0.039128531751963084; w(46)=0.037493892582280030; w(47)=0.035764540622768141; w(48)=0.033944844379410545; w(49)=0.032039400581624678; w(50)=0.030053022573989870; w(51)=0.027990728163314638; w(52)=0.025857726954024698; w(53)=0.023659407208682793; w(54)=0.021401322277669969; w(55)=0.019089176658573199; w(56)=0.016728811790177316; w(57)=0.014326191823806518; w(58)=0.011887390117010502; w(59)=0.0094185794284203876; w(60)=0.0069260419018309609; w(61)=0.0044163334569309048; w(62)=0.0018992056795136905; case(63) w(1)=0.0018398745955770841; w(2)=0.0042785083468637619; w(3)=0.0067102917659601363; w(4)=0.0091259686763266564; w(5)=0.011519376076880042; w(6)=0.013884612616115611; w(7)=0.016215878410338339; w(8)=0.018507464460161270; w(9)=0.020753761258039091; w(10)=0.022949271004889933; w(11)=0.025088620553344987; w(12)=0.027166574359097933; w(13)=0.029178047208280527; w(14)=0.031118116622219818; w(15)=0.032982034883779342; w(16)=0.034765240645355878; w(17)=0.036463370085457290; w(18)=0.038072267584349557; w(19)=0.039587995891544094; w(20)=0.041006845759666399; w(21)=0.042325345020815823; w(22)=0.043540267083027591; w(23)=0.044648638825941395; w(24)=0.045647747876292609; w(25)=0.046535149245383697; w(26)=0.047308671312268919; w(27)=0.047966421137995131; w(28)=0.048506789097883848; w(29)=0.048928452820511990; w(30)=0.049230380423747561; w(31)=0.049411833039918179; w(32)=0.049472366623931021; w(33)=0.049411833039918179; w(34)=0.049230380423747561; w(35)=0.048928452820511990; w(36)=0.048506789097883848; w(37)=0.047966421137995131; w(38)=0.047308671312268919; w(39)=0.046535149245383697; w(40)=0.045647747876292609; w(41)=0.044648638825941395; w(42)=0.043540267083027591; w(43)=0.042325345020815823; w(44)=0.041006845759666399; w(45)=0.039587995891544094; w(46)=0.038072267584349557; w(47)=0.036463370085457290; w(48)=0.034765240645355878; w(49)=0.032982034883779342; w(50)=0.031118116622219818; w(51)=0.029178047208280527; w(52)=0.027166574359097933; w(53)=0.025088620553344987; w(54)=0.022949271004889933; w(55)=0.020753761258039091; w(56)=0.018507464460161270; w(57)=0.016215878410338339; w(58)=0.013884612616115611; w(59)=0.011519376076880042; w(60)=0.0091259686763266564; w(61)=0.0067102917659601363; w(62)=0.0042785083468637619; w(63)=0.0018398745955770841; case(64) w(1)=0.0017832807216964329; w(2)=0.0041470332605624676; w(3)=0.0065044579689783629; w(4)=0.0088467598263639477; w(5)=0.011168139460131129; w(6)=0.013463047896718643; w(7)=0.015726030476024719; w(8)=0.017951715775697343; w(9)=0.020134823153530209; w(10)=0.022270173808383254; w(11)=0.024352702568710873; w(12)=0.026377469715054659; w(13)=0.028339672614259483; w(14)=0.030234657072402479; w(15)=0.032057928354851554; w(16)=0.033805161837141609; w(17)=0.035472213256882384; w(18)=0.037055128540240046; w(19)=0.038550153178615629; w(20)=0.039953741132720341; w(21)=0.041262563242623529; w(22)=0.042473515123653589; w(23)=0.043583724529323453; w(24)=0.044590558163756563; w(25)=0.045491627927418144; w(26)=0.046284796581314417; w(27)=0.046968182816210017; w(28)=0.047540165714830309; w(29)=0.047999388596458308; w(30)=0.048344762234802957; w(31)=0.048575467441503427; w(32)=0.048690957009139720; w(33)=0.048690957009139720; w(34)=0.048575467441503427; w(35)=0.048344762234802957; w(36)=0.047999388596458308; w(37)=0.047540165714830309; w(38)=0.046968182816210017; w(39)=0.046284796581314417; w(40)=0.045491627927418144; w(41)=0.044590558163756563; w(42)=0.043583724529323453; w(43)=0.042473515123653589; w(44)=0.041262563242623529; w(45)=0.039953741132720341; w(46)=0.038550153178615629; w(47)=0.037055128540240046; w(48)=0.035472213256882384; w(49)=0.033805161837141609; w(50)=0.032057928354851554; w(51)=0.030234657072402479; w(52)=0.028339672614259483; w(53)=0.026377469715054659; w(54)=0.024352702568710873; w(55)=0.022270173808383254; w(56)=0.020134823153530209; w(57)=0.017951715775697343; w(58)=0.015726030476024719; w(59)=0.013463047896718643; w(60)=0.011168139460131129; w(61)=0.0088467598263639477; w(62)=0.0065044579689783629; w(63)=0.0041470332605624676; w(64)=0.0017832807216964329; otherwise error('GAUSS_WEIGHTS - Fatal error! Illegal value of n.');end function x = gausspoints(n)%% function x = gausspoints(n)%% For 1 <= n <= 64, returns the abscissas x of an n point% Gauss-Legendre quadrature rule over the interval [-1,1].%x = ones(1,n);switch n case(1) x(1)=0.0; case(2) x(1)=-0.57735026918962576; x(2)=0.57735026918962576; case(3) x(1)=-0.77459666924148338; x(2)=0.0; x(3)=0.77459666924148338; case(4) x(1)=-0.86113631159405258; x(2)=-0.33998104358485626; x(3)=0.33998104358485626; x(4)=0.86113631159405258; case(5) x(1)=-0.90617984593866399; x(2)=-0.53846931010568309; x(3)=0.0; x(4)=0.53846931010568309; x(5)=0.90617984593866399; case(6) x(1)=-0.93246951420315203; x(2)=-0.66120938646626451; x(3)=-0.23861918608319691; x(4)=0.23861918608319691; x(5)=0.66120938646626451; x(6)=0.93246951420315203; case(7) x(1)=-0.94910791234275852; x(2)=-0.74153118559939444; x(3)=-0.40584515137739717; x(4)=0.0; x(5)=0.40584515137739717; x(6)=0.74153118559939444; x(7)=0.94910791234275852; case(8) x(1)=-0.96028985649753623; x(2)=-0.79666647741362674; x(3)=-0.52553240991632899; x(4)=-0.18343464249564980; x(5)=0.18343464249564980; x(6)=0.52553240991632899; x(7)=0.79666647741362674; x(8)=0.96028985649753623; case(9) x(1)=-0.96816023950762609; x(2)=-0.83603110732663579; x(3)=-0.61337143270059040; x(4)=-0.32425342340380893; x(5)=0.0; x(6)=0.32425342340380893; x(7)=0.61337143270059040; x(8)=0.83603110732663579; x(9)=0.96816023950762609; case(10) x(1)=-0.97390652851717172; x(2)=-0.86506336668898451; x(3)=-0.67940956829902441; x(4)=-0.43339539412924719; x(5)=-0.14887433898163121; x(6)=0.14887433898163121; x(7)=0.43339539412924719; x(8)=0.67940956829902441; x(9)=0.86506336668898451; x(10)=0.97390652851717172; case(11) x(1)=-0.97822865814605699; x(2)=-0.88706259976809530; x(3)=-0.73015200557404932; x(4)=-0.51909612920681182; x(5)=-0.26954315595234497; x(6)=0.0; x(7)=0.26954315595234497; x(8)=0.51909612920681182; x(9)=0.73015200557404932; x(10)=0.88706259976809530; x(11)=0.97822865814605699; case(12) x(1)=-0.98156063424671925; x(2)=-0.90411725637047486; x(3)=-0.76990267419430469; x(4)=-0.58731795428661745; x(5)=-0.36783149899818019; x(6)=-0.12523340851146892; x(7)=0.12523340851146892; x(8)=0.36783149899818019; x(9)=0.58731795428661745; x(10)=0.76990267419430469; x(11)=0.90411725637047486; x(12)=0.98156063424671925; case(13) x(1)=-0.98418305471858815; x(2)=-0.91759839922297797; x(3)=-0.80157809073330991; x(4)=-0.64234933944034022; x(5)=-0.44849275103644685; x(6)=-0.23045831595513479; x(7)=0.0; x(8)=0.23045831595513479; x(9)=0.44849275103644685; x(10)=0.64234933944034022; x(11)=0.80157809073330991; x(12)=0.91759839922297797; x(13)=0.98418305471858815; case(14) x(1)=-0.98628380869681234; x(2)=-0.92843488366357352; x(3)=-0.82720131506976499; x(4)=-0.68729290481168547; x(5)=-0.51524863635815409; x(6)=-0.31911236892788976; x(7)=-0.10805494870734366; x(8)=0.10805494870734366; x(9)=0.31911236892788976; x(10)=0.51524863635815409; x(11)=0.68729290481168547; x(12)=0.82720131506976499; x(13)=0.92843488366357352; x(14)=0.98628380869681234; case(15) x(1)=-0.98799251802048543; x(2)=-0.93727339240070590; x(3)=-0.84820658341042722; x(4)=-0.72441773136017005; x(5)=-0.57097217260853885; x(6)=-0.39415134707756337; x(7)=-0.20119409399743452; x(8)=0.0; x(9)=0.20119409399743452; x(10)=0.39415134707756337; x(11)=0.57097217260853885; x(12)=0.72441773136017005; x(13)=0.84820658341042722; x(14)=0.93727339240070590; x(15)=0.98799251802048543; case(16) x(1)=-0.98940093499164993; x(2)=-0.94457502307323258; x(3)=-0.86563120238783174; x(4)=-0.75540440835500303; x(5)=-0.61787624440264375; x(6)=-0.45801677765722739; x(7)=-0.28160355077925891; x(8)=-0.095012509837637440; x(9)=0.095012509837637440; x(10)=0.28160355077925891; x(11)=0.45801677765722739; x(12)=0.61787624440264375; x(13)=0.75540440835500303; x(14)=0.86563120238783174; x(15)=0.94457502307323258; x(16)=0.98940093499164993; case(17) x(1)=-0.99057547531441734; x(2)=-0.95067552176876776; x(3)=-0.88023915372698590; x(4)=-0.78151400389680141; x(5)=-0.65767115921669077; x(6)=-0.51269053708647697; x(7)=-0.35123176345387632; x(8)=-0.17848418149584786; x(9)=0.0; x(10)=0.17848418149584786; x(11)=0.35123176345387632; x(12)=0.51269053708647697; x(13)=0.65767115921669077; x(14)=0.78151400389680141; x(15)=0.88023915372698590; x(16)=0.95067552176876776; x(17)=0.99057547531441734; case(18) x(1)=-0.99156516842093095; x(2)=-0.95582394957139776; x(3)=-0.89260246649755574; x(4)=-0.80370495897252312; x(5)=-0.69168704306035321; x(6)=-0.55977083107394753; x(7)=-0.41175116146284265; x(8)=-0.25188622569150551; x(9)=-0.084775013041735301; x(10)=0.084775013041735301; x(11)=0.25188622569150551; x(12)=0.41175116146284265; x(13)=0.55977083107394753; x(14)=0.69168704306035321; x(15)=0.80370495897252312; x(16)=0.89260246649755574; x(17)=0.95582394957139776; x(18)=0.99156516842093095; case(19) x(1)=-0.99240684384358440; x(2)=-0.96020815213483003; x(3)=-0.90315590361481790; x(4)=-0.82271465653714282; x(5)=-0.72096617733522938; x(6)=-0.60054530466168102; x(7)=-0.46457074137596095; x(8)=-0.31656409996362983; x(9)=-0.16035864564022538; x(10)=0.0; x(11)=0.16035864564022538; x(12)=0.31656409996362983; x(13)=0.46457074137596095; x(14)=0.60054530466168102; x(15)=0.72096617733522938; x(16)=0.82271465653714282; x(17)=0.90315590361481790; x(18)=0.96020815213483003; x(19)=0.99240684384358440; case(20) x(1)=-0.99312859918509492; x(2)=-0.96397192727791379; x(3)=-0.91223442825132591; x(4)=-0.83911697182221882; x(5)=-0.74633190646015079; x(6)=-0.63605368072651503; x(7)=-0.51086700195082710; x(8)=-0.37370608871541956; x(9)=-0.22778585114164508; x(10)=-0.076526521133497334; x(11)=0.076526521133497334; x(12)=0.22778585114164508; x(13)=0.37370608871541956; x(14)=0.51086700195082710; x(15)=0.63605368072651503; x(16)=0.74633190646015079; x(17)=0.83911697182221882; x(18)=0.91223442825132591; x(19)=0.96397192727791379; x(20)=0.99312859918509492; case(21) x(1)=-0.99375217062038950; x(2)=-0.96722683856630629; x(3)=-0.92009933415040083; x(4)=-0.85336336458331728; x(5)=-0.76843996347567791; x(6)=-0.66713880419741232; x(7)=-0.55161883588721981; x(8)=-0.42434212020743878; x(9)=-0.28802131680240110; x(10)=-0.14556185416089509; x(11)=0.0; x(12)=0.14556185416089509; x(13)=0.28802131680240110; x(14)=0.42434212020743878; x(15)=0.55161883588721981; x(16)=0.66713880419741232; x(17)=0.76843996347567791; x(18)=0.85336336458331728; x(19)=0.92009933415040083; x(20)=0.96722683856630629; x(21)=0.99375217062038950; case(22) x(1)=-0.99429458548239929; x(2)=-0.97006049783542873; x(3)=-0.92695677218717400; x(4)=-0.86581257772030014; x(5)=-0.78781680597920816; x(6)=-0.69448726318668278; x(7)=-0.58764040350691159; x(8)=-0.46935583798675703; x(9)=-0.34193582089208423; x(10)=-0.20786042668822129; x(11)=-0.069739273319722221; x(12)=0.069739273319722221; x(13)=0.20786042668822129; x(14)=0.34193582089208423; x(15)=0.46935583798675703; x(16)=0.58764040350691159; x(17)=0.69448726318668278; x(18)=0.78781680597920816; x(19)=0.86581257772030014; x(20)=0.92695677218717400; x(21)=0.97006049783542873; x(22)=0.99429458548239929; case(23) x(1)=-0.99476933499755212; x(2)=-0.97254247121811523; x(3)=-0.93297108682601610; x(4)=-0.87675235827044167; x(5)=-0.80488840161883989; x(6)=-0.71866136313195019; x(7)=-0.61960987576364616; x(8)=-0.50950147784600755; x(9)=-0.39030103803029083; x(10)=-0.26413568097034493; x(11)=-0.13325682429846611; x(12)=0.0; x(13)=0.13325682429846611; x(14)=0.26413568097034493; x(15)=0.39030103803029083; x(16)=0.50950147784600755; x(17)=0.61960987576364616; x(18)=0.71866136313195019; x(19)=0.80488840161883989; x(20)=0.87675235827044167; x(21)=0.93297108682601610; x(22)=0.97254247121811523; x(23)=0.99476933499755212; case(24) x(1)=-0.99518721999702136; x(2)=-0.97472855597130950; x(3)=-0.93827455200273276; x(4)=-0.88641552700440103; x(5)=-0.82000198597390292; x(6)=-0.74012419157855436; x(7)=-0.64809365193697557; x(8)=-0.54542147138883954; x(9)=-0.43379350762604514; x(10)=-0.31504267969616337; x(11)=-0.19111886747361631; x(12)=-0.064056892862605626; x(13)=0.064056892862605626; x(14)=0.19111886747361631; x(15)=0.31504267969616337; x(16)=0.43379350762604514; x(17)=0.54542147138883954; x(18)=0.64809365193697557; x(19)=0.74012419157855436; x(20)=0.82000198597390292; x(21)=0.88641552700440103; x(22)=0.93827455200273276; x(23)=0.97472855597130950; x(24)=0.99518721999702136; case(25) x(1)=-0.99555696979049810; x(2)=-0.97666392145951751; x(3)=-0.94297457122897434; x(4)=-0.89499199787827537; x(5)=-0.83344262876083400; x(6)=-0.75925926303735763; x(7)=-0.67356636847346836; x(8)=-0.57766293024122297; x(9)=-0.47300273144571496; x(10)=-0.36117230580938784; x(11)=-0.24386688372098843; x(12)=-0.12286469261071040; x(13)=0.0; x(14)=0.12286469261071040; x(15)=0.24386688372098843; x(16)=0.36117230580938784; x(17)=0.47300273144571496; x(18)=0.57766293024122297; x(19)=0.67356636847346836; x(20)=0.75925926303735763; x(21)=0.83344262876083400; x(22)=0.89499199787827537; x(23)=0.94297457122897434; x(24)=0.97666392145951751; x(25)=0.99555696979049810; case(26) x(1)=-0.99588570114561693; x(2)=-0.97838544595647099; x(3)=-0.94715906666171425; x(4)=-0.90263786198430707; x(5)=-0.84544594278849802; x(6)=-0.77638594882067886; x(7)=-0.69642726041995726; x(8)=-0.60669229301761806; x(9)=-0.50844071482450572; x(10)=-0.40305175512348631; x(11)=-0.29200483948595690; x(12)=-0.17685882035689018; x(13)=-0.059230093429313207; x(14)=0.059230093429313207; x(15)=0.17685882035689018; x(16)=0.29200483948595690; x(17)=0.40305175512348631; x(18)=0.50844071482450572; x(19)=0.60669229301761806; x(20)=0.69642726041995726; x(21)=0.77638594882067886; x(22)=0.84544594278849802; x(23)=0.90263786198430707; x(24)=0.94715906666171425; x(25)=0.97838544595647099; x(26)=0.99588570114561693; case(27) x(1)=-0.99617926288898857; x(2)=-0.97992347596150122; x(3)=-0.95090055781470501; x(4)=-0.90948232067749110; x(5)=-0.85620790801829449; x(6)=-0.79177163907050823; x(7)=-0.71701347373942370; x(8)=-0.63290797194649514; x(9)=-0.54055156457945689; x(10)=-0.44114825175002688; x(11)=-0.33599390363850890; x(12)=-0.22645936543953686; x(13)=-0.11397258560952997; x(14)=0.0; x(15)=0.11397258560952997; x(16)=0.22645936543953686; x(17)=0.33599390363850890; x(18)=0.44114825175002688; x(19)=0.54055156457945689; x(20)=0.63290797194649514; x(21)=0.71701347373942370; x(22)=0.79177163907050823; x(23)=0.85620790801829449; x(24)=0.90948232067749110; x(25)=0.95090055781470501; x(26)=0.97992347596150122; x(27)=0.99617926288898857; case(28) x(1)=-0.99644249757395445; x(2)=-0.98130316537087275; x(3)=-0.95425928062893820; x(4)=-0.91563302639213207; x(5)=-0.86589252257439505; x(6)=-0.80564137091717917; x(7)=-0.73561087801363177; x(8)=-0.65665109403886496; x(9)=-0.56972047181140172; x(10)=-0.47587422495511826; x(11)=-0.37625151608907871; x(12)=-0.27206162763517808; x(13)=-0.16456928213338077; x(14)=-0.055079289884034270; x(15)=0.055079289884034270; x(16)=0.16456928213338077; x(17)=0.27206162763517808; x(18)=0.37625151608907871; x(19)=0.47587422495511826; x(20)=0.56972047181140172; x(21)=0.65665109403886496; x(22)=0.73561087801363177; x(23)=0.80564137091717917; x(24)=0.86589252257439505; x(25)=0.91563302639213207; x(26)=0.95425928062893820; x(27)=0.98130316537087275; x(28)=0.99644249757395445; case(29) x(1)=-0.99667944226059659; x(2)=-0.98254550526141317; x(3)=-0.95728559577808773; x(4)=-0.92118023295305879; x(5)=-0.87463780492010279; x(6)=-0.81818548761525244; x(7)=-0.75246285173447713; x(8)=-0.67821453760268652; x(9)=-0.59628179713822782; x(10)=-0.50759295512422764; x(11)=-0.41315288817400866; x(12)=-0.31403163786763993; x(13)=-0.21135228616600107; x(14)=-0.10627823013267923; x(15)=0.0; x(16)=0.10627823013267923; x(17)=0.21135228616600107; x(18)=0.31403163786763993; x(19)=0.41315288817400866; x(20)=0.50759295512422764; x(21)=0.59628179713822782; x(22)=0.67821453760268652; x(23)=0.75246285173447713; x(24)=0.81818548761525244; x(25)=0.87463780492010279; x(26)=0.92118023295305879; x(27)=0.95728559577808773; x(28)=0.98254550526141317; x(29)=0.99667944226059659; case(30) x(1)=-0.99689348407464954; x(2)=-0.98366812327974721; x(3)=-0.96002186496830751; x(4)=-0.92620004742927433; x(5)=-0.88256053579205268; x(6)=-0.82956576238276840; x(7)=-0.76777743210482619; x(8)=-0.69785049479331580; x(9)=-0.62052618298924286; x(10)=-0.53662414814201990; x(11)=-0.44703376953808918; x(12)=-0.35270472553087811; x(13)=-0.25463692616788985; x(14)=-0.15386991360858355; x(15)=-0.051471842555317696; x(16)=0.051471842555317696; x(17)=0.15386991360858355; x(18)=0.25463692616788985; x(19)=0.35270472553087811; x(20)=0.44703376953808918; x(21)=0.53662414814201990; x(22)=0.62052618298924286; x(23)=0.69785049479331580; x(24)=0.76777743210482619; x(25)=0.82956576238276840; x(26)=0.88256053579205268; x(27)=0.92620004742927433; x(28)=0.96002186496830751; x(29)=0.98366812327974721; x(30)=0.99689348407464954; case(31) x(1)=-0.99708748181947707; x(2)=-0.98468590966515248; x(3)=-0.96250392509294966; x(4)=-0.93075699789664816; x(5)=-0.88976002994827104; x(6)=-0.83992032014626734; x(7)=-0.78173314841662494; x(8)=-0.71577678458685328; x(9)=-0.64270672292426035; x(10)=-0.56324916140714926; x(11)=-0.47819378204490248; x(12)=-0.38838590160823294; x(13)=-0.29471806998170162; x(14)=-0.19812119933557063; x(15)=-0.099555312152341520; x(16)=0.0; x(17)=0.099555312152341520; x(18)=0.19812119933557063; x(19)=0.29471806998170162; x(20)=0.38838590160823294; x(21)=0.47819378204490248; x(22)=0.56324916140714926; x(23)=0.64270672292426035; x(24)=0.71577678458685328; x(25)=0.78173314841662494; x(26)=0.83992032014626734; x(27)=0.88976002994827104; x(28)=0.93075699789664816; x(29)=0.96250392509294966; x(30)=0.98468590966515248; x(31)=0.99708748181947707; case(32) x(1)=-0.99726386184948156; x(2)=-0.98561151154526834; x(3)=-0.96476225558750643; x(4)=-0.93490607593773969; x(5)=-0.89632115576605212; x(6)=-0.84936761373256997; x(7)=-0.79448379596794241; x(8)=-0.73218211874028968; x(9)=-0.66304426693021520; x(10)=-0.58771575724076233; x(11)=-0.50689990893222939; x(12)=-0.42135127613063535; x(13)=-0.33186860228212765; x(14)=-0.23928736225213707; x(15)=-0.14447196158279649; x(16)=-0.048307665687738316; x(17)=0.048307665687738316; x(18)=0.14447196158279649; x(19)=0.23928736225213707; x(20)=0.33186860228212765; x(21)=0.42135127613063535; x(22)=0.50689990893222939; x(23)=0.58771575724076233; x(24)=0.66304426693021520; x(25)=0.73218211874028968; x(26)=0.79448379596794241; x(27)=0.84936761373256997; x(28)=0.89632115576605212; x(29)=0.93490607593773969; x(30)=0.96476225558750643; x(31)=0.98561151154526834; x(32)=0.99726386184948156; case(33) x(1)=-0.99742469424645522; x(2)=-0.98645572623064249; x(3)=-0.96682290968999277; x(4)=-0.93869437261116835; x(5)=-0.90231676774343358; x(6)=-0.85800965267650406; x(7)=-0.80616235627416659; x(8)=-0.74723049644956216; x(9)=-0.68173195996974279; x(10)=-0.61024234583637903; x(11)=-0.53338990478634764; x(12)=-0.45185001727245070; x(13)=-0.36633925774807334; x(14)=-0.27760909715249703; x(15)=-0.18643929882799157; x(16)=-0.093631065854733386; x(17)=0.0; x(18)=0.093631065854733386; x(19)=0.18643929882799157; x(20)=0.27760909715249703; x(21)=0.36633925774807334; x(22)=0.45185001727245070; x(23)=0.53338990478634764; x(24)=0.61024234583637903; x(25)=0.68173195996974279; x(26)=0.74723049644956216; x(27)=0.80616235627416659; x(28)=0.85800965267650406; x(29)=0.90231676774343358; x(30)=0.93869437261116835; x(31)=0.96682290968999277; x(32)=0.98645572623064249; x(33)=0.99742469424645522; case(34) x(1)=-0.99757175379084192; x(2)=-0.98722781640630949; x(3)=-0.96870826253334428; x(4)=-0.94216239740510709; x(5)=-0.90780967771832447; x(6)=-0.86593463833456447; x(7)=-0.81688422790093366; x(8)=-0.76106487662987301; x(9)=-0.69893911321626291; x(10)=-0.63102172708052855; x(11)=-0.55787550066974664; x(12)=-0.48010654519032703; x(13)=-0.39835927775864594; x(14)=-0.31331108133946325; x(15)=-0.22566669161644948; x(16)=-0.13615235725918298; x(17)=-0.045509821953102543; x(18)=0.045509821953102543; x(19)=0.13615235725918298; x(20)=0.22566669161644948; x(21)=0.31331108133946325; x(22)=0.39835927775864594; x(23)=0.48010654519032703; x(24)=0.55787550066974664; x(25)=0.63102172708052855; x(26)=0.69893911321626291; x(27)=0.76106487662987301; x(28)=0.81688422790093366; x(29)=0.86593463833456447; x(30)=0.90780967771832447; x(31)=0.94216239740510709; x(32)=0.96870826253334428; x(33)=0.98722781640630949; x(34)=0.99757175379084192; case(35) x(1)=-0.99770656909960030; x(2)=-0.98793576444385150; x(3)=-0.97043761603922983; x(4)=-0.94534514820782733; x(5)=-0.91285426135931761; x(6)=-0.87321912502522233; x(7)=-0.82674989909222541; x(8)=-0.77381025228691256; x(9)=-0.71481450155662878; x(10)=-0.65022436466589039; x(11)=-0.58054534474976451; x(12)=-0.50632277324148862; x(13)=-0.42813754151781425; x(14)=-0.34660155443081395; x(15)=-0.26235294120929606; x(16)=-0.17605106116598957; x(17)=-0.088371343275659264; x(18)=0.0; x(19)=0.088371343275659264; x(20)=0.17605106116598957; x(21)=0.26235294120929606; x(22)=0.34660155443081395; x(23)=0.42813754151781425; x(24)=0.50632277324148862; x(25)=0.58054534474976451; x(26)=0.65022436466589039; x(27)=0.71481450155662878; x(28)=0.77381025228691256; x(29)=0.82674989909222541; x(30)=0.87321912502522233; x(31)=0.91285426135931761; x(32)=0.94534514820782733; x(33)=0.97043761603922983; x(34)=0.98793576444385150; x(35)=0.99770656909960030; case(36) x(1)=-0.99783046248408584; x(2)=-0.98858647890221224; x(3)=-0.97202769104969795; x(4)=-0.94827298439950755; x(5)=-0.91749777451565907; x(6)=-0.87992980089039713; x(7)=-0.83584716699247531; x(8)=-0.78557623013220651; x(9)=-0.72948917159355658; x(10)=-0.66800123658552106; x(11)=-0.60156765813598054; x(12)=-0.53068028592624516; x(13)=-0.45586394443342027; x(14)=-0.37767254711968922; x(15)=-0.29668499534402827; x(16)=-0.21350089231686558; x(17)=-0.12873610380938479; x(18)=-0.043018198473708607; x(19)=0.043018198473708607; x(20)=0.12873610380938479; x(21)=0.21350089231686558; x(22)=0.29668499534402827; x(23)=0.37767254711968922; x(24)=0.45586394443342027; x(25)=0.53068028592624516; x(26)=0.60156765813598054; x(27)=0.66800123658552106; x(28)=0.72948917159355658; x(29)=0.78557623013220651; x(30)=0.83584716699247531; x(31)=0.87992980089039713; x(32)=0.91749777451565907; x(33)=0.94827298439950755; x(34)=0.97202769104969795; x(35)=0.98858647890221224; x(36)=0.99783046248408584; case(37) x(1)=-0.99794458247791365; x(2)=-0.98918596321431919; x(3)=-0.97349303005648574; x(4)=-0.95097234326209482; x(5)=-0.92178143741246374; x(6)=-0.88612496215548608; x(7)=-0.84425298734055597; x(8)=-0.79645920050990229; x(9)=-0.74307883398196526; x(10)=-0.68448630913095936; x(11)=-0.62109260840892448; x(12)=-0.55334239186158178; x(13)=-0.48171087780320555; x(14)=-0.40670050931832611; x(15)=-0.32883742988370700; x(16)=-0.24866779279136576; x(17)=-0.16675393023985198; x(18)=-0.083670408954769902; x(19)=0.0; x(20)=0.083670408954769902; x(21)=0.16675393023985198; x(22)=0.24866779279136576; x(23)=0.32883742988370700; x(24)=0.40670050931832611; x(25)=0.48171087780320555; x(26)=0.55334239186158178; x(27)=0.62109260840892448; x(28)=0.68448630913095936; x(29)=0.74307883398196526; x(30)=0.79645920050990229; x(31)=0.84425298734055597; x(32)=0.88612496215548608; x(33)=0.92178143741246374; x(34)=0.95097234326209482; x(35)=0.97349303005648574; x(36)=0.98918596321431919; x(37)=0.99794458247791365; case(38) x(1)=-0.99804993053568762; x(2)=-0.98973945426638557; x(3)=-0.97484632859015351; x(4)=-0.95346633093352960; x(5)=-0.92574133204858440; x(6)=-0.89185573900463222; x(7)=-0.85203502193236219; x(8)=-0.80654416760531682; x(9)=-0.75568590375397068; x(10)=-0.69979868037918436; x(11)=-0.63925441582968171; x(12)=-0.57445602104780708; x(13)=-0.50583471792793110; x(14)=-0.43384716943237648; x(15)=-0.35897244047943501; x(16)=-0.28170880979016526; x(17)=-0.20257045389211670; x(18)=-0.12208402533786742; x(19)=-0.040785147904578240; x(20)=0.040785147904578240; x(21)=0.12208402533786742; x(22)=0.20257045389211670; x(23)=0.28170880979016526; x(24)=0.35897244047943501; x(25)=0.43384716943237648; x(26)=0.50583471792793110; x(27)=0.57445602104780708; x(28)=0.63925441582968171; x(29)=0.69979868037918436; x(30)=0.75568590375397068; x(31)=0.80654416760531682; x(32)=0.85203502193236219; x(33)=0.89185573900463222; x(34)=0.92574133204858440; x(35)=0.95346633093352960; x(36)=0.97484632859015351; x(37)=0.98973945426638557; x(38)=0.99804993053568762; case(39) x(1)=-0.99814738306643291; x(2)=-0.99025153685468598; x(3)=-0.97609870933347105; x(4)=-0.95577521232465228; x(5)=-0.92940914848673823; x(6)=-0.89716711929299289; x(7)=-0.85925293799990615; x(8)=-0.81590629743014310; x(9)=-0.76740124293106350; x(10)=-0.71404443589453468; x(11)=-0.65617321343201091; x(12)=-0.59415345495727799; x(13)=-0.52837726866043747; x(14)=-0.45926051230913605; x(15)=-0.38724016397156146; x(16)=-0.31277155924818592; x(17)=-0.23632551246183577; x(18)=-0.15838533999783780; x(19)=-0.079443804608755478; x(20)=0.0; x(21)=0.079443804608755478; x(22)=0.15838533999783780; x(23)=0.23632551246183577; x(24)=0.31277155924818592; x(25)=0.38724016397156146; x(26)=0.45926051230913605; x(27)=0.52837726866043747; x(28)=0.59415345495727799; x(29)=0.65617321343201091; x(30)=0.71404443589453468; x(31)=0.76740124293106350; x(32)=0.81590629743014310; x(33)=0.85925293799990615; x(34)=0.89716711929299289; x(35)=0.92940914848673823; x(36)=0.95577521232465228; x(37)=0.97609870933347105; x(38)=0.99025153685468598; x(39)=0.99814738306643291; case(40) x(1)=-0.99823770971055920; x(2)=-0.99072623869945701; x(3)=-0.97725994998377426; x(4)=-0.95791681921379166; x(5)=-0.93281280827867653; x(6)=-0.90209880696887430; x(7)=-0.86595950321225950; x(8)=-0.82461223083331166; x(9)=-0.77830565142651939; x(10)=-0.72731825518992710; x(11)=-0.67195668461417955; x(12)=-0.61255388966798024; x(13)=-0.54946712509512820; x(14)=-0.48307580168617871; x(15)=-0.41377920437160500; x(16)=-0.34199409082575847; x(17)=-0.26815218500725368; x(18)=-0.19269758070137110; x(19)=-0.11608407067525521; x(20)=-0.038772417506050822; x(21)=0.038772417506050822; x(22)=0.11608407067525521; x(23)=0.19269758070137110; x(24)=0.26815218500725368; x(25)=0.34199409082575847; x(26)=0.41377920437160500; x(27)=0.48307580168617871; x(28)=0.54946712509512820; x(29)=0.61255388966798024; x(30)=0.67195668461417955; x(31)=0.72731825518992710; x(32)=0.77830565142651939; x(33)=0.82461223083331166; x(34)=0.86595950321225950; x(35)=0.90209880696887430; x(36)=0.93281280827867653; x(37)=0.95791681921379166; x(38)=0.97725994998377426; x(39)=0.99072623869945701; x(40)=0.99823770971055920; case(41) x(1)=-0.99832158857477144; x(2)=-0.99116710969901631; x(3)=-0.97833867356108338; x(4)=-0.95990689173034623; x(5)=-0.93597698749785383; x(6)=-0.90668594475810117; x(7)=-0.87220151169244141; x(8)=-0.83272120040136133; x(9)=-0.78847114504740937; x(10)=-0.73970480306992618; x(11)=-0.68670150203495129; x(12)=-0.62976483907219632; x(13)=-0.56922094161021587; x(14)=-0.50541659919940603; x(15)=-0.43871727705140709; x(16)=-0.36950502264048144; x(17)=-0.29817627734182487; x(18)=-0.22513960563342278; x(19)=-0.15081335486399216; x(20)=-0.075623258989162997; x(21)=0.0; x(22)=0.075623258989162997; x(23)=0.15081335486399216; x(24)=0.22513960563342278; x(25)=0.29817627734182487; x(26)=0.36950502264048144; x(27)=0.43871727705140709; x(28)=0.50541659919940603; x(29)=0.56922094161021587; x(30)=0.62976483907219632; x(31)=0.68670150203495129; x(32)=0.73970480306992618; x(33)=0.78847114504740937; x(34)=0.83272120040136133; x(35)=0.87220151169244141; x(36)=0.90668594475810117; x(37)=0.93597698749785383; x(38)=0.95990689173034623; x(39)=0.97833867356108338; x(40)=0.99116710969901631; x(41)=0.99832158857477144; case(42) x(1)=-0.99839961899006242; x(2)=-0.99157728834086092; x(3)=-0.97934250806374819; x(4)=-0.96175936533820449; x(5)=-0.93892355735498818; x(6)=-0.91095972490412745; x(7)=-0.87802056981217274; x(8)=-0.84028598326181690; x(9)=-0.79796205325548741; x(10)=-0.75127993568948049; x(11)=-0.70049459055617121; x(12)=-0.64588338886924783; x(13)=-0.58774459748510932; x(14)=-0.52639574993119229; x(15)=-0.46217191207042193; x(16)=-0.39542385204297506; x(17)=-0.32651612446541151; x(18)=-0.25582507934287908; x(19)=-0.18373680656485455; x(20)=-0.11064502720851987; x(21)=-0.036948943165351776; x(22)=0.036948943165351776; x(23)=0.11064502720851987; x(24)=0.18373680656485455; x(25)=0.25582507934287908; x(26)=0.32651612446541151; x(27)=0.39542385204297506; x(28)=0.46217191207042193; x(29)=0.52639574993119229; x(30)=0.58774459748510932; x(31)=0.64588338886924783; x(32)=0.70049459055617121; x(33)=0.75127993568948049; x(34)=0.79796205325548741; x(35)=0.84028598326181690; x(36)=0.87802056981217274; x(37)=0.91095972490412745; x(38)=0.93892355735498818; x(39)=0.96175936533820449; x(40)=0.97934250806374819; x(41)=0.99157728834086092; x(42)=0.99839961899006242; case(43) x(1)=-0.99847233224250771; x(2)=-0.99195955759324415; x(3)=-0.98027822098025533; x(4)=-0.96348661301407999; x(5)=-0.94167195684763786; x(6)=-0.91494790720613873; x(7)=-0.88345376521861686; x(8)=-0.84735371620931505; x(9)=-0.80683596413693864; x(10)=-0.76211174719495512; x(11)=-0.71341423526895705; x(12)=-0.66099731375149813; x(13)=-0.60513425963960094; x(14)=-0.54611631666008472; x(15)=-0.48425117678573472; x(16)=-0.41986137602926925; x(17)=-0.35328261286430381; x(18)=-0.28486199803291363; x(19)=-0.21495624486051821; x(20)=-0.14392980951071331; x(21)=-0.072152990874586235; x(22)=0.0; x(23)=0.072152990874586235; x(24)=0.14392980951071331; x(25)=0.21495624486051821; x(26)=0.28486199803291363; x(27)=0.35328261286430381; x(28)=0.41986137602926925; x(29)=0.48425117678573472; x(30)=0.54611631666008472; x(31)=0.60513425963960094; x(32)=0.66099731375149813; x(33)=0.71341423526895705; x(34)=0.76211174719495512; x(35)=0.80683596413693864; x(36)=0.84735371620931505; x(37)=0.88345376521861686; x(38)=0.91494790720613873; x(39)=0.94167195684763786; x(40)=0.96348661301407999; x(41)=0.98027822098025533; x(42)=0.99195955759324415; x(43)=0.99847233224250771; case(44) x(1)=-0.99854020063677422; x(2)=-0.99231639213851581; x(3)=-0.98115183307791397; x(4)=-0.96509965042249314; x(5)=-0.94423950911819410; x(6)=-0.91867525998417577; x(7)=-0.88853423828604320; x(8)=-0.85396659500471038; x(9)=-0.81514453964513501; x(10)=-0.77226147924875590; x(11)=-0.72553105366071700; x(12)=-0.67518607066612237; x(13)=-0.62147734590357585; x(14)=-0.56467245318547077; x(15)=-0.50505439138820232; x(16)=-0.44292017452541148; x(17)=-0.37857935201470713; x(18)=-0.31235246650278581; x(19)=-0.24456945692820125; x(20)=-0.17556801477551679; x(21)=-0.10569190170865325; x(22)=-0.035289236964135359; x(23)=0.035289236964135359; x(24)=0.10569190170865325; x(25)=0.17556801477551679; x(26)=0.24456945692820125; x(27)=0.31235246650278581; x(28)=0.37857935201470713; x(29)=0.44292017452541148; x(30)=0.50505439138820232; x(31)=0.56467245318547077; x(32)=0.62147734590357585; x(33)=0.67518607066612237; x(34)=0.72553105366071700; x(35)=0.77226147924875590; x(36)=0.81514453964513501; x(37)=0.85396659500471038; x(38)=0.88853423828604320; x(39)=0.91867525998417577; x(40)=0.94423950911819410; x(41)=0.96509965042249314; x(42)=0.98115183307791397; x(43)=0.99231639213851581; x(44)=0.99854020063677422; case(45) x(1)=-0.99860364518193664; x(2)=-0.99264999844720374; x(3)=-0.98196871503454057; x(4)=-0.96660831039689460; x(5)=-0.94664169099562906; x(6)=-0.92216393671900039; x(7)=-0.89329167175324174; x(8)=-0.86016247596066423; x(9)=-0.82293422050208634; x(10)=-0.78178431259390629; x(11)=-0.73690884894549035; x(12)=-0.68852168077120053; x(13)=-0.63685339445322336; x(14)=-0.58215021256935319; x(15)=-0.52467282046291607; x(16)=-0.46469512391963510; x(17)=-0.40250294385854191; x(18)=-0.33839265425060216; x(19)=-0.27266976975237756; x(20)=-0.20564748978326375; x(21)=-0.13764520598325303; x(22)=-0.068986980163144172; x(23)=0.0; x(24)=0.068986980163144172; x(25)=0.13764520598325303; x(26)=0.20564748978326375; x(27)=0.27266976975237756; x(28)=0.33839265425060216; x(29)=0.40250294385854191; x(30)=0.46469512391963510; x(31)=0.52467282046291607; x(32)=0.58215021256935319; x(33)=0.63685339445322336; x(34)=0.68852168077120053; x(35)=0.73690884894549035; x(36)=0.78178431259390629; x(37)=0.82293422050208634; x(38)=0.86016247596066423; x(39)=0.89329167175324174; x(40)=0.92216393671900039; x(41)=0.94664169099562906; x(42)=0.96660831039689460; x(43)=0.98196871503454057; x(44)=0.99264999844720374; x(45)=0.99860364518193664; case(46) x(1)=-0.99866304213381798; x(2)=-0.99296234890617436; x(3)=-0.98273366980416686; x(4)=-0.96802139185399194; x(5)=-0.94889236344608980; x(6)=-0.92543379880675395; x(7)=-0.89775271153394197; x(8)=-0.86597539486685806; x(9)=-0.83024683706606605; x(10)=-0.79073005707527426; x(11)=-0.74760535961566605; x(12)=-0.70106951202040570; x(13)=-0.65133484620199772; x(14)=-0.59862828971271515; x(15)=-0.54319033026180264; x(16)=-0.48527391838816466; x(17)=-0.42514331328282840; x(18)=-0.36307287702099571; x(19)=-0.29934582270187002; x(20)=-0.23425292220626977; x(21)=-0.16809117946710353; x(22)=-0.10116247530558424; x(23)=-0.033772190016052042; x(24)=0.033772190016052042; x(25)=0.10116247530558424; x(26)=0.16809117946710353; x(27)=0.23425292220626977; x(28)=0.29934582270187002; x(29)=0.36307287702099571; x(30)=0.42514331328282840; x(31)=0.48527391838816466; x(32)=0.54319033026180264; x(33)=0.59862828971271515; x(34)=0.65133484620199772; x(35)=0.70106951202040570; x(36)=0.74760535961566605; x(37)=0.79073005707527426; x(38)=0.83024683706606605; x(39)=0.86597539486685806; x(40)=0.89775271153394197; x(41)=0.92543379880675395; x(42)=0.94889236344608980; x(43)=0.96802139185399194; x(44)=0.98273366980416686; x(45)=0.99296234890617436; x(46)=0.99866304213381798; case(47) x(1)=-0.99871872858421211; x(2)=-0.99325521098776863; x(3)=-0.98345100307162371; x(4)=-0.96934678732656450; x(5)=-0.95100396925770844; x(6)=-0.92850269301236065; x(7)=-0.90194132943852536; x(8)=-0.87143601579689632; x(9)=-0.83712013989990212; x(10)=-0.79914375416774194; x(11)=-0.75767291844543863; x(12)=-0.71288897340906430; x(13)=-0.66498774739033273; x(14)=-0.61417869995637361; x(15)=-0.56068400593466419; x(16)=-0.50473758386357792; x(17)=-0.44658407310485570; x(18)=-0.38647776408466714; x(19)=-0.32468148633773590; x(20)=-0.26146545921497457; x(21)=-0.19710611027911181; x(22)=-0.13188486655451490; x(23)=-0.066086923916355675; x(24)=0.0; x(25)=0.066086923916355675; x(26)=0.13188486655451490; x(27)=0.19710611027911181; x(28)=0.26146545921497457; x(29)=0.32468148633773590; x(30)=0.38647776408466714; x(31)=0.44658407310485570; x(32)=0.50473758386357792; x(33)=0.56068400593466419; x(34)=0.61417869995637361; x(35)=0.66498774739033273; x(36)=0.71288897340906430; x(37)=0.75767291844543863; x(38)=0.79914375416774194; x(39)=0.83712013989990212; x(40)=0.87143601579689632; x(41)=0.90194132943852536; x(42)=0.92850269301236065; x(43)=0.95100396925770844; x(44)=0.96934678732656450; x(45)=0.98345100307162371; x(46)=0.99325521098776863; x(47)=0.99871872858421211; case(48) x(1)=-0.99877100725242612; x(2)=-0.99353017226635076; x(3)=-0.98412458372282686; x(4)=-0.97059159254624725; x(5)=-0.95298770316043086; x(6)=-0.93138669070655433; x(7)=-0.90587913671556967; x(8)=-0.87657202027424789; x(9)=-0.84358826162439353; x(10)=-0.80706620402944263; x(11)=-0.76715903251574034; x(12)=-0.72403413092381465; x(13)=-0.67787237963266391; x(14)=-0.62886739677651362; x(15)=-0.57722472608397270; x(16)=-0.52316097472223303; x(17)=-0.46690290475095840; x(18)=-0.40868648199071673; x(19)=-0.34875588629216074; x(20)=-0.28736248735545558; x(21)=-0.22476379039468906; x(22)=-0.16122235606889172; x(23)=-0.097004699209462699; x(24)=-0.032380170962869362; x(25)=0.032380170962869362; x(26)=0.097004699209462699; x(27)=0.16122235606889172; x(28)=0.22476379039468906; x(29)=0.28736248735545558; x(30)=0.34875588629216074; x(31)=0.40868648199071673; x(32)=0.46690290475095840; x(33)=0.52316097472223303; x(34)=0.57722472608397270; x(35)=0.62886739677651362; x(36)=0.67787237963266391; x(37)=0.72403413092381465; x(38)=0.76715903251574034; x(39)=0.80706620402944263; x(40)=0.84358826162439353; x(41)=0.87657202027424789; x(42)=0.90587913671556967; x(43)=0.93138669070655433; x(44)=0.95298770316043086; x(45)=0.97059159254624725; x(46)=0.98412458372282686; x(47)=0.99353017226635076; x(48)=0.99877100725242612; case(49) x(1)=-0.99882015060663538; x(2)=-0.99378866194416779; x(3)=-0.98475789591421300; x(4)=-0.97176220090155538; x(5)=-0.95485365867413723; x(6)=-0.93410029475581015; x(7)=-0.90958565582807329; x(8)=-0.88140844557300891; x(9)=-0.84968211984416570; x(10)=-0.81453442735985543; x(11)=-0.77610689434544664; x(12)=-0.73455425423740270; x(13)=-0.69004382442513211; x(14)=-0.64275483241923766; x(15)=-0.59287769410890071; x(16)=-0.54061324699172607; x(17)=-0.48617194145249204; x(18)=-0.42977299334157652; x(19)=-0.37164350126228489; x(20)=-0.31201753211974876; x(21)=-0.25113517861257727; x(22)=-0.18924159246181359; x(23)=-0.12658599726967205; x(24)=-0.063420684982686786; x(25)=0.0; x(26)=0.063420684982686786; x(27)=0.12658599726967205; x(28)=0.18924159246181359; x(29)=0.25113517861257727; x(30)=0.31201753211974876; x(31)=0.37164350126228489; x(32)=0.42977299334157652; x(33)=0.48617194145249204; x(34)=0.54061324699172607; x(35)=0.59287769410890071; x(36)=0.64275483241923766; x(37)=0.69004382442513211; x(38)=0.73455425423740270; x(39)=0.77610689434544664; x(40)=0.81453442735985543; x(41)=0.84968211984416570; x(42)=0.88140844557300891; x(43)=0.90958565582807329; x(44)=0.93410029475581015; x(45)=0.95485365867413723; x(46)=0.97176220090155538; x(47)=0.98475789591421300; x(48)=0.99378866194416779; x(49)=0.99882015060663538; case(50) x(1)=-0.99886640442007105; x(2)=-0.99403196943209071; x(3)=-0.98535408404800588; x(4)=-0.97286438510669207; x(5)=-0.95661095524280794; x(6)=-0.93665661894487793; x(7)=-0.91307855665579189; x(8)=-0.88596797952361305; x(9)=-0.85542976942994608; x(10)=-0.82158207085933595; x(11)=-0.78455583290039926; x(12)=-0.74449430222606854; x(13)=-0.70155246870682225; x(14)=-0.65589646568543936; x(15)=-0.60770292718495024; x(16)=-0.55715830451465005; x(17)=-0.50445814490746420; x(18)=-0.44980633497403879; x(19)=-0.39341431189756513; x(20)=-0.33550024541943736; x(21)=-0.27628819377953199; x(22)=-0.21600723687604176; x(23)=-0.15489058999814590; x(24)=-0.093174701560086141; x(25)=-0.031098338327188876; x(26)=0.031098338327188876; x(27)=0.093174701560086141; x(28)=0.15489058999814590; x(29)=0.21600723687604176; x(30)=0.27628819377953199; x(31)=0.33550024541943736; x(32)=0.39341431189756513; x(33)=0.44980633497403879; x(34)=0.50445814490746420; x(35)=0.55715830451465005; x(36)=0.60770292718495024; x(37)=0.65589646568543936; x(38)=0.70155246870682225; x(39)=0.74449430222606854; x(40)=0.78455583290039926; x(41)=0.82158207085933595; x(42)=0.85542976942994608; x(43)=0.88596797952361305; x(44)=0.91307855665579189; x(45)=0.93665661894487793; x(46)=0.95661095524280794; x(47)=0.97286438510669207; x(48)=0.98535408404800588; x(49)=0.99403196943209071; x(50)=0.99886640442007105; case(51) x(1)=-0.99890999084890350; x(2)=-0.99426126043675257; x(3)=-0.98591599173590300; x(4)=-0.97390336801932387; x(5)=-0.95826784861390819; x(6)=-0.93906754400296238; x(7)=-0.91637386230978023; x(8)=-0.89027121802952730; x(9)=-0.86085671118229237; x(10)=-0.82823976382306483; x(11)=-0.79254171209938121; x(12)=-0.75389535448537553; x(13)=-0.71244445757703664; x(14)=-0.66834322117537009; x(15)=-0.62175570460072327; x(16)=-0.57285521635130384; x(17)=-0.52182366936618584; x(18)=-0.46885090428604106; x(19)=-0.41413398322630388; x(20)=-0.35787645668840951; x(21)=-0.30028760633533194; x(22)=-0.24158166644779870; x(23)=-0.18197702695707755; x(24)=-0.12169542101888877; x(25)=-0.060961100150578725; x(26)=0.0; x(27)=0.060961100150578725; x(28)=0.12169542101888877; x(29)=0.18197702695707755; x(30)=0.24158166644779870; x(31)=0.30028760633533194; x(32)=0.35787645668840951; x(33)=0.41413398322630388; x(34)=0.46885090428604106; x(35)=0.52182366936618584; x(36)=0.57285521635130384; x(37)=0.62175570460072327; x(38)=0.66834322117537009; x(39)=0.71244445757703664; x(40)=0.75389535448537553; x(41)=0.79254171209938121; x(42)=0.82823976382306483; x(43)=0.86085671118229237; x(44)=0.89027121802952730; x(45)=0.91637386230978023; x(46)=0.93906754400296238; x(47)=0.95826784861390819; x(48)=0.97390336801932387; x(49)=0.98591599173590300; x(50)=0.99426126043675257; x(51)=0.99890999084890350; case(52) x(1)=-0.99895111110395028; x(2)=-0.99447759092921603; x(3)=-0.98644619565154984; x(4)=-0.97488388422174450; x(5)=-0.95983182693308655; x(6)=-0.94134385364135906; x(7)=-0.91948612891642454; x(8)=-0.89433689053449532; x(9)=-0.86598616284606759; x(10)=-0.83453543232673453; x(11)=-0.80009728343046832; x(12)=-0.76279499519374496; x(13)=-0.72276209974998319; x(14)=-0.68014190422716770; x(15)=-0.63508697769524592; x(16)=-0.58775860497957907; x(17)=-0.53832620928582744; x(18)=-0.48696674569809608; x(19)=-0.43386406771876167; x(20)=-0.37920826911609367; x(21)=-0.32319500343480783; x(22)=-0.26602478360500183; x(23)=-0.20790226415636606; x(24)=-0.14903550860694918; x(25)=-0.089635244648900565; x(26)=-0.029914109797338766; x(27)=0.029914109797338766; x(28)=0.089635244648900565; x(29)=0.14903550860694918; x(30)=0.20790226415636606; x(31)=0.26602478360500183; x(32)=0.32319500343480783; x(33)=0.37920826911609367; x(34)=0.43386406771876167; x(35)=0.48696674569809608; x(36)=0.53832620928582744; x(37)=0.58775860497957907; x(38)=0.63508697769524592; x(39)=0.68014190422716770; x(40)=0.72276209974998319; x(41)=0.76279499519374496; x(42)=0.80009728343046832; x(43)=0.83453543232673453; x(44)=0.86598616284606759; x(45)=0.89433689053449532; x(46)=0.91948612891642454; x(47)=0.94134385364135906; x(48)=0.95983182693308655; x(49)=0.97488388422174450; x(50)=0.98644619565154984; x(51)=0.99447759092921603; x(52)=0.99895111110395028; case(53) x(1)=-0.99898994777632823; x(2)=-0.99468191930800708; x(3)=-0.98694703502337152; x(4)=-0.97581023371498458; x(5)=-0.96130969462313633; x(6)=-0.94349535346444188; x(7)=-0.92242860304281213; x(8)=-0.89818205787542663; x(9)=-0.87083929755824135; x(10)=-0.84049457654580138; x(11)=-0.80725249841689548; x(12)=-0.77122765492553231; x(13)=-0.73254423080751025; x(14)=-0.69133557560136672; x(15)=-0.64774374391651007; x(16)=-0.60191900571376933; x(17)=-0.55401932827706788; x(18)=-0.50420983165713344; x(19)=-0.45266221946184579; x(20)=-0.39955418695395298; x(21)=-0.34506880849572236; x(22)=-0.28939390645162621; x(23)=-0.23272140372427259; x(24)=-0.17524666215532575; x(25)=-0.11716780907195515; x(26)=-0.058685054300259465; x(27)=0.0; x(28)=0.058685054300259465; x(29)=0.11716780907195515; x(30)=0.17524666215532575; x(31)=0.23272140372427259; x(32)=0.28939390645162621; x(33)=0.34506880849572236; x(34)=0.39955418695395298; x(35)=0.45266221946184579; x(36)=0.50420983165713344; x(37)=0.55401932827706788; x(38)=0.60191900571376933; x(39)=0.64774374391651007; x(40)=0.69133557560136672; x(41)=0.73254423080751025; x(42)=0.77122765492553231; x(43)=0.80725249841689548; x(44)=0.84049457654580138; x(45)=0.87083929755824135; x(46)=0.89818205787542663; x(47)=0.92242860304281213; x(48)=0.94349535346444188; x(49)=0.96130969462313633; x(50)=0.97581023371498458; x(51)=0.98694703502337152; x(52)=0.99468191930800708; x(53)=0.99898994777632823; case(54) x(1)=-0.99902666686734098; x(2)=-0.99487511701833888; x(3)=-0.98742063739734356; x(4)=-0.97668632885790324; x(5)=-0.96270764578592358; x(6)=-0.94553097516499585; x(7)=-0.92521335986665149; x(8)=-0.90182228628470158; x(9)=-0.87543545406556894; x(10)=-0.84614051597077295; x(11)=-0.81403478591356784; x(12)=-0.77922491534625402; x(13)=-0.74182653880918432; x(14)=-0.70196388971917292; x(15)=-0.65976938763198312; x(16)=-0.61538319833112737; x(17)=-0.56895276819520943; x(18)=-0.52063233438593307; x(19)=-0.47058241248138228; x(20)=-0.41896926325520453; x(21)=-0.36596434037219118; x(22)=-0.31174372083446823; x(23)=-0.25648752006999730; x(24)=-0.20037929360621357; x(25)=-0.14360542731625615; x(26)=-0.086354518263248215; x(27)=-0.028816748199341778; x(28)=0.028816748199341778; x(29)=0.086354518263248215; x(30)=0.14360542731625615; x(31)=0.20037929360621357; x(32)=0.25648752006999730; x(33)=0.31174372083446823; x(34)=0.36596434037219118; x(35)=0.41896926325520453; x(36)=0.47058241248138228; x(37)=0.52063233438593307; x(38)=0.56895276819520943; x(39)=0.61538319833112737; x(40)=0.65976938763198312; x(41)=0.70196388971917292; x(42)=0.74182653880918432; x(43)=0.77922491534625402; x(44)=0.81403478591356784; x(45)=0.84614051597077295; x(46)=0.87543545406556894; x(47)=0.90182228628470158; x(48)=0.92521335986665149; x(49)=0.94553097516499585; x(50)=0.96270764578592358; x(51)=0.97668632885790324; x(52)=0.98742063739734356; x(53)=0.99487511701833888; x(54)=0.99902666686734098; case(55) x(1)=-0.99906141956481854; x(2)=-0.99505797784741188; x(3)=-0.98786894119888920; x(4)=-0.97751573550398921; x(5)=-0.96403132859313520; x(6)=-0.94745886804121074; x(7)=-0.92785142472079170; x(8)=-0.90527180074400003; x(9)=-0.87979232241989551; x(10)=-0.85149460661715447; x(11)=-0.82046929855932091; x(12)=-0.78681578112762237; x(13)=-0.75064185634802191; x(14)=-0.71206339998663784; x(15)=-0.67120399031982640; x(16)=-0.62819451224992814; x(17)=-0.58317273802603210; x(18)=-0.53628288590834330; x(19)=-0.48767515818747410; x(20)=-0.43750526003717459; x(21)=-0.38593390074097943; x(22)=-0.33312627889002389; x(23)=-0.27925155320080654; x(24)=-0.22448230064784548; x(25)=-0.16899396364687321; x(26)=-0.11296428805932927; x(27)=-0.056572753818336776; x(28)=0.0; x(29)=0.056572753818336776; x(30)=0.11296428805932927; x(31)=0.16899396364687321; x(32)=0.22448230064784548; x(33)=0.27925155320080654; x(34)=0.33312627889002389; x(35)=0.38593390074097943; x(36)=0.43750526003717459; x(37)=0.48767515818747410; x(38)=0.53628288590834330; x(39)=0.58317273802603210; x(40)=0.62819451224992814; x(41)=0.67120399031982640; x(42)=0.71206339998663784; x(43)=0.75064185634802191; x(44)=0.78681578112762237; x(45)=0.82046929855932091; x(46)=0.85149460661715447; x(47)=0.87979232241989551; x(48)=0.90527180074400003; x(49)=0.92785142472079170; x(50)=0.94745886804121074; x(51)=0.96403132859313520; x(52)=0.97751573550398921; x(53)=0.98786894119888920; x(54)=0.99505797784741188; x(55)=0.99906141956481854; case(56) x(1)=-0.99909434380146558; x(2)=-0.99523122608106975; x(3)=-0.98829371554016151; x(4)=-0.97830170914025638; x(5)=-0.96528590190549018; x(6)=-0.94928647956196264; x(7)=-0.93035288024749630; x(8)=-0.90854362042065549; x(9)=-0.88392610832782754; x(10)=-0.85657643376274864; x(11)=-0.82657913214288165; x(12)=-0.79402692289386650; x(13)=-0.75902042270512890; x(14)=-0.72166783445018808; x(15)=-0.68208461269447046; x(16)=-0.64039310680700689; x(17)=-0.59672218277066332; x(18)=-0.55120682485553462; x(19)=-0.50398771838438171; x(20)=-0.45521081487845958; x(21)=-0.40502688092709128; x(22)=-0.35359103217495452; x(23)=-0.30106225386722067; x(24)=-0.24760290943433720; x(25)=-0.19337823863527526; x(26)=-0.13855584681037624; x(27)=-0.083305186822435374; x(28)=-0.027797035287275437; x(29)=0.027797035287275437; x(30)=0.083305186822435374; x(31)=0.13855584681037624; x(32)=0.19337823863527526; x(33)=0.24760290943433720; x(34)=0.30106225386722067; x(35)=0.35359103217495452; x(36)=0.40502688092709128; x(37)=0.45521081487845958; x(38)=0.50398771838438171; x(39)=0.55120682485553462; x(40)=0.59672218277066332; x(41)=0.64039310680700689; x(42)=0.68208461269447046; x(43)=0.72166783445018808; x(44)=0.75902042270512890; x(45)=0.79402692289386650; x(46)=0.82657913214288165; x(47)=0.85657643376274864; x(48)=0.88392610832782754; x(49)=0.90854362042065549; x(50)=0.93035288024749630; x(51)=0.94928647956196264; x(52)=0.96528590190549018; x(53)=0.97830170914025638; x(54)=0.98829371554016151; x(55)=0.99523122608106975; x(56)=0.99909434380146558; case(57) x(1)=-0.99912556562526285; x(2)=-0.99539552367843031; x(3)=-0.98869657765022205; x(4)=-0.97904722670946871; x(5)=-0.96647608517188668; x(6)=-0.95102062644787674; x(7)=-0.93272696106710170; x(8)=-0.91164967852139121; x(9)=-0.88785167888222133; x(10)=-0.86140398326204694; x(11)=-0.83238552115043912; x(12)=-0.80088289454721824; x(13)=-0.76699011935945020; x(14)=-0.73080834474452332; x(15)=-0.69244555119951774; x(16)=-0.65201622828097689; x(17)=-0.60964103290871537; x(18)=-0.56544642926923676; x(19)=-0.51956431139118761; x(20)=-0.47213160951797571; x(21)=-0.42328988145156395; x(22)=-0.37318489008659446; x(23)=-0.32196616839537864; x(24)=-0.26978657316183877; x(25)=-0.21680182879612404; x(26)=-0.16317006259126425; x(27)=-0.10905133280878780; x(28)=-0.054607151001646824; x(29)=0.0; x(30)=0.054607151001646824; x(31)=0.10905133280878780; x(32)=0.16317006259126425; x(33)=0.21680182879612404; x(34)=0.26978657316183877; x(35)=0.32196616839537864; x(36)=0.37318489008659446; x(37)=0.42328988145156395; x(38)=0.47213160951797571; x(39)=0.51956431139118761; x(40)=0.56544642926923676; x(41)=0.60964103290871537; x(42)=0.65201622828097689; x(43)=0.69244555119951774; x(44)=0.73080834474452332; x(45)=0.76699011935945020; x(46)=0.80088289454721824; x(47)=0.83238552115043912; x(48)=0.86140398326204694; x(49)=0.88785167888222133; x(50)=0.91164967852139121; x(51)=0.93272696106710170; x(52)=0.95102062644787674; x(53)=0.96647608517188668; x(54)=0.97904722670946871; x(55)=0.98869657765022205; x(56)=0.99539552367843031; x(57)=0.99912556562526285; case(58) x(1)=-0.99915520040738661; x(2)=-0.99555147659729090; x(3)=-0.98907900824844264; x(4)=-0.97975501469435031; x(5)=-0.96760620250292409; x(6)=-0.95266755751886909; x(7)=-0.93498213758825935; x(8)=-0.91460092856435254; x(9)=-0.89158269202203018; x(10)=-0.86599379407480748; x(11)=-0.83790801333937332; x(12)=-0.80740632791308814; x(13)=-0.77457668174965275; x(14)=-0.73951373102004227; x(15)=-0.70231857115390811; x(16)=-0.66309844533212527; x(17)=-0.62196643526307911; x(18)=-0.57904113513022503; x(19)=-0.53444630964884759; x(20)=-0.48831053721671846; x(21)=-0.44076683918683957; x(22)=-0.39195229633075315; x(23)=-0.34200765359799526; x(24)=-0.29107691431110919; x(25)=-0.23930692496615345; x(26)=-0.18684695183576132; x(27)=-0.13384825059546686; x(28)=-0.080463630214142729; x(29)=-0.026847012365942356; x(30)=0.026847012365942356; x(31)=0.080463630214142729; x(32)=0.13384825059546686; x(33)=0.18684695183576132; x(34)=0.23930692496615345; x(35)=0.29107691431110919; x(36)=0.34200765359799526; x(37)=0.39195229633075315; x(38)=0.44076683918683957; x(39)=0.48831053721671846; x(40)=0.53444630964884759; x(41)=0.57904113513022503; x(42)=0.62196643526307911; x(43)=0.66309844533212527; x(44)=0.70231857115390811; x(45)=0.73951373102004227; x(46)=0.77457668174965275; x(47)=0.80740632791308814; x(48)=0.83790801333937332; x(49)=0.86599379407480748; x(50)=0.89158269202203018; x(51)=0.91460092856435254; x(52)=0.93498213758825935; x(53)=0.95266755751886909; x(54)=0.96760620250292409; x(55)=0.97975501469435031; x(56)=0.98907900824844264; x(57)=0.99555147659729090; x(58)=0.99915520040738661; case(59) x(1)=-0.99918335390929468; x(2)=-0.99569964038324596; x(3)=-0.98944236513373093; x(4)=-0.98042757395671569; x(5)=-0.96868022168178153; x(6)=-0.95423300937695106; x(7)=-0.93712619035345386; x(8)=-0.91740743878815528; x(9)=-0.89513171174347209; x(10)=-0.87036109429288226; x(11)=-0.84316462581687220; x(12)=-0.81361810728821157; x(13)=-0.78180388986236091; x(14)=-0.74781064527864023; x(15)=-0.71173311867719773; x(16)=-0.67367186450493723; x(17)=-0.63373296623885010; x(18)=-0.59202774070403014; x(19)=-0.54867242780839638; x(20)=-0.50378786655771798; x(21)=-0.45749915825326669; x(22)=-0.40993531781041897; x(23)=-0.36122891416979481; x(24)=-0.31151570080301370; x(25)=-0.26093423734281171; x(26)=-0.20962550339203654; x(27)=-0.15773250558785797; x(28)=-0.10539987901634414; x(29)=-0.052773484088310004; x(30)=0.0; x(31)=0.052773484088310004; x(32)=0.10539987901634414; x(33)=0.15773250558785797; x(34)=0.20962550339203654; x(35)=0.26093423734281171; x(36)=0.31151570080301370; x(37)=0.36122891416979481; x(38)=0.40993531781041897; x(39)=0.45749915825326669; x(40)=0.50378786655771798; x(41)=0.54867242780839638; x(42)=0.59202774070403014; x(43)=0.63373296623885010; x(44)=0.67367186450493723; x(45)=0.71173311867719773; x(46)=0.74781064527864023; x(47)=0.78180388986236091; x(48)=0.81361810728821157; x(49)=0.84316462581687220; x(50)=0.87036109429288226; x(51)=0.89513171174347209; x(52)=0.91740743878815528; x(53)=0.93712619035345386; x(54)=0.95423300937695106; x(55)=0.96868022168178153; x(56)=0.98042757395671569; x(57)=0.98944236513373093; x(58)=0.99569964038324596; x(59)=0.99918335390929468; case(60) x(1)=-0.99921012322743602; x(2)=-0.99584052511883817; x(3)=-0.98978789522222172; x(4)=-0.98106720175259819; x(5)=-0.96970178876505273; x(6)=-0.95572225583999611; x(7)=-0.93916627611642325; x(8)=-0.92007847617762755; x(9)=-0.89851031081004594; x(10)=-0.87451992264689832; x(11)=-0.84817198478592963; x(12)=-0.81953752616214576; x(13)=-0.78869373993226405; x(14)=-0.75572377530658569; x(15)=-0.72071651335573040; x(16)=-0.68376632738135544; x(17)=-0.64497282848947707; x(18)=-0.60444059704851036; x(19)=-0.56227890075394454; x(20)=-0.51860140005856975; x(21)=-0.47352584176170711; x(22)=-0.42717374158307839; x(23)=-0.37967005657679798; x(24)=-0.33114284826844819; x(25)=-0.28172293742326169; x(26)=-0.23154355137602934; x(27)=-0.18073996487342542; x(28)=-0.12944913539694500; x(29)=-0.077809333949536569; x(30)=-0.025959772301247799; x(31)=0.025959772301247799; x(32)=0.077809333949536569; x(33)=0.12944913539694500; x(34)=0.18073996487342542; x(35)=0.23154355137602934; x(36)=0.28172293742326169; x(37)=0.33114284826844819; x(38)=0.37967005657679798; x(39)=0.42717374158307839; x(40)=0.47352584176170711; x(41)=0.51860140005856975; x(42)=0.56227890075394454; x(43)=0.60444059704851036; x(44)=0.64497282848947707; x(45)=0.68376632738135544; x(46)=0.72071651335573040; x(47)=0.75572377530658569; x(48)=0.78869373993226405; x(49)=0.81953752616214576; x(50)=0.84817198478592963; x(51)=0.87451992264689832; x(52)=0.89851031081004594; x(53)=0.92007847617762755; x(54)=0.93916627611642325; x(55)=0.95572225583999611; x(56)=0.96970178876505273; x(57)=0.98106720175259819; x(58)=0.98978789522222172; x(59)=0.99584052511883817; x(60)=0.99921012322743602; case(61) x(1)=-0.99923559763136347; x(2)=-0.99597459981512023; x(3)=-0.99011674523251705; x(4)=-0.98167601128403708; x(5)=-0.97067425883318291; x(6)=-0.95714015191298409; x(7)=-0.94110898668136115; x(8)=-0.92262258138295526; x(9)=-0.90172916247400117; x(10)=-0.87848323721488103; x(11)=-0.85294545084766345; x(12)=-0.82518242810865995; x(13)=-0.79526659928235965; x(14)=-0.76327601117231220; x(15)=-0.72929412344946511; x(16)=-0.69340959089449116; x(17)=-0.65571603209507087; x(18)=-0.61631178519792172; x(19)=-0.57529965135083062; x(20)=-0.53278662650292527; x(21)=-0.48888362226225212; x(22)=-0.44370517653853160; x(23)=-0.39736915472575661; x(24)=-0.34999644220406683; x(25)=-0.30171062896303071; x(26)=-0.25263768716905350; x(27)=-0.20290564251805850; x(28)=-0.15264424023081530; x(29)=-0.10198460656227407; x(30)=-0.051058906707974349; x(31)=0.0; x(32)=0.051058906707974349; x(33)=0.10198460656227407; x(34)=0.15264424023081530; x(35)=0.20290564251805850; x(36)=0.25263768716905350; x(37)=0.30171062896303071; x(38)=0.34999644220406683; x(39)=0.39736915472575661; x(40)=0.44370517653853160; x(41)=0.48888362226225212; x(42)=0.53278662650292527; x(43)=0.57529965135083062; x(44)=0.61631178519792172; x(45)=0.65571603209507087; x(46)=0.69340959089449116; x(47)=0.72929412344946511; x(48)=0.76327601117231220; x(49)=0.79526659928235965; x(50)=0.82518242810865995; x(51)=0.85294545084766345; x(52)=0.87848323721488103; x(53)=0.90172916247400117; x(54)=0.92262258138295526; x(55)=0.94110898668136115; x(56)=0.95714015191298409; x(57)=0.97067425883318291; x(58)=0.98167601128403708; x(59)=0.99011674523251705; x(60)=0.99597459981512023; x(61)=0.99923559763136347; case(62) x(1)=-0.99925985930877703; x(2)=-0.99610229631626713; x(3)=-0.99042997118929035; x(4)=-0.98225594909723665; x(5)=-0.97160072337165181; x(6)=-0.95849117297392709; x(7)=-0.94296040139232850; x(8)=-0.92504763563620376; x(9)=-0.90479812252109347; x(10)=-0.88226301283189736; x(11)=-0.85749923151207092; x(12)=-0.83056933360400485; x(13)=-0.80154134610397637; x(14)=-0.77048859605541932; x(15)=-0.73748952528315675; x(16)=-0.70262749222229706; x(17)=-0.66599056133547945; x(18)=-0.62767128064688518; x(19)=-0.58776644795308734; x(20)=-0.54637686630025110; x(21)=-0.50360708934475596; x(22)=-0.45956515724011340; x(23)=-0.41436232371712605; x(24)=-0.36811277504656453; x(25)=-0.32093334159419400; x(26)=-0.27294320269672634; x(27)=-0.22426358560416553; x(28)=-0.17501745924901563; x(29)=-0.12532922361589681; x(30)=-0.075324395496234333; x(31)=-0.025129291421820615; x(32)=0.025129291421820615; x(33)=0.075324395496234333; x(34)=0.12532922361589681; x(35)=0.17501745924901563; x(36)=0.22426358560416553; x(37)=0.27294320269672634; x(38)=0.32093334159419400; x(39)=0.36811277504656453; x(40)=0.41436232371712605; x(41)=0.45956515724011340; x(42)=0.50360708934475596; x(43)=0.54637686630025110; x(44)=0.58776644795308734; x(45)=0.62767128064688518; x(46)=0.66599056133547945; x(47)=0.70262749222229706; x(48)=0.73748952528315675; x(49)=0.77048859605541932; x(50)=0.80154134610397637; x(51)=0.83056933360400485; x(52)=0.85749923151207092; x(53)=0.88226301283189736; x(54)=0.90479812252109347; x(55)=0.92504763563620376; x(56)=0.94296040139232850; x(57)=0.95849117297392709; x(58)=0.97160072337165181; x(59)=0.98225594909723665; x(60)=0.99042997118929035; x(61)=0.99610229631626713; x(62)=0.99925985930877703; case(63) x(1)=-0.99928298402912378; x(2)=-0.99622401277797011; x(3)=-0.99072854689218947; x(4)=-0.98280881059372723; x(5)=-0.97248403469757002; x(6)=-0.95977944975894193; x(7)=-0.94472613404100980; x(8)=-0.92736092062184321; x(9)=-0.90772630277853156; x(10)=-0.88587032850785343; x(11)=-0.86184648236412372; x(12)=-0.83571355431950284; x(13)=-0.80753549577345676; x(14)=-0.77738126299037234; x(15)=-0.74532464831784742; x(16)=-0.71144409958484581; x(17)=-0.67582252811498609; x(18)=-0.63854710582136539; x(19)=-0.59970905187762524; x(20)=-0.55940340948628501; x(21)=-0.51772881329003325; x(22)=-0.47478724799480440; x(23)=-0.43068379879511160; x(24)=-0.38552639421224789; x(25)=-0.33942554197458440; x(26)=-0.29249405858625144; x(27)=-0.24484679324595336; x(28)=-0.19660034679150668; x(29)=-0.14787278635787197; x(30)=-0.098783356446945280; x(31)=-0.049452187116159627; x(32)=0.0; x(33)=0.049452187116159627; x(34)=0.098783356446945280; x(35)=0.14787278635787197; x(36)=0.19660034679150668; x(37)=0.24484679324595336; x(38)=0.29249405858625144; x(39)=0.33942554197458440; x(40)=0.38552639421224789; x(41)=0.43068379879511160; x(42)=0.47478724799480440; x(43)=0.51772881329003325; x(44)=0.55940340948628501; x(45)=0.59970905187762524; x(46)=0.63854710582136539; x(47)=0.67582252811498609; x(48)=0.71144409958484581; x(49)=0.74532464831784742; x(50)=0.77738126299037234; x(51)=0.80753549577345676; x(52)=0.83571355431950284; x(53)=0.86184648236412372; x(54)=0.88587032850785343; x(55)=0.90772630277853156; x(56)=0.92736092062184321; x(57)=0.94472613404100980; x(58)=0.95977944975894193; x(59)=0.97248403469757002; x(60)=0.98280881059372723; x(61)=0.99072854689218947; x(62)=0.99622401277797011; x(63)=0.99928298402912378; case(64) x(1)=-0.99930504173577214; x(2)=-0.99634011677195528; x(3)=-0.99101337147674432; x(4)=-0.98333625388462596; x(5)=-0.97332682778991096; x(6)=-0.96100879965205372; x(7)=-0.94641137485840282; x(8)=-0.92956917213193958; x(9)=-0.91052213707850281; x(10)=-0.88931544599511411; x(11)=-0.86599939815409282; x(12)=-0.84062929625258036; x(13)=-0.81326531512279756; x(14)=-0.78397235894334141; x(15)=-0.75281990726053190; x(16)=-0.71988185017161083; x(17)=-0.68523631305423324; x(18)=-0.64896547125465734; x(19)=-0.61115535517239325; x(20)=-0.57189564620263403; x(21)=-0.53127946401989455; x(22)=-0.48940314570705296; x(23)=-0.44636601725346409; x(24)=-0.40227015796399160; x(25)=-0.35722015833766812; x(26)=-0.31132287199021096; x(27)=-0.26468716220876742; x(28)=-0.21742364374000708; x(29)=-0.16964442042399282; x(30)=-0.12146281929612055; x(31)=-0.072993121787799039; x(32)=-0.024350292663424433; x(33)=0.024350292663424433; x(34)=0.072993121787799039; x(35)=0.12146281929612055; x(36)=0.16964442042399282; x(37)=0.21742364374000708; x(38)=0.26468716220876742; x(39)=0.31132287199021096; x(40)=0.35722015833766812; x(41)=0.40227015796399160; x(42)=0.44636601725346409; x(43)=0.48940314570705296; x(44)=0.53127946401989455; x(45)=0.57189564620263403; x(46)=0.61115535517239325; x(47)=0.64896547125465734; x(48)=0.68523631305423324; x(49)=0.71988185017161083; x(50)=0.75281990726053190; x(51)=0.78397235894334141; x(52)=0.81326531512279756; x(53)=0.84062929625258036; x(54)=0.86599939815409282; x(55)=0.88931544599511411; x(56)=0.91052213707850281; x(57)=0.92956917213193958; x(58)=0.94641137485840282; x(59)=0.96100879965205372; x(60)=0.97332682778991096; x(61)=0.98333625388462596; x(62)=0.99101337147674432; x(63)=0.99634011677195528; x(64)=0.99930504173577214; otherwise error('GAUSS_WEIGHTS - Fatal error! Illegal value of n.');endfunction uqII = fetoquadgen(xiI, xiqII, xeI, xeII, connectI, ufeI)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % transforms ufe from FE-coefficient of mesh-I to quadrature-grid % representation of mesh-II. % ufe is a full FE coefficient vector, having values for all nodes in % mesh-I, including the domain-boundary nodes% % Inputs% ======% ufeI(:) : full FE coefficient vector, having values for all nodes % in a mesh-I, including domain-boundary nodes.% xiI(:) : parent basis nodes; xi(i) = coordinate of i-th % parent basis node of mesh-I% xiqII(:) : quadrature points of mesh-II% connect(:,:) : nodal connectivity; connect(i,j) = index of basis node% corresponding to local node j of element i% xq(:,:) : quadrature pts in physical space; % xq(i,j) = coordinate of i-th point in j-th element% % Output% ====== % uqII(:,:) : quadrature-grid representatin of ufe on mesh-II;% uqII(i,j) = value at i-th quadrature point of % j-th element% Generate global coordinates of quadrature points of mesh-II:nspII = length(xiqII); % # of quadrature points of mesh-II xqII = getphysicalquadpts(xeII, xiqII);numelI = length(xeI) - 1; % # of elements of mesh-InumelII = length(xeII) - 1; % # of elements of mesh-IIpI = length(xiI) - 1; % element order of mesh-I% Evaluate at quad points of mesh-II in each element% Initialize outputuqII = zeros(nspII,numelII);for eII = 1:numelII % loop over elements of mesh-II % Physical coordintaes of quadrature-points inside element eII in mesh-II xqIIeII = xqII(:,eII); % length nspII for iqII = 1:length(xqIIeII) % loop over q-points of eII xsampleII = xqIIeII(iqII); % Now loop over elements of mesh-I to scan which element (can be % only one) contains quadrature point xsampleII for eI = 1:numelI if iswithin(xsampleII, xeI(eI), xeI(eI+1)) == 1 % get the FE-nodal solution for that element ueI = ufeI(connectI(eI,:)); xl = xeI(eI); xr = xeI(eI+1); l = xr - xl; Nl = (xr - xsampleII)/l; Nr = (xsampleII - xl)/l; % affine transformation: [xl xr] --> [-1 1] xisample = Nr - Nl; % Nl*(-1) + Nr*(+1) % Tabulate parent basis at quadrature point xsampleII: [phi, ~] = getshapefunc(pI, xisample); % Compute the FE solution at quadrature point xsampleII uqII(iqII,eII) = dot(phi, ueI); % since xsampleII can be contained by just one element. % Once found, hence just "break" the loop to move on % to the next quadrature point break else continue end end endendendfunction r = iswithin(x, low, hi)% returns logical true if low < x < hir = (x >= low) & (x <= hi);endfunction uq = fetoquad(xi, xiq, connect, ufe)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % transforms ufe from FE-coefficient to quadrature-grid representation% ufe is a full FE coefficient vector, having values for all nodes in % the mesh, including domain-boundary nodes% % Inputs% ======% ufe(:) : full FE coefficient vector, having values for all nodes % in a mesh, including domain-boundary nodes.% xi(:) : parent basis nodes; xi(i) = coordinate of i-th % parent basis node% xiq(:) : quadrature points% connect(:,:) : nodal connectivity; connect(i,j) = index of basis node% corresponding to local node j of element i% % Output% ====== % uq(:,:) : quadrature-grid representatin of ufe;% uq(i,j) = value at i-th quadrature point of j-th element% Tabulate parent basis at quadrature points:nparntnodes = length(xi);p = nparntnodes - 1;nsp = length(xiq);numel = size(connect,1);[phiq, ~] = getshapefunc(p, xiq);% Evaluate at quad points in each elementuq = zeros(nsp,numel);for e = 1:numel ue = ufe(connect(e,:)); for iq = 1:nsp uq(iq, e) = dot(phiq(iq,:), ue); endendendfunction [excq, Vxcq] = exhangecorrelation(type, rhoq)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % Calculates exchange-correlation (LDA density and potential so far) % from the electron density rhoq. rhoq must be positive% %% Inputs% ====== % rhoq(:,:) : quadrature-grid representatin of rho; % rho(i,j) = rho value at i-th quadrature point % of j-th element% Output% ====== % Vxcq(:,:) : quadrature-grid representatin of Vxc;% Vxc(i,j) = value at i-th quadrature point of j-th elementtype = lower(type); % LDA density and potential so farswitch type case 'lda' % The Local Density Approximation (LDA) % Parameters: y0 = -0.10498; b = 3.72744; c = 12.9352; A = 0.0621814; Q = sqrt(4.0*c - b^2); rs = (3.0./(4.0*pi*rhoq)).^(1.0/3.0); y = sqrt(rs); Y = computeY(y, b, c); Y0 = computeY(y0, b, c); % Vosko-Wilk-Nusair correlation term: ec = A/2.0*(log(y.^2./Y) + 2.0*b/Q.*atan(Q./(2.0*y + b)) - ... b*y0./Y0.*(log((y - y0).^2./Y) + 2.0*(b + 2.0*y0)./ ... Q.*atan(Q./(2.0*y + b)))); Vc = ec - A/6.0*(c*(y - y0) - b*y0*y)./((y - y0).*Y); % Electron gas exchange term: ex = -3.0./(4.0*pi)*(3.0*pi^2*rhoq).^(1.0/3.0); Vx = 4.0*ex./3.0; excq = ex + ec; Vxcq = Vx + Vc; otherwise error('Wrong exchange-correlation type entered. Use lda instead');endrhonegval = any(rhoq(:) < 0.0);if rhonegval disp('Density is negative!'); indcs = find(rhoq(:) < 0.0); excq(indcs) = 0.0; Vxcq(indcs) = 0.0;endreturnendfunction Y = computeY(y, b, c)Y = y.^2 + b.*y + c;endfunction [T_s, E_ee, E_en, E_xc, E_tot] = computetotalenergy(xe, xq, wtq, ... rhooutq, Veffq, Vhq, excq, ... Z, E_band)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Purpose% ======= % Variational, quadratically convergent form of total energy% Returns the total energy as well as its parts%% Inputs% ======% xe(:) : xe(i/i+1) = coord of left/right boundary of ith element% xq(:,:) : quadrature pts in physical space; % xq(i,j) = coordinate of i-th point in j-th element% wtq(:) : Quadrature weights% rhooutq(:,:) : radial density at quadrature points; rhoq(i,j) = value% at i-th quadrature point of j-th element% veffq(:, :) :% V_h_q(:, :) :% e_xc_q(:, :) :% E_band :% Z :% % Output% ======% xqsqrd = xq.^2;T_s = E_band + 4.0*pi*integrateonmesh(xe, wtq, Veffq.*rhooutq.*xqsqrd);E_ee = -2.0*pi*integrateonmesh(xe, wtq, Vhq.*rhooutq.*xqsqrd);E_en = 4.0*pi*Z*integrateonmesh(xe, wtq, rhooutq.*xq);E_xc = -4.0*pi*integrateonmesh(xe, wtq, excq.*rhooutq.*xqsqrd);E_tot = T_s + E_ee + E_en + E_xc;returnend- p收敛性分析：
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % Purpose% ========% % Script file to test convergence of ground state energy % with p-refinement of Spectral FEM basis functions%clc;clear all;close all%% Set timer%tstart = cputime;% global numdofsSch;% ==========================================% Problem data (Chemistry part of the input)% ==========================================% % Set up Problem Parameters:% fprintf('\n');fprintf('----> Setting up the Problem Parameter . . .\n');fprintf('\n');Zion = 14;qtot = 0.0; % net charge; nonzero for ionic calculationnelec = Zion - qtot;problemdata = struct('Z',Zion, 'nelec',nelec);% % Quadrature%global quadtype;quadtype = 1; % Type of quadrature: 1 = Gauss, 2 = Lobatto% % 1. Quadrature for radial Schrodinger eq.:% nspSch = 53;% nspSch = pSch + 1; % Number of quadrature points/weights; % nsp = p + 1 => diagonal overlap (S)-matrix. This % leads to exact integration of H-matrix but % "underintegrates" S-matrix. Nonetheless, % optimal convergence is preservedquaddatainputSch = struct('quadtype',quadtype, 'nsp',nspSch);% % 2. Quadrature for radial Poisson eq.:% nspPois = nspSch;% nspPois = pPois + 1; % Number of quadrature points/weights; % nsp = p + 1 => diagonal overlap (S)-matrix. This % leads to exact integration of H-matrix but % "underintegrates" S-matrix. Nonetheless, % optimal convergence is preservedquaddatainputPois = struct('quadtype',quadtype, 'nsp',nspPois);% % SCF% maxscfiter = 200;scftol = 1.0E-6;scftolmax = 5.0E-5;mixingparam = 0.5;scfopts = struct('maxiter',maxscfiter, 'scftol',scftol, 'scftolmax',scftolmax, ... 'mixingparam',mixingparam);maxeigiter = 2000;eigtol = 1.0E-10;eigdisp = 0;eigopts = struct('maxit',maxeigiter, 'tol',eigtol, 'disp',eigdisp);% % Domain:% rmin = 0.0;rmax = 10.0;%% Mesh:% domain = [rmin rmax];toplot = 'on';% % 1. Mesh for radial Schrodinger eq.:% meshtypeSch = 1; % type: 1 = uniform, 2 = logarithmic, 3 = exponentialaSch = 10.0;numelSch = 40;pSch = [1:10]';% % 2. Mesh for radial Poisson eq.:% meshtypePois = meshtypeSch; % type: 1 = uniform, 2 = logarithmic, 3 = exponentialaPois = aSch;pPois = pSch;numelPois = numelSch;% % Initialize:% Etotp = zeros(length(pSch),1);Ekinp = zeros(length(pSch),1);Ecoulp = zeros(length(pSch),1);Eenucp = zeros(length(pSch),1); Excp = zeros(length(pSch),1);totdof = zeros(length(pSch),1);for run = 1:length(pSch) meshoptsSch = struct('element_order',pSch(run), 'numel',numelSch, 'domain',domain, ... 'verbose',toplot, 'element_type','spectral');% meshdatainputSch = struct('meshtype',meshtypeSch, 'param',aSch, ... 'meshopts',meshoptsSch); meshoptsPois = struct('element_order',pPois(run), 'numel',numelPois, 'domain',domain, ... 'verbose',toplot, 'element_type','spectral');% meshdatainputPois = struct('meshtype',meshtypePois, 'param',aPois, ... 'meshopts',meshoptsPois); fprintf('\n'); fprintf('Solving Radial DFT problem for Polynomial Order: %-2d \n', pSch(run)); fprintf('\n') [lam, u, scf_converged, Etotp(run), Ekinp(run), Ecoulp(run), Eenucp(run), Excp(run), history, exitflag] = radialdft(problemdata, ... meshdatainputSch, meshdatainputPois, ... quaddatainputSch, quaddatainputPois, ... scfopts, eigopts); totdof(run) = numdofsSch; if exitflag ~= 0 fprintf('\n'); fprintf('Radial Kohn-Sham Equation could not be solved with Polynomial Order: %-2d and %-5d elements \n', pSch(run), numelSch); fprintf('because MATLAB''s eigensolver (eigs) failed! \n'); fprintf('\n'); fprintf('********************************************************* \n'); timetot = cputime - tstart; fprintf('\n'); fprintf(' Total Computing Time so far: %20.3e s \n', timetot); fprintf('\n'); fprintf('********************************************************* \n'); error('p-Convergence Test Aborted!'); else fprintf('\n'); fprintf('Radial Kohn-Sham Equation Successfully solved with Polynomial Order: %-2d and %-5d elements \n', pSch(run), numelSch); fprintf('p-Convergence Test Continuing . . . \n'); fprintf('\n'); endend% % Plot Total energy:% width = 6; % Width in inchesheight = 4; % Height in inchesalw = 5.0; % AxesLineWidthfsz = 18; % Fontsizelw = 2.0; % LineWidthfigureset(gcf,'InvertHardcopy','on');set(gcf,'PaperUnits', 'inches');pos = get(gcf, 'Position');set(gcf, 'Position', [pos(1) pos(2) width*1000, height*1000]);set(gca, 'FontSize', fsz, 'LineWidth', alw);set(get(gca,'xlabel'),'FontSize', fsz, 'FontWeight', 'Bold');set(get(gca,'ylabel'),'FontSize', fsz, 'FontWeight', 'Bold');set(get(gca,'title'),'FontSize', fsz, 'FontWeight', 'Bold');hold on;% plot(totdof, Etotp, 'r', 'LineWidth', 1.25)% xlabel('# of DOF (\equiv # of FE Basis Functions)','Interpreter','tex');ylabel('Converged Total Energy (Ha)','Interpreter','tex');box on; axis square;set(gca,'LineWidth',lw);set(gca,'FontSize',fsz);set(gca,'FontWeight','Bold');set(gcf,'color','w');timetot = cputime - tstart;fprintf('*************************************************************** \n');fprintf('\n');fprintf(' Time to complete the Simulation = %20.3e s\n', timetot);fprintf('\n');fprintf('*************************************************************** \n');- h收敛性分析：
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Purpose% ========% % Script file to test convergence of ground state energy % with h-refinement of Spectral FEM basis functions%clc;clear all;close all%% Set timer%tstart = cputime;% global numdofsSch;% ==========================================% Problem data (Chemistry part of the input)% ==========================================% % Set up Problem Parameters:% fprintf('\n');fprintf('----> Setting up the Problem Parameter . . .\n');fprintf('\n');Zion = 14;qtot = 0.0; % net charge; nonzero for ionic calculationnelec = Zion - qtot;problemdata = struct('Z',Zion, 'nelec',nelec);% % Quadrature%global quadtype;quadtype = 1; % Type of quadrature: 1 = Gauss, 2 = Lobatto% % 1. Quadrature for radial Schrodinger eq.:% nspSch = 53;% nspSch = pSch + 1; % Number of quadrature points/weights; % nsp = p + 1 => diagonal overlap (S)-matrix. This % leads to exact integration of H-matrix but % "underintegrates" S-matrix. Nonetheless, % optimal convergence is preservedquaddatainputSch = struct('quadtype',quadtype, 'nsp',nspSch);% % 2. Quadrature for radial Poisson eq.:% nspPois = nspSch;% nspPois = pPois + 1; % Number of quadrature points/weights; % nsp = p + 1 => diagonal overlap (S)-matrix. This % leads to exact integration of H-matrix but % "underintegrates" S-matrix. Nonetheless, % optimal convergence is preservedquaddatainputPois = struct('quadtype',quadtype, 'nsp',nspPois);% % SCF% maxscfiter = 200;scftol = 1.0E-6;scftolmax = 5.0E-5;mixingparam = 0.7;scfopts = struct('maxiter',maxscfiter, 'scftol',scftol, 'scftolmax',scftolmax, ... 'mixingparam',mixingparam);maxeigiter = 2000;eigtol = 1.0E-10;eigdisp = 0;eigopts = struct('maxit',maxeigiter, 'tol',eigtol, 'disp',eigdisp);% % Domain:% rmin = 0.0;rmax = 10.0;%% Mesh:% domain = [rmin rmax];toplot = 'on';% % 1. Mesh for radial Schrodinger eq.:% meshtypeSch = 1; % type: 1 = uniform, 2 = logarithmic, 3 = exponentialaSch = 10.0;numelSch = 5*(2.^[0:4]);pSch = 10;% % 2. Mesh for radial Poisson eq.:% meshtypePois = meshtypeSch; % type: 1 = uniform, 2 = logarithmic, 3 = exponentialaPois = aSch;pPois = pSch;numelPois = numelSch;% % Initialize:% Etoth = zeros(length(numelSch),1);Ekinh = zeros(length(numelSch),1);Ecoulh = zeros(length(numelSch),1);Eenuch = zeros(length(numelSch),1); Exch = zeros(length(numelSch),1);totdof = zeros(length(numelSch),1); for run = 1:length(numelSch) meshoptsSch = struct('element_order',pSch, 'numel',numelSch(run), 'domain',domain, ... 'verbose',toplot, 'element_type','spectral');% meshdatainputSch = struct('meshtype',meshtypeSch, 'param',aSch, ... 'meshopts',meshoptsSch); meshoptsPois = struct('element_order',pPois, 'numel',numelPois(run), 'domain',domain, ... 'verbose',toplot, 'element_type','spectral');% meshdatainputPois = struct('meshtype',meshtypePois, 'param',aPois, ... 'meshopts',meshoptsPois); fprintf('\n'); fprintf('Solving Radial DFT problem for # of elements = %-5d \n', numelSch(run)); fprintf('\n') [lam, u, scf_converged, Etoth(run), Ekinh(run), Ecoulh(run), Eenuch(run), Exch(run), history, exitflag] = radialdft(problemdata, ... meshdatainputSch, meshdatainputPois, ... quaddatainputSch, quaddatainputPois, ... scfopts, eigopts); totdof(run) = numdofsSch; if exitflag ~= 0 fprintf('\n'); fprintf('Radial Kohn-Sham Equation could not be solved with Polynomial Order: %-2d and %-5d elements \n', pSch, numelSch(run)); fprintf('because MATLAB''s eigensolver (eigs) failed! \n'); fprintf('\n'); timetot = cputime - tstart; fprintf('********************************************************* \n'); fprintf('\n'); fprintf(' Total Computing Time so far: %20.3e s \n', timetot); fprintf('\n'); fprintf('********************************************************* \n'); error('h-Convergence Test Aborted!'); else fprintf('\n'); fprintf('Radial Kohn-Sham Equation Successfully solved with Polynomial Order: %-2d and %-5d elements \n', pSch, numelSch(run)); fprintf('h-Convergence Test Continuing . . . \n'); fprintf('\n'); endend% % Plot Total energy:% width = 6; % Width in inchesheight = 4; % Height in inchesalw = 5.0; % AxesLineWidthfsz = 18; % Fontsizelw = 2.0; % LineWidthfigureset(gcf,'InvertHardcopy','on');set(gcf,'PaperUnits', 'inches');pos = get(gcf, 'Position');set(gcf, 'Position', [pos(1) pos(2) width*1000, height*1000]);set(gca, 'FontSize', fsz, 'LineWidth', alw);set(get(gca,'xlabel'),'FontSize', fsz, 'FontWeight', 'Bold');set(get(gca,'ylabel'),'FontSize', fsz, 'FontWeight', 'Bold');set(get(gca,'title'),'FontSize', fsz, 'FontWeight', 'Bold');hold on;% plot(totdof, Etoth, 'r', 'LineWidth', 1.25)% xlabel('# of DOF (\equiv # of FE Basis Functions)','Interpreter','tex');ylabel('Converged Total Energy (Ha)','Interpreter','tex');box on; axis square;set(gca,'LineWidth',lw);set(gca,'FontSize',fsz);set(gca,'FontWeight','Bold');set(gcf,'color','w');timetot = cputime - tstart;fprintf('*************************************************************** \n');fprintf('\n');fprintf(' Time to complete the Simulation = %20.3e s\n', timetot);fprintf('\n');fprintf('*************************************************************** \n');函数用法简介：
这是一个单原子径向Kohn-Sham方程matlab求解器。 主函数spectatom.m中的变量Zion表示原子序数（1-92）。
spectatom.m中设置网格参数和积分参数后调用radialdft.m函数。网格参数包括uniform/logarithmic/exponential三个选项，积分参数保括Gauss-Legendre/
Gauss-Legendre-Lobatto两个选项。
radialdft.m函数的初始化：电荷密度（每次托马斯费马近似），以及一些在KSscfiteration.m函数用到的参量的初始化。
KSscfiteration.m用于SCF迭代计算，在每一次迭代中，它首先要通过给定的密度解决径向泊松问题（泊松函数求解：solveradialpoisson.m和radialpoissonelemeqns.m），随后，这个解（哈里斯势能）被用于计算哈密顿矩阵。
函数radialschroedingerelemeqns.m和solveradialschroed.m分别构建和解决了径向Kohn-Sham特征值问题（本质上和径向薛定谔问题是一致的）。
被用于构建输出密度的特征对的计算在函数computetotalenergy.m中进行，并且通过计算输入和输出密度的的差距来评估收敛性。如果没有达到这一程度，那么SCF迭代就会继续。
可以尝试改变主函数spectatom.m中的原子序数Zion和函数getatomicstate.m中相应的基元配置（单元数目，形函数阶，网格类型），看看最后的结果有什么不同。
函数主要的输出就是基态能量，原子基态能量的标准数据可以在这个网页中查找：https://www.nist.gov/pml/data/atomic-total-energies-and-eigenvalues-html
参考文献:
Ondrej Certik, et al., `dftatom: A robust and general Schrodinger and
Dirac solver for atomic structure calculations,’ Computer Physics
Communications; Volume 184, Issue 7, July 2013, Pages 1777-1791
DOI: https://doi.org/10.1016/j.cpc.2013.02.014Chao Yang, et al., `KSSOLV - A MATLAB toolbox for solving the Kohn-Sham
equations,’ ACM Transactions on Mathematical Software (TOMS),
TOMS Homepage archive, Volume 36 Issue 2, March 2009, Article No. 10
DOI: http://doi.acm.org/10.1145/1499096.1499099
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