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

¨ä¤¤ $a_1$ ¤Î $a_2$ ¬°½u©Ê°Ñ¼Æ¡A$\lambda_2$ ¤Î $\lambda_2$ ¬°«D½u©Ê°Ñ¼Æ¡A¦]¦¹§Ú­Ì¥i¥H¹ï¥¦­Ì¤À¶}³B²z¡A¦p¤U¡G

¦¹¤èªkªº¦n³B¬O¡G³Ì¤p¥­¤èªk¯à°÷¦b«D½u©Ê°Ñ¼Æ¡]$\lambda_2$ ¤Î $\lambda_2$¡^©T©wªº±¡ªp¤U¡A¤@¦¸§ä¨ì³Ì¦nªº½u©Ê°Ñ¼Æ¡]$a_1$ ¤Î $a_2$¡^ªº­È¡A¦]¬°·j´MªÅ¶¡ªººû«×¥Ñ 4¡]$a_1$¡B$a_2$¡B$\lambda_1$¡B$\lambda_2$¡^¡A­°¬° 2¡]$\lambda_1$¡B$\lambda_2$¡^¡A³Ì¨Î¤Æ·|§ó¦³®Ä²v¡C

¦pªG­n¨Ï¥Î¤W­z²V¦¨¡]Hybrid¡^ªº¤èªk¡A¨ç¦¡ errorMeasure1 ¶·§ï¼g¦¨ errorMeasure2¡A¦p¤U¡G

Example 1: 10-¦±½uÀÀ¦X»P°jÂk¤ÀªR/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);

¨ä¤¤ lambda ¬O«D½u©Ê°Ñ¼Æ¦V¶q¡A¥u¥]§t $\lambda_2$ ¤Î $\lambda_2$¡Adata ¤´¬OÆ[¹î¨ìªº¸ê®ÆÂI¡Aa «h¬O§Q¥Î³Ì¤p¥­¤èªkºâ¥Xªº³Ì¨Î½u©Ê°Ñ¼Æ¦V¶q¡]§Y $a_1$ ¤Î $a_2$ ÀH¶Ç¤Jªº $\lambda_2$ ¤Î $\lambda_2$ ¦ÓÅÜ¡^¡A¦Ó¶Ç¦^ªº squareError ¤´¬OÁ`¥­¤è»~®t¡C¦¹¨ç¼Æ©M¡@errorMeasure1¡@ªº³Ì¤j¤£¦P¦b©ó§Ú­Ì¥u¶·¶Ç¤J½u©Ê°Ñ¼Æ¡]$\lambda_2$ ¤Î $\lambda_2$¡^¡A³Ì¨Î½u©Ê°Ñ¼Æ¡]$a_1$ ¤Î $a_2$¡^ªº­È«h¦b¨ç¦¡¤¤¥Ñ³Ì¤p¥­¤èªk¨M©w¡C

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Example 2: 10-¦±½uÀÀ¦X»P°jÂk¤ÀªR/nonlinearFit02.mload data.txt lambda0 = [0 0]; tic lambda = fminsearch(@(x)errorMeasure2(x, data), lambda0); fprintf('­pºâ®É¶¡ = %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('»~®t¥­¤è©M = %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-¦±½uÀÀ¦X»P°jÂk¤ÀªR\errorMeasure2.m', 6)" style="font-weight:bold">errorMeasure2</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-¦±½uÀÀ¦X»P°jÂk¤ÀªR\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-¦±½uÀÀ¦X»P°jÂk¤ÀªR\nonlinearFit02.m', 4)" style="font-weight:bold">nonlinearFit02</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-¦±½uÀÀ¦X»P°jÂk¤ÀªR\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-¦±½uÀÀ¦X»P°jÂk¤ÀªR\errorMeasure2.m', 6)" style="font-weight:bold">errorMeasure2</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-¦±½uÀÀ¦X»P°jÂk¤ÀªR\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-¦±½uÀÀ¦X»P°jÂk¤ÀªR\nonlinearFit02.m', 4)" style="font-weight:bold">nonlinearFit02</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-¦±½uÀÀ¦X»P°jÂk¤ÀªR\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-¦±½uÀÀ¦X»P°jÂk¤ÀªR\errorMeasure2.m', 6)" style="font-weight:bold">errorMeasure2</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-¦±½uÀÀ¦X»P°jÂk¤ÀªR\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-¦±½uÀÀ¦X»P°jÂk¤ÀªR\nonlinearFit02.m', 4)" style="font-weight:bold">nonlinearFit02</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-¦±½uÀÀ¦X»P°jÂk¤ÀªR\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-¦±½uÀÀ¦X»P°jÂk¤ÀªR\errorMeasure2.m', 6)" style="font-weight:bold">errorMeasure2</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-¦±½uÀÀ¦X»P°jÂk¤ÀªR\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-¦±½uÀÀ¦X»P°jÂk¤ÀªR\nonlinearFit02.m', 4)" style="font-weight:bold">nonlinearFit02</a> (<a href="matlab: opentoline('D:\users\jang\books\matlabProgramming4guru\example\10-¦±½uÀÀ¦X»P°jÂk¤ÀªR\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>)] ­pºâ®É¶¡ = 0.153491 »~®t¥­¤è©M = 1.477226e-01

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