## 10-4 ?炵??ц看姝革?浣跨敤娣锋?娉?

竊┮磞瓃よ猭琌┦よ猭ウр癸┮Τ把计常琌跌く场跌獶絬┦把计琌и临秈˙э▆碞琌盢絬┦籔獶絬┦把计だ秨ノぃよ猭ㄓ矪瞶ㄒτē计厩家

$$y= a_1 e^{\lambda_1 x} + a_2 e^{\lambda_2 x}$$

ㄤい $a_1$ の $a_2$ 絬┦把计$\lambda_2$ の $\lambda_2$ 獶絬┦把计и癸ウだ秨矪瞶

• 絬┦把计$a_1$ の $a_2$ㄏノ程キよ猭オ埃┪\
• 獶絬┦把计$\lambda_2$ の $\lambda_2$ㄏノ Simplex ℡Α穓碝猭 fminsearch

よ猭矪琌程キよ猭镑獶絬┦把计$\lambda_2$ の $\lambda_2$㏕﹚薄猵Ωт程絬┦把计$a_1$ の $a_2$穓碝丁蝴パ 4$a_1$$a_2$$\lambda_1$$\lambda_2$ 2$\lambda_1$$\lambda_2$程ㄎて穦Τ瞯

狦璶ㄏノ瓃睼ΘHybridよ猭ㄧΑ errorMeasure1 斗э糶Θ errorMeasure2

Example 1: 10-Ρ絬览籔癹耴だ猂/errorMeasure2.mfunction squaredError = errorMeasure2(lambda, data) if nargin<1; return; end x = data(:,1); y = data(:,2); A = [exp(lambda(1)*x) exp(lambda(2)*x)]; a = A\y; y2 = a(1)*exp(lambda(1)*x)+a(2)*exp(lambda(2)*x); squaredError = sum((y-y2).^2);

ㄤい lambda 琌獶絬┦把计秖 $\lambda_2$ の $\lambda_2$data ご琌芠诡戈翴a 玥琌ノ程キよ猭衡程ㄎ絬┦把计秖 $a_1$ の $a_2$ 繦肚 $\lambda_2$ の $\lambda_2$ τ跑τ肚 squareError ご琌羆キよ粇畉ㄧ计㎝errorMeasure1程ぃи斗肚絬┦把计$\lambda_2$ の $\lambda_2$程ㄎ絬┦把计$a_1$ の $a_2$玥ㄧΑいパ程キよ猭∕﹚

Example 2: 10-Ρ絬览籔癹耴だ猂/nonlinearFit02.mload data.txt lambda0 = [0 0]; tic lambda = fminsearch(@(x)errorMeasure2(x, data), lambda0); fprintf('璸衡丁 = %g\n', toc); x = data(:, 1); y = data(:, 2); A = [exp(lambda(1)*x) exp(lambda(2)*x)]; a = A\y; y2 = A*a; plot(x, y, 'ro', x, y2, 'b-'); legend('Sample data', 'Regression curve'); fprintf('粇畉キよ㎝ = %d\n', sum((y-y2).^2));[Warning: Rank deficient, rank = 1, tol = 2.136826e-14.] [> In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('errorMeasure2', 'D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\errorMeasure2.m', 6)" style="font-weight:bold">errorMeasure2</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\errorMeasure2.m',6,0)">line 6</a>) In %4287963446>@(x)errorMeasure2(x,data) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('fminsearch', 'C:\Program Files\MATLAB\R2015a\toolbox\matlab\optimfun\fminsearch.p', 190)" style="font-weight:bold">fminsearch</a> (<a href="matlab: opentoline('C:\Program Files\MATLAB\R2015a\toolbox\matlab\optimfun\fminsearch.p',190,0)">line 190</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('nonlinearFit02', 'D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\nonlinearFit02.m', 4)" style="font-weight:bold">nonlinearFit02</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\nonlinearFit02.m',4,0)">line 4</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('goWriteOutputFile>dummyFunction', 'd:\users\jang\books\goWriteOutputFile.m', 85)" style="font-weight:bold">goWriteOutputFile>dummyFunction</a> (<a href="matlab: opentoline('d:\users\jang\books\goWriteOutputFile.m',85,0)">line 85</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('goWriteOutputFile', 'd:\users\jang\books\goWriteOutputFile.m', 55)" style="font-weight:bold">goWriteOutputFile</a> (<a href="matlab: opentoline('d:\users\jang\books\goWriteOutputFile.m',55,0)">line 55</a>)] [Warning: Rank deficient, rank = 1, tol = 2.137360e-14.] [> In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('errorMeasure2', 'D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\errorMeasure2.m', 6)" style="font-weight:bold">errorMeasure2</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\errorMeasure2.m',6,0)">line 6</a>) In %4287963446>@(x)errorMeasure2(x,data) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('fminsearch', 'C:\Program Files\MATLAB\R2015a\toolbox\matlab\optimfun\fminsearch.p', 309)" style="font-weight:bold">fminsearch</a> (<a href="matlab: opentoline('C:\Program Files\MATLAB\R2015a\toolbox\matlab\optimfun\fminsearch.p',309,0)">line 309</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('nonlinearFit02', 'D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\nonlinearFit02.m', 4)" style="font-weight:bold">nonlinearFit02</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\nonlinearFit02.m',4,0)">line 4</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('goWriteOutputFile>dummyFunction', 'd:\users\jang\books\goWriteOutputFile.m', 85)" style="font-weight:bold">goWriteOutputFile>dummyFunction</a> (<a href="matlab: opentoline('d:\users\jang\books\goWriteOutputFile.m',85,0)">line 85</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('goWriteOutputFile', 'd:\users\jang\books\goWriteOutputFile.m', 55)" style="font-weight:bold">goWriteOutputFile</a> (<a href="matlab: opentoline('d:\users\jang\books\goWriteOutputFile.m',55,0)">line 55</a>)] [Warning: Rank deficient, rank = 1, tol = 2.136960e-14.] [> In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('errorMeasure2', 'D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\errorMeasure2.m', 6)" style="font-weight:bold">errorMeasure2</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\errorMeasure2.m',6,0)">line 6</a>) In %4287963446>@(x)errorMeasure2(x,data) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('fminsearch', 'C:\Program Files\MATLAB\R2015a\toolbox\matlab\optimfun\fminsearch.p', 350)" style="font-weight:bold">fminsearch</a> (<a href="matlab: opentoline('C:\Program Files\MATLAB\R2015a\toolbox\matlab\optimfun\fminsearch.p',350,0)">line 350</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('nonlinearFit02', 'D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\nonlinearFit02.m', 4)" style="font-weight:bold">nonlinearFit02</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\nonlinearFit02.m',4,0)">line 4</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('goWriteOutputFile>dummyFunction', 'd:\users\jang\books\goWriteOutputFile.m', 85)" style="font-weight:bold">goWriteOutputFile>dummyFunction</a> (<a href="matlab: opentoline('d:\users\jang\books\goWriteOutputFile.m',85,0)">line 85</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('goWriteOutputFile', 'd:\users\jang\books\goWriteOutputFile.m', 55)" style="font-weight:bold">goWriteOutputFile</a> (<a href="matlab: opentoline('d:\users\jang\books\goWriteOutputFile.m',55,0)">line 55</a>)] [Warning: Rank deficient, rank = 1, tol = 2.137093e-14.] [> In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('errorMeasure2', 'D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\errorMeasure2.m', 6)" style="font-weight:bold">errorMeasure2</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\errorMeasure2.m',6,0)">line 6</a>) In %4287963446>@(x)errorMeasure2(x,data) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('fminsearch', 'C:\Program Files\MATLAB\R2015a\toolbox\matlab\optimfun\fminsearch.p', 365)" style="font-weight:bold">fminsearch</a> (<a href="matlab: opentoline('C:\Program Files\MATLAB\R2015a\toolbox\matlab\optimfun\fminsearch.p',365,0)">line 365</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('nonlinearFit02', 'D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\nonlinearFit02.m', 4)" style="font-weight:bold">nonlinearFit02</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-Ρ絬览籔癹耴だ猂\nonlinearFit02.m',4,0)">line 4</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('goWriteOutputFile>dummyFunction', 'd:\users\jang\books\goWriteOutputFile.m', 85)" style="font-weight:bold">goWriteOutputFile>dummyFunction</a> (<a href="matlab: opentoline('d:\users\jang\books\goWriteOutputFile.m',85,0)">line 85</a>) In <a href="matlab:matlab.internal.language.introspective.errorDocCallback('goWriteOutputFile', 'd:\users\jang\books\goWriteOutputFile.m', 55)" style="font-weight:bold">goWriteOutputFile</a> (<a href="matlab: opentoline('d:\users\jang\books\goWriteOutputFile.m',55,0)">line 55</a>)] 璸衡丁 = 0.153491 粇畉キよ㎝ = 1.477226e-01

贺睼Θ猭玻ネ耕粇畉キよ㎝┮惠璸衡丁ゑ耕祏

Hint
• ㄒいMATLAB 穦玻ネ牡癟硂琌パ程ㄎて筁祘い瞷 $\mathbf{A}$  rank ㄤ︽计旧璓 $a=\mathbf{A}$\$y$ 瞷 Rank deficient 牡癟璶程и浪跌览瓜眔骸種挡狦┛菠硂程ㄎて筁祘┮玻ネ既牡癟
• 饼浪跌瓃癹耴筁祘笆礶陪ボ MATLAB 跌怠块 fitdemo

MATLAB祘Α砞璸秈顶絞