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%% construct a network.net.nIn=1;                              %the input layer has 1 ANN.net.nHidden=10;                         %the hidden has 10 ANN.net.nOut=1;                             %the output layer has 1ANN.w=2*(rand(net.nHidden,net.nIn)-1/2);    %the weight coefficient of the hidden layer.b=2*(rand(net.nHidden,1)-1/2);          %the thresholdnet.w1=[w,b];                           %the weight coefficient and the threshold are linked up.W=2*(rand(net.nOut,net.nHidden)-1/2);   %the weight coefficient of the output layer.B=2*(rand(net.nOut,1)-1/2);             %the thresholdnet.w2=[W,B];                           %the weight coefficient and the threshold are linked up.%% set the parametersmc=0.01;                                %set the momentum termeta=0.001;                              %set the learining ratemaxiter=50000;                          %set the iteration times%% set the training samples.trainIn=[0:pi/4:2*pi];                  %the input of the training samples.trainOut=sin(trainIn);                  %the ouput of the training samples.trainnum=9;                             %the amount of the training samples.SampIn=[trainIn;ones(1,trainnum)];      %the input of the network, and the input of the threshold is a constan 1.expectedOut=trainOut;                   %the expected ouput is the output of the training samples.errRec=zeros(1,maxiter);                %used to store the error of the training output.%% set the testing samplestestIn=[0:pi/180:2*pi];                 %the input of the testing samples.testOut=sin(testIn);                    %the output of the testing sanples.testnum=361;                            %the amount of the testing samples.%% the training procedurefor i=1:maxiter;hid_input=net.w1*SampIn;                %calculate the weighting sum of the hidden layerhid_out=tansig(hid_input);              %calculate the output of the hidden layer.ou_input1=[hid_out;ones(1,trainnum)];   %the input of the output layer, and the input of the threshold is a constan 1.ou_input2=net.w2*ou_input1;             %calculate the weighting sum of the output layer.out_out=2*tansig(ou_input2);            %calculate the output of the output layer.err=expectedOut-out_out;                %caiculate the error vectorsse=sumsqr(err);                        %calculate the square sum of the error.errRec(i)=sse;                          %store the error%% the back-propagation of errorDELTA=err.*dtansig(ou_input2,out_out/2);                    %the gradient of between the hidden layer and the output layerdelta=net.w2(:,1:end-1)'*DELTA.*dtansig(hid_input,hid_out); %the gradient of between the input layer and the hidden layerdWEX=DELTA*ou_input1';                                      %the delta of the weight coefficient of the output layerdwex=delta*SampIn';                                         %the delta of the weight coefficient of the hidden layerif i==1                                                     %if it is the first time to revise the coefficient, we do not use the momentum term    net.w2=net.w2+eta*dWEX;    net.w1=net.w1+eta*dwex;else                                                        %else we use the momentum term.    net.w2=net.w2+(1-mc)*eta*dWEX+mc*dWEXOld;    net.w1=net.w1+(1-mc)*eta*dwex+mc*dwexOld;enddWEXOld=dWEX;                                               %record the delta of the last revisiondwexOld=dwex;end%% the display of the resultssubplot(1,2,1);plot(errRec);                               %plot the errortitle('error curve');xlabel('iteration times');ylabel('error');realIn=[testIn;ones(1,testnum)];            %the input of the testing samplesrealhid_input=net.w1*realIn;                %calculate the weighting sum of the hidden layerrealhid_out=tansig(realhid_input);          %calculate the output of the hidden layer.realou_input1=[realhid_out;ones(1,testnum)];%the input of the output layer, and the input of the threshold is a constan 1.realou_input2=net.w2*realou_input1;         %calculate the weighting sum of the output layer.realout_out=2*tansig(realou_input2);        %calculate the output of the output layer.realerr=testOut-realout_out;                %caiculate the error vectorrealsse=sumsqr(realerr);                    %calculate the square sum of the error.subplot(1,2,2);plot(testIn,realout_out,testIn,sin(testIn));%plot the standard sin and the output of the testing.axis([0 2*pi -1.1 1.1]);                    %set the coordinate range.set(gca,'XTick',pi/4:pi/4:2*pi);grid on;title('the testing output and the standard output');