數(shù)值分析A實驗報告

數(shù)值分析實驗報告工程物理系二〇一六年一月十日

實驗3.1（主元的選取與算法的穩(wěn)定性）問題提出：Gauss消去法是我們在線性代數(shù)中已經(jīng)熟悉的。但由于計算機的數(shù)值運算是在一個有限的浮點數(shù)集合上進(jìn)行的，如何才能確保Gauss消去法作為數(shù)值算法的穩(wěn)定性呢？Gauss消去法從理論算法到數(shù)值算法，其關(guān)鍵是主元的選擇。主元的選擇從數(shù)學(xué)理論上看起來平凡，它卻是數(shù)值分析中十分典型的問題。實驗內(nèi)容：考慮線性方程組編制一個能自動選取主元，又能手動選取主元的求解線性方程組的Gauss消去過程。實驗要求：（1）取矩陣，則方程有解取n=10計算矩陣的條件數(shù)。分別用順序GAUSS消元法、列主元GAUSS消元法、完全主元GAUSS消元法，結(jié)果如何？

（2）現(xiàn)選擇程序中手動選取主元的功能。每步消去過程總選取按模最小或按模盡可能小的元素作為主元，觀察并記錄計算結(jié)果。若每步消去過程總選取按模最大的元素作為主元，結(jié)果又如何？分析實驗的結(jié)果。（3）取矩陣階數(shù)n=20或者更大，重復(fù)上述實驗過程，觀察記錄并分析不同的問題及消去過程中選擇不同的主元時計算結(jié)果的差異，說明主元素的選取在消去過程中的作用。（4）選取其他你感興趣的問題或者隨機生成矩陣，計算其條件數(shù)。重復(fù)上述實驗，觀察記錄并分析實驗結(jié)果。3.1.1程序清單formatlong;n=input('矩陣的階數(shù)：n=');sp_M=input('矩陣的種類（1:Hilbert;2:隨機矩陣；3:本題給出的矩陣；4：幻方矩陣）：sp_M=');switchsp_Mcase(1);A=hilb(n);case(2);A=round(8*rand(n));case(3);A=6*diag(ones(1,n),0)+8*diag(ones(1,n-1),-1)+diag(ones(1,n-1),1);case(4);A=magic(n);end;b=A*ones(n,1);p=input('計算條件數(shù)的p-范數(shù)，p=');cond_A=cond(A,p)Any1=zeros(1,n);Any20=zeros(n,1);Any21=zeros(n,1);Any12=eye(n);[m,n]=size(A);Ab=[Ab];Pro=input('計算方法（1：順序高斯消元法；2,：列主元高斯消元法；3：完全主元高斯消元法；4：手動選主元法，Pro=');Abfori=1:n-1switchProcase(1);case(2);[aii,ip]=max(abs(Ab(i:n,i)));ip=ip+i-1;Any1=Ab(ip,:);Ab(ip,:)=Ab(i,:);Ab(i,:)=Any1;case(3);[Y,I]=max(max(abs(Ab(i:n,i:n))));%顯示最大值列號I=I+i-1;[x1,r]=max(max(abs(Ab(i:n,i:n)')));%顯示最大值行號r=r+i-1;Any2=Ab(:,I);Ab(:,I)=Ab(:,i);Ab(:,i)=Any2;%Ab陣I列與i列互換Any1=Ab(r,:);Ab(r,:)=Ab(i,:);Ab(i,:)=Any1;%Ab陣r行與i行互換Any21=Any12(:,I);Any12(:,I)=Any12(:,i);Any12(:,i)=Any21;%列交換跟蹤case(4);ip=input(['第',num2str(i),'步消元，請輸入第',num2str(i),'列所選元素所處行數(shù)：']);Any1=Ab(ip,:);Ab(ip,:)=Ab(i,:);Ab(i,:)=Any1;end;aii=Ab(i,i);fork=i+1:nif(aii~=0)Ab(k,i:n+1)=Ab(k,i:n+1)-(Ab(k,i)/aii)*Ab(i,i:n+1);elsebreak;end;end;Abend;x=zeros(n,1);x(n)=Ab(n,n+1)/Ab(n,n);fori=n-1:-1:1if(Pro==3)x(i)=(Ab(i,n+1)-Ab(i,i+1:n)*x(i+1:n))/Ab(i,i);x=Any12^-1*x;elsex(i)=(Ab(i,n+1)-Ab(i,i+1:n)*x(i+1:n))/Ab(i,i);end;endx3.1.2實驗結(jié)果及分析（1）Cond(A,1)=2.557500000000000×103Cond(A,2)=1.727556024913821×103Cond(A,inf)=2.557500000000000×103未知數(shù)順序高斯消元法列主元高斯消元法完全主元高斯消元法x11.00000000000000011x21.00000000000000011x31.00000000000000011x41.00000000000000111x50.99999999999999811x61.00000000000000411x70.99999999999999311x81.00000000000001211x90.99999999999997911x101.00000000000002811(2)手動選取主元，n=10未知數(shù)主元模最小或模盡可能小主元模最最大x11.0000000000000001x21.0000000000000001x31.0000000000000001x41.0000000000000011x50.9999999999999981x61.0000000000000041x70.9999999999999931x81.0000000000000121x90.9999999999999791x101.0000000000000281結(jié)果分析：由計算結(jié)果可知，主元取模最小結(jié)果沒有主元取模最大結(jié)果好。由于最大主元與最小主元都沒有過于小，都是相對好的一個數(shù)，所以結(jié)果差距很小。造成這樣結(jié)果的原因是使用較小的數(shù)作為除數(shù)計算結(jié)果的誤差會被放大。（3）n=20Cond(A,1)=2.621437500000000×106Cond(A,2)=1.789670565812014×106Cond(A,inf)=2.621437500000001×106未知數(shù)順序高斯消元法列主元高斯消元法完全主元高斯消元法手動主元模最小或模盡可能小手動主元模最最大x11.000000000000000111.0000000000000001x21.000000000000000111.0000000000000001x31.000000000000000111.0000000000000001x41.000000000000001111.0000000000000011x50.999999999999998110.9999999999999981x61.000000000000004111.0000000000000041x70.999999999999993110.9999999999999931x81.000000000000014111.0000000000000141x90.999999999999972110.9999999999999721x101.000000000000057111.0000000000000571x110.999999999999886110.9999999999998861x121.000000000000227111.0000000000002271x130.999999999999547110.9999999999995471x141.000000000000902111.0000000000009021x150.999999999998209110.9999999999982091x161.000000000003524111.0000000000035241x170.999999999993179110.9999999999931791x181.000000000012732111.0000000000127321x190.999999999978173110.9999999999781731x201.000000000029102111.0000000000291021計算結(jié)果與二相比表明：1.選取模最大的元素作為主元會產(chǎn)生為精確的結(jié)果。2.在選取模最小元素作為主元時條件數(shù)越大誤差越大。這是可以由攝動理論中事后誤差估計是看出來的。（4）1、10階Hilbert矩陣中各種方法解的情況Cond(A,1)=3.535336877175068×1013Cond(A,2)=1.602517099202631×1013Cond(A,inf)=3.535336877175066×1013未知數(shù)手動主元模最小或模盡可能小手動主元模最最大x10.9999999984611550.999999998758705x21.0000001311885301.000000106500618x30.9999972342961560.999997743217252x41.0000249388387821.000020435292621x50.9998818352853170.999902835789981x61.0003230379745661.000266409823714x70.9994724788979130.999563856087119x81.0005077323000321.000420698156409x90.9997343797724790.999779492550673x101.0000582339522601.0000484246488472、10階隨機矩陣中各種方法解的情況Cond(A,1)=70.699729258533296Cond(A,2)=1.581788216637637×102Cond(A,inf)=1.358379386423256×102未知數(shù)手動主元模最小或模盡可能小手動主元模最最大x10.9999999999999860.999999999999998x21.0000000000000381.000000000000005x30.9999999999999620.999999999999996x41.0000000000000281.000000000000007x50.9999999999999720.999999999999999x61.0000000000000071.000000000000000x71.0000000000000140.999999999999993x81.0000000000000060.999999999999998x90.9999999999999681.000000000000002x101.0000000000000301.0000000000000033、10階幻方矩陣中各種方法解的情況Cond(A,1)=8.043128074178017×1017Cond(A,2)=7.670050592487186×1017Cond(A,inf)=8.129276476140978×1017未知數(shù)手動主元模最小或模盡可能小手動主元模最大x11.0000000122802001x20.9999999550964281x31.0000000962946981x40.9999999927228721x51.0000001239341051x60.9999999229185381x71.0000000309500471x80.9999999927228721x91.0000000007631181x101.0000000122802001一般來說，模最大元素作為主元比模最小的元素作為主元時的計算結(jié)果更精確。但一些方陣，如幻方矩陣，則是選擇模最小的元素作為主元時計算結(jié)果最精確（選模最小的元素只是一個表象，這種選主元方法優(yōu)于其他選主元方法的本質(zhì)是這種選擇方法能使消去過程不產(chǎn)生浮點數(shù)，而全是整數(shù)運算，只有在回代過程中才有可能會產(chǎn)生浮點數(shù)）。一般來說，需按模最大元素作為主元精度比較高。

實驗3.3（病態(tài)的線性方程組的求解）問題提出：理論的分析表明，求解病態(tài)的線性方程組是困難的。實際情況是否如此，會出現(xiàn)怎樣的現(xiàn)象呢？實驗內(nèi)容：考慮方程組Hx=b的求解，其中系數(shù)矩陣H為Hilbert矩陣，這是一個著名的病態(tài)問題。通過首先給定解（例如取為各個分量均為1）再計算出右端b的辦法給出確定的問題。實驗要求：（1）選擇問題的維數(shù)為6，分別用Gauss消去法、J迭代法、GS迭代法和SOR迭代法求解方程組，其各自的結(jié)果如何？將計算結(jié)果與問題的解比較，結(jié)論如何？（2）逐步增大問題的維數(shù)，仍然用上述的方法來解它們，計算的結(jié)果如何？計算的結(jié)果說明了什么？（3）討論病態(tài)問題求解的算法3.3.1程序清單formatlong;n=input('矩陣的階數(shù)：n=');Hibert_n=ones(n,n);forr=1:1:nforc=1:1:nHibert_n(r,c)=1/(r+c-1);end;end;b=Hibert_n*ones(n,1);Hibert_nb=[Hibert_nb];D=zeros(n,n);L=zeros(n,n);U=zeros(n,n);fori=2:1:nL(i,1:i-1)=-1.*Hibert_n(i,1:i-1);end;fori=1:1:n-1U(i,i+1:n)=-1.*Hibert_n(i,i+1:n);end;fori=1:1:nD(i,i)=Hibert_n(i,i);end;x=zeros(n,1);Medthod_number=input('計算方法（1：高斯消去法；2:Jacobi迭代；3：GS迭代；4：SOR迭代，Medthod_number=');Hibert_nbswitchMedthod_numbercase(1);fori=1:n-1Hibert_nbii=Hibert_nb(i,i);fork=i+1:nif(Hibert_nbii~=0)Hibert_nb(k,i:n+1)=Hibert_nb(k,i:n+1)-(Hibert_nb(k,i)/Hibert_nbii)*Hibert_nb(i,i:n+1);elsebreak;end;end;end;x=zeros(n,1);x(n)=Hibert_nb(n,n+1)/Hibert_nb(n,n);fori=n-1:-1:1x(i)=(Hibert_nb(i,n+1)-Hibert_nb(i,i+1:n)*x(i+1:n))/Hibert_nb(i,i);end;xcase(2);num_of_iter=0;norm_errorv=2;whilenorm_errorv>=10^-6xtemp=x;num_of_iter_temp=num_of_iter;x=D^-1*(L+U)*xtemp+D^-1*b;error_vector=x-xtemp;norm_errorv=norm(error_vector);num_of_iter=num_of_iter_temp+1;end;num_of_iterxcase(3);num_of_iter=0;norm_errorv=2;whilenorm_errorv>=10^-6xtemp=x;num_of_iter_temp=num_of_iter;x=(D-L)^-1*U*xtemp+(D-L)^-1*b;error_vector=x-xtemp;norm_errorv=norm(error_vector);num_of_iter=num_of_iter_temp+1;end;num_of_iterxcase(4);I=eye(n);B=I-D^-1*Hibert_n;spe_rB=max(abs(eig(B)))Wopt=1.5;num_of_iter=0;norm_errorv=2;Lw=(D-Wopt*L)^-1*((1-Wopt)*D+Wopt*U)whilenorm_errorv>=10^-6xtemp=x;num_of_iter_temp=num_of_iter;x=(D-Wopt*L)^-1*((1-Wopt)*D+Wopt*U)*xtemp+Wopt*(D-Wopt*L)^-1*b;error_vector=x-xtemp;norm_errorv=norm(error_vector);num_of_iter=num_of_iter_temp+1;end;Woptnum_of_iterxend;3.3.2運行結(jié)果及簡要分析（1）n=6未知數(shù)GaussJGSSORx10.999999999999228Inf0.9999306341337850.999998116326860x21.000000000021937Inf1.0009194791104210.999536557238050x30.999999999851792NaN0.9981016368189121.005197221859359x41.000000000385370NaN0.9973780771021130.982933473818400x50.999999999574584NaN1.0089019779771601.021622489348798x61.000000000167680NaN0.9947049210396540.990659946952854迭代次數(shù)——4871740616769由結(jié)果知GAUSS法的結(jié)果最為精確。以error<=10^-6為收斂標(biāo)準(zhǔn)。GS法和SOR法收斂，但收斂的速度比較慢，SOR法略快于GS法。J法是發(fā)散的。（2）6階未知數(shù)GaussJGSSORx10.999999999999228Inf0.9999306341337850.999998116326860x21.000000000021937Inf1.0009194791104210.999536557238050x30.999999999851792NaN0.9981016368189121.005197221859359x41.000000000385370NaN0.9973780771021130.982933473818400x50.999999999574584NaN1.0089019779771601.021622489348798x61.000000000167680NaN0.9947049210396540.990659946952854迭代次數(shù)——48717406167697階未知數(shù)GaussJGSSORx10.999999999994453Inf0.9999993847606911.000077419518485x21.000000000221598NaN0.9994111921692410.997675627422646x30.999999997864368NaN1.0054159211198081.015125014152487x41.000000008303334NaN0.9869439523676270.965402133950665x50.999999984778069NaN1.0058756746177651.026302560980251x61.000000013152188NaN1.0106338440546081.004148021358175x70.999999995682024NaN0.9916438432417530.991184902946059迭代次數(shù)——4331174992368階未知數(shù)GaussJGSSORx10.999999999966269-Inf1.0001010618346241.000110641227852x21.000000001809060NaN0.9974178826668250.997590430173341x30.999999976372675NaN1.0136096522484571.011909590979345x41.000000127868104NaN0.9794442972837040.981162475678398x50.999999655764117NaN0.9981715055106731.004262765594196x61.000000487042162NaN1.0140860365352361.005452790879320x70.999999653427125NaN1.0100862161756011.007936908023023x81.000000097774747NaN0.9869559021244590.991507742993205迭代次數(shù)——3968342379219階未知數(shù)GaussJGSSORx10.999999999760212-Inf1.0001599663510921.000073440013547x21.000000016452158-Inf0.9969056403632590.998669263275490x30.999999722685967-Inf1.0126928847973041.004744337452915x41.000001973450782-Inf0.9875795363182760.998078248613115x50.999992779505688-Inf0.9915767143208420.991737673628083x61.000014713209384-Inf1.0049764504074231.002354905151437x70.999983130886114NaN1.0115911901907401.006475193208506x81.000010174801268NaN1.0059753549458001.004999791122969x90.999997489086183NaN0.9884243695012710.992808128443414迭代次數(shù)——368215713867510階未知數(shù)GaussJGSSORx10.999999998754834Inf1.0001099892711001.000037307041751x21.000000106784973Inf0.9981823981941720.999664538362142x30.999997737861476Inf1.0056174964955880.998580945511472x41.000020479418515NaN0.9993251807758511.010945178465891x50.999902641847339NaN0.9915579706762920.985033582889975x61.000266907013337NaN0.9967960737850350.999312905811790x70.999563088470270NaN1.0052445079606501.001998084480765x81.000421401157600NaN1.0087341883658521.008630071839024x90.999779140796212NaN1.0038903039624241.003331412484509x101.000048498721852NaN0.9904417404459950.992405941672389迭代次數(shù)——347269513217411階未知數(shù)GaussJGSSORx10.999999994751196-Inf1.0000678423903170.999999366078665x21.000000546746356-Inf0.9992080278903001.000670734390764x30.999985868343700-Inf1.0003082345229980.992687197515483x41.000157549468624NaN1.0069212532444701.022009340799829x50.999063537004343NaN0.9934019559184000.981143191862049x61.003286333127798NaN0.9923264105133340.996870387749579x70.992855789229368NaN0.9995569274820530.996749665623868x81.009726486881558NaN1.0065982879089851.008306329410864x90.991930155925812NaN1.0083329414393731.007566335246938x101.003729850349020NaN1.0028836723136931.003440352977447x110.999263885025643NaN0.9902968578591940.990482851656876迭代次數(shù)——330250402616312階未知數(shù)GaussJGSSORx10.999999978015381-Inf1.0000259509399550.999954703394399x21.000002718658690-Inf1.0001743738870221.001785735264775x30.999916137983244-Inf0.9956523480373680.986627540764427x41.001124917423389-Inf1.0125908158347851.032053329265755x50.991858974446319NaN0.9961501109955520.979136034412255x61.035384713191155NaN0.9898767778078420.995292945294574x70.902314812813458NaN0.9947282800474210.991506439526719x81.175419475669440NaN1.0028964799507191.005967134258715x90.795767756284983NaN1.0084196081275921.008142963002572x101.148662570420705NaN1.0083434372718081.009201716248544x110.938525472294121NaN1.0018282708181941.001354831964326x121.011022484909254NaN0.9892065937798180.988879057012070迭代次數(shù)——316218012099613階未知數(shù)GaussJGSSORx11.000000100845780-Inf0.9999797589992360.999898394014506x20.999984107801401-Inf1.0011764560567461.003102724057934x31.000614076745963-Inf0.9911882524521380.979985275134039x40.989763393555848-Inf1.0171284922885411.041822199898047x51.091975046430242-Inf0.9994937504545630.978339148728908x60.501003423820652-Inf0.9888375948008110.994598624657415x72.740851874285036-Inf0.9907570860497240.986468393979083x8-3.036058361310595-Inf0.9986697541967931.002570102424027x97.283884268701900-Inf1.0063066472730771.006486292294313x10-5.493529661418352-Inf1.0099777355518151.011620410991678x115.270874315267408-Inf1.0081295929919911.008018704221022x12-0.618191404610789NaN1.0005391576898431.000316599383520x131.268828909804817NaN0.9876890626527210.986641262097305迭代次數(shù)——304187201714914階未知數(shù)GaussJGSSORx11.000000076434261-Inf0.9999254280475640.999821850368462x20.999989977372466-Inf1.0022742081668451.004719127001512x31.000309640223686-Inf0.9867031890927140.972637303241281x40.996220152145736-Inf1.0208247183007161.051064686323259x51.019948388320157-Inf1.0032765441908230.978784942485267x60.981737772434851-Inf0.9889493525394300.994964408389401x70.675197450814358-Inf0.9876706993474990.982038216196169x82.922896201795558-Inf0.9944160645575090.998668726968105x9-4.498100214715922NaN1.0029870007784921.003296483598423x1010.515964827807839NaN1.0092161216653711.011498111734201x11-9.428207690148314NaN1.0109778536801801.011357832701018x128.094762876614597NaN1.0075545030208791.007823563006471x13-1.740964386133816NaN0.9990553739301920.998296502569765x141.460245042732823NaN0.9860139144292810.984858091437531迭代次數(shù)——294161951466915階未知數(shù)GaussJGSSORx10.999821850368462-Inf0.9998575571816410.999887622919398x21.004719127001512NaN1.0035255617953491.002415708045205x30.972637303241281NaN0.9821110358594410.988810342812522x41.051064686323259NaN1.0236479722255971.014030721059258x50.978784942485267NaN1.0074096037750861.002548533950694x60.994964408389401NaN0.9901678030952660.998016571956734x70.982038216196169NaN0.9855812613031880.991839105511616x80.998668726968105NaN0.9904621531699090.994102989105642x91.003296483598423NaN0.9990129280149990.997668442512087x101.011498111734201NaN1.0068192011970671.002777831762777x111.011357832701018NaN1.0113120959450541.006058415843252x121.007823563006471NaN1.0113368439412851.007066865308874x130.998296502569765NaN1.0066439110125291.004799747956600x140.984858091437531NaN0.9974992219972080.999304390919186x150.999821850368462NaN0.9844255314424750.990589637285929迭代次數(shù)——286144985007516階未知數(shù)GaussJGSSORx11.000000038309608-Inf0.9998154791644970.999912245138576x20.999992356721026-Inf1.0037679427425541.001674705174921x31.000358126460529NaN0.9838633778807720.993777266155035x40.992915973395254NaN1.0162269207347921.002826246938333x51.074614485124896NaN1.0101717951463951.009452652207939x60.529206540133373NaN0.9964068272273280.999442700448315x72.896111260937623NaN0.9891272502238620.995130123070590x8-4.015764564087196NaN0.9895886176802570.992776438860727x99.670665346779543NaN0.9947899305784160.995497169367632x10-8.180072412591157NaN1.0013820248397070.999350896127904x115.338569007974306NaN1.0068625014596781.003550249866065x123.212675695583656NaN1.0097038543794491.006125996340453x13-3.852719521183188NaN1.0091571637045981.006509290332898x144.216873886870227NaN1.0050100426552411.004108083739434x15-0.004602922381234NaN0.9973850721265090.998887412656845x161.121176658583945NaN0.9865937795648230.990897211484367迭代次數(shù)——2782633351324矩陣階數(shù)678910111213141516GS1740611749379212157126951250402180118720161951449826333SOR16769923683423867532174261632099617149146695007551324結(jié)果分析：1、從6階一直算到16階，6到12階GAUSS法精度最高，從12階開始誤差急劇增大，從13階以后GAUSS法結(jié)果誤差很大。2、J法一直不收斂，原因是迭代矩陣譜半徑大于1。3、GS法和SOR（w=1.5）法一直收斂，但收斂速度較慢。4、SOR法在大多數(shù)情況下收斂速度比GS法慢。（3）病態(tài)問題求解對原方程進(jìn)行適當(dāng)變換，取非奇異矩陣P,Q使得（PAQ）y=Pb,原方程的解x=Qy。用此種方法的目的是使矩陣PAQ的條件數(shù)cond(PAQ)<=cond(A),即使得條件數(shù)有所改善，對于1~100階Hilbert矩陣一般選擇對角陣D1=D2=D,設(shè)DM為Hilbert矩陣的對角陣。設(shè)使得cond(DHD)最小的一種。這樣條件數(shù)的大小就會被改善。

4.1非線性方程組的解法問題提出：非線性方程組的求解方法很多，基本的思想是線性化。不同的方法效果如何，要靠計算的實踐來分析、比較。

實驗內(nèi)容：考慮算法

(1)牛頓法

(2)擬牛頓法

分別編寫它們的matlab程序。

實驗要求：

(1)用上述方法，分別計算兩個例子。在達(dá)到精度相同的前提下，比較迭代次數(shù)、浮點運算次數(shù)和CPU時間等。(2)取其他初值結(jié)果又如何？反復(fù)選取不同的初值比較其結(jié)果。（3）總結(jié)歸納你的實驗結(jié)果，試說明各種方法適用的問題4.1.1程序清單*f1.m文件functiony=f1(x)y(1)=12*x(1)-x(2)^2-4*x(3)-7;y(2)=x(1)^2+10*x(2)-x(3)-11;y(3)=x(2)^3+10*x(3)-8;*ff1.m文件functiony=ff1(x)y(1,:)=[12,-2*x(2),-4];y(2,:)=[2*x(1),10,-1];y(3,:)=[0,3*x(2)^2,10];*f2.m文件functiony=f2(x)y(1)=3*x(1)-cos(x(2)*x(3))-0.5;y(2)=x(1)^2-81*(x(2)+0.1)^2+sin(x(3))+1.06;y(3)=exp(-1*x(2)*x(1))+20*x(3)+1/3*(10*pi-3);*ff2.m文件functiony=ff2%f2的雅克比矩陣y(1,:)=[3,sin(x(2)*x(3)*x(3)),sin(x(2)*x(3))*x(2)];y(2,:)=[2*x(1),162*(x(2)+0.1),cos(x(3))];y(3,:)=[exp(-1*x(2)*x(1))*(-1*x(2)),exp(-1*x(2)*x(1))*(-1*x(1)),20];以函數(shù)（1）為例牛頓法x10=[1,1,1]';x20=[0,0,0]';n=15;i=1;increx=[1,1,1]';x(:,1)=x10;incx=increx;tic; while(norm(incx)>10^(-6))&(i<n)incx=-ff1(x(:,1))\f1(x(:,1))';x(:,1)=x(:,1)+incx;i=i+1;end;iterations=i-1t=tocx(:,1)擬牛頓法x10=[1,1,1]';x20=[0,0,0]';n=15;i=1;increx=[1,1,1]';x(:,1)=x10;incx=increx;A=ff1(x(:,1));F=f1(x(:,1));tic;while(norm(incx)>10^(-6))&(i<n)xt=x(:,1);yt=f1(x(:,1));x(:,1)=x(:,1)-A\f1(x(:,1))';incx=x(:,1)-xt;incy=f1(x(:,1))'-yt';A=A+(incy-A*incx)*incx'/(incx'*incx);i=i+1;end;itera