Eigen矩阵运算库

最近一直在做工程上的事情，比较多的使用了Eigen矩阵运算库。

简单说一下Eigen的特点：

(1) 使用方便、无需预编译，调用开销小

(2) 函数丰富，风格有点近似MATLAB，易上手；

(3) 速度中规中矩，比opencv快，比MKL、openBLAS慢；

Eigen3.3版本链接 http://eigen.tuxfamily.org/index.php?title=Main_Page

注：绝大部分使用说明和示例都可以在官网上找到，所以有时候不需要纠结百度到的与实际不符，可以直接官网or谷歌

使用方法很简单：下载Eigen后解压，然后包含解压路径，最后只需要在程序里引用头文件

#include <Eigen/Dense> 

基本使用方法如下：

矩阵定义

#include <Eigen/Dense>Matrix<double, 3, 3> A;               // Fixed rows and cols. Same as Matrix3d.Matrix<double, 3, Dynamic> B;         // Fixed rows, dynamic cols.Matrix<double, Dynamic, Dynamic> C;   // Full dynamic. Same as MatrixXd.Matrix<double, 3, 3, RowMajor> E;     // Row major; default is column-major.Matrix3f P, Q, R;                     // 3x3 float matrix.Vector3f x, y, z;                     // 3x1 float matrix.RowVector3f a, b, c;                  // 1x3 float matrix.VectorXd v;                           // Dynamic column vector of doubles

基本使用方法

// Basic usage// Eigen          // Matlab           // commentsx.size()          // length(x)        // vector sizeC.rows()          // size(C,1)        // number of rowsC.cols()          // size(C,2)        // number of columnsx(i)              // x(i+1)           // Matlab is 1-basedC(i,j)            // C(i+1,j+1)       //A.resize(4, 4);   // Runtime error if assertions are on.B.resize(4, 9);   // Runtime error if assertions are on.A.resize(3, 3);   // Ok; size didn't change.B.resize(3, 9);   // Ok; only dynamic cols changed.A << 1, 2, 3,     // Initialize A. The elements can also be     4, 5, 6,     // matrices, which are stacked along cols     7, 8, 9;     // and then the rows are stacked.B << A, A, A;     // B is three horizontally stacked A's.A.fill(10);       // Fill A with all 10's.

特殊矩阵生成

// Eigen                            // MatlabMatrixXd::Identity(rows,cols)       // eye(rows,cols)C.setIdentity(rows,cols)            // C = eye(rows,cols)MatrixXd::Zero(rows,cols)           // zeros(rows,cols)C.setZero(rows,cols)                // C = ones(rows,cols)MatrixXd::Ones(rows,cols)           // ones(rows,cols)C.setOnes(rows,cols)                // C = ones(rows,cols)MatrixXd::Random(rows,cols)         // rand(rows,cols)*2-1        // MatrixXd::Random returns uniform random numbers in (-1, 1).C.setRandom(rows,cols)              // C = rand(rows,cols)*2-1VectorXd::LinSpaced(size,low,high)   // linspace(low,high,size)'v.setLinSpaced(size,low,high)        // v = linspace(low,high,size)'

矩阵块操作

// Matrix slicing and blocks. All expressions listed here are read/write.// Templated size versions are faster. Note that Matlab is 1-based (a size N// vector is x(1)...x(N)).// Eigen                           // Matlabx.head(n)                          // x(1:n)x.head<n>()                        // x(1:n)x.tail(n)                          // x(end - n + 1: end)x.tail<n>()                        // x(end - n + 1: end)x.segment(i, n)                    // x(i+1 : i+n)x.segment<n>(i)                    // x(i+1 : i+n)P.block(i, j, rows, cols)          // P(i+1 : i+rows, j+1 : j+cols)P.block<rows, cols>(i, j)          // P(i+1 : i+rows, j+1 : j+cols)P.row(i)                           // P(i+1, :)P.col(j)                           // P(:, j+1)P.leftCols<cols>()                 // P(:, 1:cols)P.leftCols(cols)                   // P(:, 1:cols)P.middleCols<cols>(j)              // P(:, j+1:j+cols)P.middleCols(j, cols)              // P(:, j+1:j+cols)P.rightCols<cols>()                // P(:, end-cols+1:end)P.rightCols(cols)                  // P(:, end-cols+1:end)P.topRows<rows>()                  // P(1:rows, :)P.topRows(rows)                    // P(1:rows, :)P.middleRows<rows>(i)              // P(i+1:i+rows, :)P.middleRows(i, rows)              // P(i+1:i+rows, :)P.bottomRows<rows>()               // P(end-rows+1:end, :)P.bottomRows(rows)                 // P(end-rows+1:end, :)P.topLeftCorner(rows, cols)        // P(1:rows, 1:cols)P.topRightCorner(rows, cols)       // P(1:rows, end-cols+1:end)P.bottomLeftCorner(rows, cols)     // P(end-rows+1:end, 1:cols)P.bottomRightCorner(rows, cols)    // P(end-rows+1:end, end-cols+1:end)P.topLeftCorner<rows,cols>()       // P(1:rows, 1:cols)P.topRightCorner<rows,cols>()      // P(1:rows, end-cols+1:end)P.bottomLeftCorner<rows,cols>()    // P(end-rows+1:end, 1:cols)P.bottomRightCorner<rows,cols>()   // P(end-rows+1:end, end-cols+1:end)

矩阵元素交换以及转置等

// Of particular note is Eigen's swap function which is highly optimized.// Eigen                           // MatlabR.row(i) = P.col(j);               // R(i, :) = P(:, i)R.col(j1).swap(mat1.col(j2));      // R(:, [j1 j2]) = R(:, [j2, j1])// Views, transpose, etc; all read-write except for .adjoint().// Eigen                           // MatlabR.adjoint()                        // R'R.transpose()                      // R.' or conj(R')R.diagonal()                       // diag(R)x.asDiagonal()                     // diag(x)R.transpose().colwise().reverse(); // rot90(R)R.conjugate()                      // conj(R)

矩阵四则运算

// All the same as Matlab, but matlab doesn't have *= style operators.// Matrix-vector.  Matrix-matrix.   Matrix-scalar.y  = M*x;          R  = P*Q;        R  = P*s;a  = b*M;          R  = P - Q;      R  = s*P;a *= M;            R  = P + Q;      R  = P/s;                   R *= Q;          R  = s*P;                   R += Q;          R *= s;                   R -= Q;          R /= s;

单个元素操作

// Vectorized operations on each element independently// Eigen                  // MatlabR = P.cwiseProduct(Q);    // R = P .* QR = P.array() * s.array();// R = P .* sR = P.cwiseQuotient(Q);   // R = P ./ QR = P.array() / Q.array();// R = P ./ QR = P.array() + s.array();// R = P + sR = P.array() - s.array();// R = P - sR.array() += s;           // R = R + sR.array() -= s;           // R = R - sR.array() < Q.array();    // R < QR.array() <= Q.array();   // R <= QR.cwiseInverse();         // 1 ./ PR.array().inverse();      // 1 ./ PR.array().sin()           // sin(P)R.array().cos()           // cos(P)R.array().pow(s)          // P .^ sR.array().square()        // P .^ 2R.array().cube()          // P .^ 3R.cwiseSqrt()             // sqrt(P)R.array().sqrt()          // sqrt(P)R.array().exp()           // exp(P)R.array().log()           // log(P)R.cwiseMax(P)             // max(R, P)R.array().max(P.array())  // max(R, P)R.cwiseMin(P)             // min(R, P)R.array().min(P.array())  // min(R, P)R.cwiseAbs()              // abs(P)R.array().abs()           // abs(P)R.cwiseAbs2()             // abs(P.^2)R.array().abs2()          // abs(P.^2)(R.array() < s).select(P,Q);  // (R < s ? P : Q)

矩阵缩减

// Reductions.int r, c;// Eigen                  // MatlabR.minCoeff()              // min(R(:))R.maxCoeff()              // max(R(:))s = R.minCoeff(&r, &c)    // [s, i] = min(R(:)); [r, c] = ind2sub(size(R), i);s = R.maxCoeff(&r, &c)    // [s, i] = max(R(:)); [r, c] = ind2sub(size(R), i);R.sum()                   // sum(R(:))R.colwise().sum()         // sum(R)R.rowwise().sum()         // sum(R, 2) or sum(R')'R.prod()                  // prod(R(:))R.colwise().prod()        // prod(R)R.rowwise().prod()        // prod(R, 2) or prod(R')'R.trace()                 // trace(R)R.all()                   // all(R(:))R.colwise().all()         // all(R)R.rowwise().all()         // all(R, 2)R.any()                   // any(R(:))R.colwise().any()         // any(R)R.rowwise().any()         // any(R, 2)

矩阵点乘及归一化

// Dot products, norms, etc.// Eigen                  // Matlabx.norm()                  // norm(x).    Note that norm(R) doesn't work in Eigen.x.squaredNorm()           // dot(x, x)   Note the equivalence is not true for complexx.dot(y)                  // dot(x, y)x.cross(y)                // cross(x, y) Requires #include <Eigen/Geometry>

矩阵类型转换

//// Type conversion// Eigen                           // MatlabA.cast<double>();                  // double(A)A.cast<float>();                   // single(A)A.cast<int>();                     // int32(A)A.real();                          // real(A)A.imag();                          // imag(A)// if the original type equals destination type, no work is done// Note that for most operations Eigen requires all operands to have the same type:MatrixXf F = MatrixXf::Zero(3,3);A += F;                // illegal in Eigen. In Matlab A = A+F is allowedA += F.cast<double>(); // F converted to double and then added (generally, conversion happens on-the-fly)

内存映射创建矩阵

// Eigen can map existing memory into Eigen matrices.float array[3];Vector3f::Map(array).fill(10);            // create a temporary Map over array and sets entries to 10int data[4] = {1, 2, 3, 4};Matrix2i mat2x2(data);                    // copies data into mat2x2Matrix2i::Map(data) = 2*mat2x2;           // overwrite elements of data with 2*mat2x2MatrixXi::Map(data, 2, 2) += mat2x2;      // adds mat2x2 to elements of data (alternative syntax if size is not know at compile time)

解方程

// Solve Ax = b. Result stored in x. Matlab: x = A \ b.x = A.ldlt().solve(b));  // A sym. p.s.d.    #include <Eigen/Cholesky>x = A.llt() .solve(b));  // A sym. p.d.      #include <Eigen/Cholesky>x = A.lu()  .solve(b));  // Stable and fast. #include <Eigen/LU>x = A.qr()  .solve(b));  // No pivoting.     #include <Eigen/QR>x = A.svd() .solve(b));  // Stable, slowest. #include <Eigen/SVD>// .ldlt() -> .matrixL() and .matrixD()// .llt()  -> .matrixL()// .lu()   -> .matrixL() and .matrixU()// .qr()   -> .matrixQ() and .matrixR()// .svd()  -> .matrixU(), .singularValues(), and .matrixV()

特征值

// Eigenvalue problems// Eigen                          // MatlabA.eigenvalues();                  // eig(A);EigenSolver<Matrix3d> eig(A);     // [vec val] = eig(A)eig.eigenvalues();                // diag(val)eig.eigenvectors();               // vec// For self-adjoint matrices use SelfAdjointEigenSolver<>

求广义逆矩阵
Eigen中并没有求广义逆的函数，这里用SVD实现，数据类型大家可以看需修改为doubel

using Eigen::Dynamic;using Eigen::Matrix;using Eigen::RowMajor;typedef Matrix<float, Dynamic, Dynamic, RowMajor> MatXf;MatXf pinv(MatXf x){    JacobiSVD<MatXf> svd(x,ComputeFullU | ComputeFullV);    float  pinvtoler=1.e-8; //tolerance    MatXf singularValues_inv = svd.singularValues();    for ( long i=0; i<x.cols(); ++i) {        if ( singularValues_inv(i) > pinvtoler )            singularValues_inv(i)=1.0/singularValues_inv(i);        else singularValues_inv(i)=0;    }    return svd.matrixV()*singularValues_inv.asDiagonal()*svd.matrixU().transpose();}