第四章MATLAB的數(shù)值計(jì)算功能Chapter4:NumericalcomputationofMATLAB一、多項(xiàng)式(Polynomial)`1．多項(xiàng)式的表達(dá)與創(chuàng)建(ExpressionandCreatingofpolynomial)(1)多項(xiàng)式的表達(dá)（expressionofpolynomial）_Matlab用行矢量表達(dá)多項(xiàng)式系數(shù)(Coefficient)，各元素按變量的降冪順序排列，如多項(xiàng)式為：P(x)=a0xn+a1xn-1+a2xn-2…an-1x+an則其系數(shù)矢量(Vectorofcoefficient)為：P=[a0a1…an-1an]如將根矢量(Vectorofroot)表示為：ar=[ar1ar2…arn]則根矢量與系數(shù)矢量之間關(guān)系為：（x-ar1）(x-ar2)…(x-arn)=a0xn+a1xn-1+a2xn-2…an-1x+an(2)多項(xiàng)式的創(chuàng)建（polynomialcreating）a)系數(shù)矢量的直接輸入法利用poly2sym函數(shù)直接輸入多項(xiàng)式的系數(shù)矢量，就可方便的建立符號(hào)形式的多項(xiàng)式。例：創(chuàng)建多項(xiàng)式x3-4x2+3x+2poly2sym([1-432])ans=x^3-4*x^2+3*x+2POLYConvertrootstopolynomial.POLY(A),whenAisanNbyNmatrix,isarowvectorwithN+1elementswhicharethecoefficientsofthecharacteristicpolynomial,DET(lambda*EYE(SIZE(A))-A).POLY(V),whenVisavector,isavectorwhoseelementsarethecoefficientsofthepolynomialwhoserootsaretheelementsofV.Forvectors,ROOTSandPOLYareinversefunctionsofeachother,uptoordering,scaling,androundofferror.b)由根矢量創(chuàng)建多項(xiàng)式通過(guò)調(diào)用函數(shù)p=poly(ar)產(chǎn)生多項(xiàng)式的系數(shù)矢量,再利用poly2sym函數(shù)就可方便的建立符號(hào)形式的多項(xiàng)式。注：（1）根矢量元素為n，則多項(xiàng)式系數(shù)矢量元素為n+1；（2）函數(shù)poly2sym(pa)把多項(xiàng)式系數(shù)矢量表達(dá)成符號(hào)形式的多項(xiàng)式，缺省情況下自變量符號(hào)為x，可以指定自變量。（3）使用簡(jiǎn)單繪圖函數(shù)ezplot可以直接繪制符號(hào)形式多項(xiàng)式的曲線(xiàn)。例1：由根矢量創(chuàng)建多項(xiàng)式。將多項(xiàng)式(x-6)(x-3)(x-8)表示為系數(shù)形式a=[638]%根矢量pa=poly(a)%求系數(shù)矢量ppa=poly2sym(pa)%以符號(hào)形式表示原多項(xiàng)式ezplot(ppa,[-50,50])pa=1-1790-144ppa=x^3-17*x^2+90*x-144注：含復(fù)數(shù)根的根矢量所創(chuàng)建的多項(xiàng)式要注意：（1）要形成實(shí)系數(shù)多項(xiàng)式，根矢量中的復(fù)數(shù)根必須共軛成對(duì)；（2）含復(fù)數(shù)根的根矢量所創(chuàng)建的多項(xiàng)式系數(shù)矢量中，可能帶有很小的虛部，此時(shí)可采用取實(shí)部的命令(real)把虛部濾掉。進(jìn)行多項(xiàng)式的求根運(yùn)算時(shí)，有兩種方法，一是直接調(diào)用求根函數(shù)roots，poly和roots互為逆函數(shù)。另一種是先把多項(xiàng)式轉(zhuǎn)化為伴隨矩陣，然后再求其特征值，該特征值即是多項(xiàng)式的根。例3：由給定復(fù)數(shù)根矢量求多項(xiàng)式系數(shù)矢量。r=[-0.5-0.3+0.4i-0.3-0.4i];p=poly(r)pr=real(p)ppr=poly2sym(pr)p=1.00001.10000.55000.1250pr=1.00001.10000.55000.1250ppr=x^3+11/10*x^2+11/20*x+1/8c)特征多項(xiàng)式輸入法用poly函數(shù)可實(shí)現(xiàn)由矩陣的特征多項(xiàng)式系數(shù)創(chuàng)建多項(xiàng)式。條件：特征多項(xiàng)式系數(shù)矢量的第一個(gè)元素必須為一。例2：求三階方陣A的特征多項(xiàng)式系數(shù)，并轉(zhuǎn)換為多項(xiàng)式形式。a=[638;756;135]Pa=poly(a)%求矩陣的特征多項(xiàng)式系數(shù)矢量Ppa=poly2sym(pa)Pa=1.0000-16.000038.0000-83.0000Ppa=x^3-17*x^2+90*x-144注：n階方陣的特征多項(xiàng)式系數(shù)矢量一定是n+1階的。注：（1）要形成實(shí)系數(shù)多項(xiàng)式，根矢量中的復(fù)數(shù)根必須共軛成對(duì)；（2）含復(fù)數(shù)根的根矢量所創(chuàng)建的多項(xiàng)式系數(shù)矢量中，可能帶有很小的虛部，此時(shí)可采用取實(shí)部的命令(real)把虛部濾掉。進(jìn)行多項(xiàng)式的求根運(yùn)算時(shí)，有兩種方法，一是直接調(diào)用求根函數(shù)roots，poly和roots互為逆函數(shù)。另一種是先把多項(xiàng)式轉(zhuǎn)化為伴隨矩陣，然后再求其特征值，該特征值即是多項(xiàng)式的根。例4：將多項(xiàng)式的系數(shù)表示形式轉(zhuǎn)換為根表現(xiàn)形式。求x3-6x2-72x-27的根a=[1-6-72-27]r=roots(a)r=12.1229-5.7345-0.3884MATLAB約定，多項(xiàng)式系數(shù)矢量用行矢量表示，根矢量用列矢量表示。>>1.多項(xiàng)式的乘除運(yùn)算(Multiplicationanddivisionofpolynomial)多項(xiàng)式乘法用函數(shù)conv(a,b)實(shí)現(xiàn)，除法用函數(shù)deconv(a,b)實(shí)現(xiàn)。例1：a(s)=s2+2s+3,b(s)=4s2+5s+6,計(jì)算a(s)與b(s)的乘積。a=[123];b=[456];c=conv(a,b)cs=poly2sym(c,’s’)c=413282718cs=4*s^4+13*s^3+28*s^2+27*s+18例2：展開(kāi)(s2+2s+2)(s+4)(s+1)（多個(gè)多項(xiàng)式相乘）c=conv([1,2,2],conv([1,4],[1,1]))cs=poly2sym(c,’s’)%（指定變量為s）c=1716188cs=s^4+7*s^3+16*s^2+18*s+8例2：求多項(xiàng)式s^4+7*s^3+16*s^2+18*s+8分別被(s+4),(s+3)除后的結(jié)果。c=[1716188];[q1,r1]=deconv(c,[1,4])%q—商矢量，r—余數(shù)矢量[q2,r2]=deconv(c,[1,3])cc=conv(q2,[1,3])%對(duì)除(s+3)結(jié)果檢驗(yàn)test=((c-r2)==cc)q1=1342r1=00000q2=1446r2=000000--10cc=1771618818test=11111111.其他常用的的多項(xiàng)式運(yùn)運(yùn)算命令(Otheercoomputtatiooncoommanndoffpollynommial))pa=polyyval((p,s))按數(shù)組運(yùn)運(yùn)算規(guī)則計(jì)計(jì)算給定ss時(shí)多項(xiàng)式式p的值。pm=polyyvalmm(p,ss)按矩陣運(yùn)運(yùn)算規(guī)則計(jì)計(jì)算給定ss時(shí)多項(xiàng)式式p的值。[r,p,k]]=ressiduee(b,aa)部分分式式展開(kāi)，bb,a分別別是分子分分母多項(xiàng)式式系數(shù)矢量量，r,pp,k分別別是留數(shù)、極極點(diǎn)和直項(xiàng)項(xiàng)矢量p=polyffit(xx,y,nn)用n階多項(xiàng)式式擬合x(chóng)，y矢量給定定的數(shù)據(jù)。polyderr(p)多項(xiàng)式微微分。注：對(duì)于多項(xiàng)項(xiàng)式b(ss)與不重重根的n階多項(xiàng)式式a(s))之比，其其部分分式式展開(kāi)為：：式中：p1,pp2,…,pn稱(chēng)為極點(diǎn)(polles)，r1,r2,…,rn稱(chēng)為留數(shù)(ressiduees)，k(s))稱(chēng)為直項(xiàng)(dirrecttermms)，假如a((s)含有有m重根pj，則相應(yīng)部部分應(yīng)寫(xiě)成成：RESIDUEEParrtiall-fraactioonexxpanssion(ressiduees).[R,,P,K]]=RRESIDDUE(BB,A)finddsthhereesiduues,poleesannddiirecttterrmoffappartiialffracttionexpaansioonofftheerattioooftwwopoolynoomiallsB((s)/AA(s)..Iftherrearrenoomulltipllerooots,,B(s)R(11)RR(2)R(n))-----=----------+----------+....++----------+K(ss)A(ss)s-P(1))s-P(2))s-P((n)VectorssBaandAAspeecifyytheecoeefficcienttsofftheenummerattoraandddenomminattorppolynnomiaalsiindeescenndinggpowwersofss.TTherresidduesareretturneedinntheecollumnvecttorRR,thhepoolellocattionssincoluumnvvectoorP,,anddtheedirrecttermmsinnrowwvecctorK.TThennumbeeroffpollesiisn=leengthh(A)--1=lenggth(RR)=lenggth(PP).TTheddirecctteermcoeffficiientvecttoriisemmptyifllengtth(B))<llengtth(A)),ottherwwiselenngth((K)==lenngth((B)-llengtth(A))+1.IfP(j))=....==P(jj+m-11)issappoleofmmultppliciitymm,thhenttheeexpannsionninccludeesteermsoftthefformR(j)RR(j+11)R((j+m--1)---------+---------------++....+---------------s-P(jj)(s--P(jj))^22(s-P((j))^^m[B,A]==RESSIDUEE(R,PP,K),,witth3inpuutarrgumeentsand2ouutputtarggumennts,convvertsstheeparrtiallfraactioonexxpanssionbackktothepolyynomiialswithhcoeefficcienttsinnBaandAA.例3：對(duì)(3x4+2x33+5x2+4x++6)/((x5+3x4+4x3+2x2+7x++2)做部分分分式展開(kāi)a=[13342722];b=[322546];[r,s,k]]=ressiduee(b,aa)r=1.12274++1.11513ii1.12274--1.11513ii-0.02232--0.00722ii-0.02232++0.00722ii0.79916s=-1.76680++1.22673ii-1.76680--1.22673ii0.41176++1.11130ii0.41176--1.11130ii-0.29991k=[](分分母階數(shù)高高于分子階階數(shù)時(shí)，kk將是空矩矩陣，表示示無(wú)此項(xiàng))例5：對(duì)一組實(shí)驗(yàn)數(shù)據(jù)據(jù)進(jìn)行多項(xiàng)項(xiàng)式最小二二乘擬合(leasstsqquareefitt)x=[122345];%實(shí)驗(yàn)數(shù)據(jù)據(jù)y=[5.543..11128290..74498.44];p=polyffit(xx,y,33)%%做三階多多項(xiàng)式擬合合x(chóng)2=1:.11:5;y2=polyyval((p,x22);%%根據(jù)給定定值計(jì)算多多項(xiàng)式結(jié)果果plot(x,,y,’o’,x2,,y2)二、線(xiàn)性代數(shù)((LineearAAlgebbra)解線(xiàn)性方程(LLineaareqquatiion)就是找出出是否存在在一個(gè)唯一一的矩陣xx，使得a,bb滿(mǎn)足關(guān)系系：ax=b或或xa==bMALAB中xx=a\bb是方程程ax==b的解，x=b//a是方程程式xa==b的解。通常線(xiàn)性方程多多寫(xiě)成axx=b，“\”較多用，兩兩者的關(guān)系系為：(b/a)’==(a’\b’)系數(shù)矩陣a可能能是m行n列的，有有三種情況況：*方陣系統(tǒng):((Squaaremmatriix)m=n可求出精精確解（aa必須是非非奇異(nonnsinggularr)，即滿(mǎn)秩(fulllraank)）*超定系統(tǒng)：(OOverddeterrminddsysstem))m>n可求出最最小二乘解解*欠定系統(tǒng)：(UUnderrdeteerminndsyystemm)m<n可嘗試找找出含有最最少m個(gè)基解或或最小范數(shù)數(shù)解MATLAB對(duì)對(duì)不同形式式的參數(shù)矩矩陣，采用用不同的運(yùn)運(yùn)算法則來(lái)來(lái)處理，它它會(huì)自動(dòng)檢檢測(cè)參數(shù)矩矩陣，以區(qū)區(qū)別下面幾幾種形式：：*三角矩陣(TrrianggularrMattrix))*對(duì)稱(chēng)正定矩陣((symmmetriicalposiitiveedettermiinedmatrrix)*非奇異方陣(NNonsiingullarmmatriix)*超定系統(tǒng)(Ovverdeetermmindsysttem)*欠定系統(tǒng)(Unnderddeterrminddsysstem))方陣系統(tǒng)：(SSquarrearrray))最常見(jiàn)的是系數(shù)數(shù)矩陣為方方陣a，常數(shù)項(xiàng)項(xiàng)b為列矢量量,其解x可寫(xiě)成x==a\b,,x和b大小相同同。例1：求方陣系統(tǒng)的根根。a=[11677;5139;17188]b=[16134]’x=a\ba=1167511391718b=16134x=3.997635.44455-8.66303例2：假如a,b為為兩個(gè)大小小相同的矩矩陣，求方方陣系統(tǒng)的的根。a=[4559;;18195;144133]b=[155122;31519;;76110]x=a\bC=a*xa=4591811951413b=15123115197610x=-3.66750-00.733332.997083.7725011.46667-2.11292-0.3325000.066671.11958C=1.0000055.0000012.000003.00000155.0000019.000007.000066.0000010.00000若方陣a的各個(gè)個(gè)行矢量線(xiàn)線(xiàn)性相關(guān)(linnearcorrrelattion))，則稱(chēng)方方陣a為奇異矩矩陣。這時(shí)時(shí)線(xiàn)性方程程將有無(wú)窮窮多組解。若若方陣是奇奇異矩陣，則則反斜線(xiàn)運(yùn)運(yùn)算因子將將發(fā)出警告告信息。2．超定系統(tǒng)(OOverddeterrminddsysstem))實(shí)驗(yàn)數(shù)據(jù)較多，尋尋求他們的的曲線(xiàn)擬合合。如在t內(nèi)測(cè)得一一組數(shù)據(jù)yy：tyy0.00..820.30..720.80..630.601.60..552.20..50這些數(shù)據(jù)顯然有有衰減指數(shù)數(shù)趨勢(shì)：y((t)~cc1+c2e-t此方程意為y矢矢量可以由由兩個(gè)矢量量逐步逼近近而得，一一個(gè)是單行行的常數(shù)矢矢量，一個(gè)個(gè)是由指數(shù)數(shù)e-t項(xiàng)構(gòu)成成，兩個(gè)參參數(shù)c1和c2可用最小小二乘法求求得，它們們表示實(shí)驗(yàn)驗(yàn)數(shù)據(jù)與方方程y(tt)~c11+c2e-t之間距距離的最小小平方和。例1：求上述數(shù)據(jù)的最最小二乘解解。將數(shù)據(jù)據(jù)帶入方程程式y(tǒng)(tt)~c11+c2e-t中，可可得到含有有兩個(gè)未知知數(shù)的6個(gè)等式，可可寫(xiě)成6行2列的矩陣陣e.t=[000.30.81.111..622.2]’;y=[0.8220..720.6330..600.5550..50]’;e=[oness(sizze(t)))eexp(--t)]%求6個(gè)y(t))方程的系系數(shù)矩陣c=e\y%%求方程的的解e=1.0000011.000001.0000000.740081.0000000.449931.0000000.332291.0000000.201191.0000000.11008c=0.447440.3434帶入方程得：yy(t)~~0.47744+00.34334e-tt用此方程可繪制制曲線(xiàn)：t=[000.30.81.111..622.2]’;y=[0.8220..720.6330..600.5550..50]’;t1=[0:00.1:22.5]’;y1==[onees(siize(tt1)),,exp((-t1))]*cplot(t11,y1,,’b’,t,yy,’ro’)如果一個(gè)矩陣的的行矢量是是線(xiàn)性相關(guān)關(guān)的，則它它的最小二二乘解并不不唯一，因因此，a\\b運(yùn)算將將給出警告告，并產(chǎn)生生含有最少少元素的基基解。3.欠定系統(tǒng)統(tǒng):(Undeerdettermiindssysteem)欠定系統(tǒng)為線(xiàn)性性相關(guān)系統(tǒng)統(tǒng)，其解都都不唯一，MATLAB會(huì)計(jì)算一組構(gòu)成通解的基解，而方程的特解則用QR分解法決定。兩種解法：最少少元素解aa\b，最最小范數(shù)解解pinvv(a)**b.例：用兩種方方法求解欠欠定系統(tǒng)。對(duì)a和矢量b分別用用a\b和pinvv(a)**b求解:a=[1111;;11--1]b=[106]’p=a\bq=pinv((a)*bba=111111-1b=106p=8.0000002.00000q=4.000004.000002.00000三．逆矩陣及行列式式(Reveersaandddeterrminaantoofmaatrixx)1．方陣的逆和行列列式(Reveersaandddeterrminaantoofsqquareemattrix))若a是方陣，且為非非奇異陣，則則方程axx=I和xa==I有相同同的解X。X稱(chēng)為a的逆矩陣陣，記做aa-1，在MATTLAB中中用inv函數(shù)來(lái)計(jì)計(jì)算矩陣的的逆。計(jì)算算方陣的行行列式則用用det函數(shù)數(shù)。DETDeteerminnant..DET(X)isttheddeterrminaantoofthhesqquareemattrixX.UUseCCONDinstteadofDDETttoteestfformmatriixsiingullaritty.INVMatrrixiinverrse.INV(X)isttheiinverrseoofthhesqquareemattrixX.AwaarninngmeessaggeisspriinteddifXissbaddlysscaleedorrnearrlysinggularr.例：計(jì)算方陣的的行列式和和逆矩陣。a=[3--311;-35-2;11-221]];b=[14135;55112;;611455];d1=det((a)x1=inv((a)d2=det((b)x2=inv((b)d1=1x1=1.0000011.000001.000001.0000022.000003.000001.0000033.000006.00000d2=-13551x2=0.11207-00.00337-0.11118-0.00348-00.029960.11058-0.0047400.087730.003772．廣義逆矩陣（偽偽逆）(Geneeraliizedinveersematrrix)一般非方陣無(wú)逆逆矩陣和行行列式，方方程ax==I和xa=II至少有一一個(gè)無(wú)解，這這種矩陣可可以求得特特殊的逆矩矩陣，成為為廣義逆矩矩陣(gennerallizeddinvverseemattrix))（或偽逆pseeudoiinverrse）。矩陣陣amn存在廣廣義逆矩陣陣xnm，使得得ax=IImn，MATLLAB用pinvv函數(shù)來(lái)計(jì)計(jì)算廣義逆逆矩陣。例：計(jì)算廣義逆矩矩陣。a=[8114;113;;96]x=pinv((a)b=x*ac=a*xd=c*a%%d=a**x*a==ae=x*c%%e=x**a*x==xa=81141396x=-0.00661-00.040020.117430.1104500.04006-0.00974b=1.00000-00.00000-0.0000011.00000c=0.9933400.247720.003170.2247200.08117-0.111770.00317-00.117770.99849d=8.00000144.000001.0000033.000009.0000066.00000e=-0.00661-00.040020.117430.104500.04006-0.00974PINVPseuudoinnversse.X=PINNV(A))prooduceesamatrrixXXofthesameedimmensiionsasAA'soothaatA**X*A=A,,X*AA*X==XaandAA*XaandXX*AaareHHermiitiann.ThhecoomputtatioonissbassedoonSVVD(A))anddanyysinngulaarvaaluesslesssthhanaatollerannceaarettreattedaaszeero.TheedeffaultttollerannceiisMAAX(SIIZE(AA))**NORRM(A))*EEPS.PINV(A,,TOL))useesthhetooleraanceTOLinstteadofttheddefauult.四．矩陣分解(Maatrixxdeccompoositiion)MATLAB求求解線(xiàn)性方方程的過(guò)程程基于三種種分解法則則：（１）Choleesky分分解，針對(duì)對(duì)對(duì)稱(chēng)正定定矩陣；（２）高斯消元元法，針對(duì)一一般矩陣；；（３）正交化，針對(duì)一一般矩陣（行行數(shù)≠列數(shù)）這三種分解運(yùn)算算分別由cchol,,lu和qr三個(gè)函函數(shù)來(lái)分解解.Choleskky分解(CholleskyyDeccompoositiion)僅適用于對(duì)稱(chēng)和和上三角矩矩陣?yán)篶holeesky分分解。a=pascaal(6))b=chol((a)a=11111112345613611015221141022035556151533570122616215561262552b=1111110123450013611000014110000015000000011CHOLCholleskyyfacctoriizatiion.CHOL(X))useesonnlyttheddiagoonalanduppeertrrianggleoofX..Theelowwerttrianngulaarissasssumeddtobetthe((compplexconjjugatte)ttranssposeeoftheuppeer.IfXXisposiitiveedeffinitte,tthenR=CHOLL(X)prodducessanuppeertrrianggularrRsoothaatR''*R==X.IfXXisnotposiitiveedeffinitte,aanerrrormesssageisprinnted..[R,p]==CHOOL(X)),wiithttwoooutpuutarrgumeents,,nevverpproduucesanerrormmessaage.IfXisspossitivvedeefiniite,thennpiis0andRisstheesammeassaboove.BuutiffXiisnootpoositiiveddefinnite,,theenpisaapossitivveinntegeer.WheenXisffull,,Riisannuppperttrianngulaarmaatrixxofordeerq=p--1sothattR'**R=X(1::q,1::q).WhennXiisspparsee,Risaanupppertriaangullarmmatriixoffsizzeq--by-nnsothatttheeL-sshapeedreegionnofthefirsstqrowssanddfirrstqqcollumnssofR'*RRagrreewwiththosseoffX.2．LU分解(LLUfaactorrizattion)).用lu函數(shù)完成LLU分解，將將矩陣分解解為上、下下兩個(gè)三角角陣，其調(diào)調(diào)用格式為為：[l,u]=llu(a))l代表下下三角陣，u代表上三角陣。例：LU分解。a=[472422;11440;300388411][l,u]=llu(a))a=47224221144403033841l=1.00000000.2234011.0000000.6638300.590091.00000u=47.00000244.0000022.000000388.38330-5.114890030.00000LULUffactoorizaationn.[L,U]==LU((X)sstoreesannuppperttrianngulaarmaatrixxinUannda"psyychollogiccallyylowwerttrianngulaarmaatrixx"(ii.e.aprroducctofflowwerttrianngulaaranndpeermuttatioonmaatricces)inLL,soothaatX=L**U.XXcannberecttanguular..[L,U,P]]=LLU(X))retturnssuniitloowertriaangullarmmatriixL,,uppperttrianngulaarmaatrixxU,andpermmutattionmatrrixPPsothattP**X=L*U..3．QR分解(OOrthoogonaal-trrianggularrdeccompoositiion).函數(shù)調(diào)用格式：：[q,rr]=qrr(a),,q代表表正規(guī)正交交矩陣，rr代表三角角形矩陣。原原始陣a不必一定定是方陣。如如果矩陣aa是m×n階的的，則矩陣陣q是m×m階的的，矩陣rr是m×n階的的。例：QR分解..A=[22462020;30364644;339884552]];[q,r]=qqr(A))q=-0.44082-00.72009-0.55601-0.55566-00.289980.77786-0.7723600.62996-0.22829r=-53.88981-444.60227--66.33289-344.101140-388.556640.55823-255.909970011.88800222.48996QROrthhogonnal-ttrianngulaardeecompposittion..[Q,R]==QR((A)pproduucesanuupperrtriianguularmatrrixRRofthesameedimmensiionaasAandaunnitarrymaatrixxQssothhatAA=QQ*R.[Q,R,E]]=QQR(A))prooduceesapermmutattionmatrrixEE,annupppertriianguularRanndaunittaryQsoothaatA**E=Q*R..ThhecoolumnnperrmutaationnEiischhosennsothattabss(diaag(R)))issdeccreassing..[Q,R]==QR((A,0))prooduceesthhe"eeconoomyssize""deccompoositiion.IfAAism-byy-nwwithm>n,tthenonlyytheefirrstnncollumnssofQarrecoomputted.4.特征值與特特征矢量(Eigeenvalluesandeigeenvecctorss).MATLAB中中使用函數(shù)數(shù)eig計(jì)算算特征值和和特征矢量量，有兩種種調(diào)用方法法：*e=eig(aa),其其中e是包含特特征值的矢矢量；*[v,d]=eiig(a)),其中v是一個(gè)與與a相同的n××n階矩陣陣，它的每每一列是矩矩陣a的一個(gè)特特征值所對(duì)對(duì)應(yīng)的特征征矢量，dd為對(duì)角陣陣，其對(duì)角角元素即為為矩陣a的特征值值。例：計(jì)算特征值值和特征矢矢量。a=[342515;18359;44122199]e=eig(aa)[v,d]=eeig(aa)a=3422515183359412219e=68.5506615.55122-6.00187v=-0.66227-00.44009-0.33105-0.4496900.67886-0.00717-0.66044-00.587750.99479d=68.55066000155.51222000-6.00187EIGEigeenvalluesandeigeenvecctorss.E=EIGG(X)isaavecctorconttainiingttheeeigennvaluuesoofasquaaremmatriixX..[V,D]==EIGG(X)prodducessaddiagoonalmatrrixDDofeigeenvalluesandafuullmmatriixVwhossecoolumnnsarrethhecoorressponddingeigeenvecctorsssothattX*VV=VV*D.[V,D]==EIGG(X,''nobaalancce')perfformsstheecommputaationnwitthbaalanccingdissableed,wwhichhsommetimmesggivessmorreacccuraaterresulltsfforccertaainprooblemmswiithuunusuualsscaliing.IfXXissymmmetriic,EEIG(XX,'noobalaance'')isignooredsincceXisaalreaadybbalannced..5.奇異值分解.((Sinngulaarvaaluedecoompossitioon).如存在兩個(gè)矢量量u,v及一一常數(shù)c,使得矩陣陣A滿(mǎn)足：Avv=cu,,A’u=cvv稱(chēng)c為奇異值，稱(chēng)uu,v為奇奇異矢量。將奇異值寫(xiě)寫(xiě)成對(duì)角方方陣∑，而相對(duì)對(duì)應(yīng)的奇異異矢量作為為列矢量則則可寫(xiě)成兩兩個(gè)正交矩矩陣U，V，使得：AV=UU∑，A‘U=V∑因?yàn)閁，V正交，所所以可得奇奇異值表達(dá)達(dá)式：A=U∑V’’。一個(gè)m行n列的的矩陣A經(jīng)奇異值值分解，可可求得m行m列的U,m行n列的矩陣陣∑和n行n列的矩陣陣V.。奇異值分解用ssvd函數(shù)數(shù)實(shí)現(xiàn)，調(diào)調(diào)用格式為為；[u,s,v]]=svdd(a)SVDSinggularrvallueddecommposiitionn.[U,S,V]]=SSVD(XX)prroduccesaadiaagonaalmaatrixxS,oftthessamedimeensioonassXaandwwithnonnnegattivediaggonalleleementtsinndeccreassingordeer,aanduunitaarymmatriicesUanndVsotthatX=U*S**V'.S=SVDD(X)retuurnsaveectorrconntainningthesinggularrvallues..[U,S,V]]=SSVD(XX,0)prodducessthee"ecconommysiize"decoompossitioon.IIfXismm-by--nwiithmm>nn,thhenoonlythefirsstncoluumnsofUUareecommputeedanndSisnn-by--n.例：奇異值分解。a=[855;73;446]];[u,s,v]]=svdd(a)%s為奇異值值對(duì)角方陣陣u=-0.66841-00.18226-0.77061-0.55407-00.522280.66591-0.4489500.832270.22589s=13.776490033.0866500v=-0.88148-00.57997-0.5579700.81448五．?dāng)?shù)據(jù)分析(DaataAAnalyyaia))MATLAB對(duì)對(duì)數(shù)據(jù)分析析有兩條約約定：（1）若輸入量X是矢矢量，則不不論是行矢矢量還是列列矢量，運(yùn)運(yùn)算是對(duì)整整個(gè)矢量進(jìn)進(jìn)行的；（2）若輸入量X是是數(shù)組，（或或稱(chēng)矩陣），則則命令運(yùn)算算是按列進(jìn)進(jìn)行的。即即默認(rèn)每個(gè)個(gè)列是有一一個(gè)變量的的不同“觀(guān)察“所得的數(shù)數(shù)據(jù)組成。1.基本統(tǒng)計(jì)命命令（表4-1）例：做各種基本統(tǒng)計(jì)計(jì)運(yùn)算。A=[5-110-660;22633-3;;-955-10011;;-221700-199;-16-444]Amax=maax(A))%找A各列的最最大元素Amin=miin(A))%找A各列的最最小元素Amed=meediann(A)%找A各列的中中位元素Amean=mmean((A)%找A各列的平平均值A(chǔ)std=sttd(A))%求A各列的標(biāo)標(biāo)準(zhǔn)差A(yù)prod=pprod((A)%求A各列元素素的積Asum=suum(A))%求A各列元素素的和S=cumsuum(A))%求A各列元素素的累積和和P=cumprrod(AA)%求A各列元素素的累積j積I=sort((A)%使A的各列元元素按遞增增排列A=5-110-60263--3-95-10111-221170-119-16-44Amax=51173111Amin=-22-110-10-119Amed=-16-40Amean==-5.0000044.80000-3.44000-11.40000Astd=10.8839799.628815.00794111.14990Aprod==-19880--30600000Asum=-25224-17--7S=5-110-607--4-3--3-21-138-24118-13-111-25224-17--7P=5-110--60110-660-1180-990-30001880019880-5100000-19880--30600000I=-22-110-10-119-95-6--3-16-40260451173111>>求矩陣元素的最最大值、最最小值可用用：Amax=maax(maaxA)))或Amaax=maax(A((:))，Amin=mmin(mmin(AA))或或Amiin=miin(A((:))2．協(xié)方差陣和相相關(guān)陣(CovaarianncemmatriixanndCoorrellatiooncooeffiiciennts).（表4—2）例：計(jì)算協(xié)方差和相相關(guān)陣。x=rand((10,33);y=rand((10,33);cx=cov((x)%求協(xié)方差差陣cy=cov((y)cxy=covv(x,yy)%求兩隨機(jī)機(jī)變量的協(xié)協(xié)方差px=corrrcoeff(x)%求相關(guān)陣陣pxy=corrrcoeef(x,,y)%求兩隨機(jī)機(jī)變量的（2×2）相關(guān)系系數(shù)cx=0.00483-00.006660.00146-0.0006600.028830.001540.0014600.015540.00978cy=0.1117700.00773-0.001270.0007300.02339-0.00230-0.00127-00.023300.00772cxy=0.0055000.002230.0002300.06997px=1.00000-00.178830.22118-0.1178311.000000.229340.2211800.293341.00000pxy=1.0000000.037720.0037211.00000COVCoovariianceemattrix..COV(X),,ifXissavvectoor,rreturrnstthevvariaance..Foormaatricces,wherreeaachrrowiisannobsservaationn,anndeaachccolummnavariiablee,COOV(X))isthecovaarianncemmatriix.DIAGG(COVV(X)))isaveectorrofvariianceesfooreaachccolummn,aandSSQRT((DIAGG(COVV(X))))issavvectooroffstaandarrddeeviattionss.COV(X,YY),wwhereeXaandYYareevecctorssofequaalleengthh,issequuivallenttoCCOV([[X(:))Y(::)])..COV(X)orCCOV(XX,Y)normmalizzesbby(NN-1)wherreNistthennumbeeroffobsservaationns.ThissmakkesCCOV(XX)thhebeestuunbiaasedestiimateeofthecovariianceemattrixifttheoobserrvatiionsarefrommannormaaldiistriibutiion.CORRCOEEFCoorrellatiooncooeffiiciennts.R=CORRCCOEF((X)ccalcuulateesamatrrixRRofcorrrelattioncoeffficiientssforranarraayX,,inwhiccheaachrrowiisannobsservaationnanddeacchcoolumnnisavaariabble.R=CORRCCOEF((X,Y)),whhereXanndYarecoluumnvvectoors,istthessameasR=CORRCCOEF(([XYY]).IfCCisthecovaarianncemmatriix,CC=CCOV(XX),tthenCORRRCOEFF(X)istthemmatriixwhhose(i,jj)'thheleementtisC(i,,j)/SSQRT((C(i,,i)*CC(j,jj)).2.微分與梯梯度(Diffferennceaandaapprooximaatedderivvativve,graadiennt).(表4—3))例1：按列求微分。x=[1,100,20;;2,122,23;;3,144,26;;3,166,29]]d=diff((x)%求一階微微分x=111020211223311426311629d=1231230223例2：對(duì)于（u=x22+y2和Δ2=4）求5點(diǎn)差分。[x,y]=mmeshggrid((-4:44,-3::3);u=x.^2++y.^22v4=4*deel2(uu)%求m×n階矩矩陣U的五點(diǎn)差差分矩陣u=25181331099101331825520138854458813200171055211255101771694410014491661710552112551017720138854458813200251813310991013318255v4=444444444444444444444444444444444444444444444444444444444444444MESHGRIIDXanndYarraaysffor33-Dpplotss.[X,Y]=MEESHGRRID(xx,y)trannsforrmsttheddomaiinsppeciffiedbyvvectoorsxaandyyinttoarrrayssXaandYYthaatcaanbeeuseedfoorthheevvaluaationnoffuncctionnsofftwoovarriabllesaand33-Dssurfaacepplotss.TheerowwsofftheeouttputarraayXarecopiiesoofthheveectorrxaandtheecollumnssoftheoutpputaarrayyYaareccopieesofftheevecctory.[X,Y]==MESSHGRIID(x))isanaabbreeviattionfor[X,YY]=MESHHGRIDD(x,xx).[X,Y,Z]]=MMESHGGRID((x,y,,z)pproduuces3-Darraaystthatcanbeuusedtoevaaluattefuunctiionsoftthreeevarriabllesaand33-Dvvolummetriicpllots..DEL2DiiscreeteLLaplaaciann.L=DELL2(U))wheenUisaamattrix,,isanddiscrreteapprroximmatioonoff0.255*dell^2uu=((d^2uu/dx^^2+d^2//dy^22)/4..ThhemaatrixxLiisthhesaamessizeasUUwittheaacheelemeenteequalltothediffferenncebbetweeenaanellemenntoffUaandttheaaveraageoofittsfoournneighhborss.L=DELL2(U))wheenUisaanN--Darrray,,retturnssanapprroximmatioonoff(deel^2u)/22/nwwhereeniisnddims((u).L=DELL2(U,,H),wherreHisaascaalar,,useesHastthesspaciingbbetweeenpoiintsineeachdireectioon(HH=1bbydeefaullt).L=DELL2(U,,HX,HHY)wwhenUiss2-DD,ussestthesspaciingsspeciifieddbyHXanddHY..IfHXiisascallar,itggivesstheespaacinggbettweennpoiintsintheex-ddirecctionn.IffHXisaavecctor,,itmusttbeofllengtthSIIZE(UU,2)anddspeecifiiestthexx-cooordinnatessofthepoinnts.Simmilarrly,ifHHYisasccalarr,ittgivvestthesspaciingbbetweeenppointtsinntheey-ddirecctionn.IffHYisaavecctor,,itmusttbeofllengtthSIIZE(UU,1)andspeecifiiesttheyy-cooordinnatessofthepoinnts.L=DELL2(U,,HX,HHY,HZZ,....)whhenUUisN-D,,useesthhesppacinnggiivenbyHX,,HY,,HZ,,etcc.例3：產(chǎn)生一個(gè)二元元函數(shù)偏導(dǎo)導(dǎo)數(shù)和梯度度。x=-2:0..2:2;;y=-2:0..2:2;;[xx,yy]]=messhgriid(x,,y);z=xx.*eexp(--xx.^^2-yyy.^2));[Gx,Gy]]=graadiennt(z,,0.2,,0.2));%%Gx,GGy分別是二二元函數(shù)的的偏導(dǎo)contourr(x,yy,z,''k'),,holddon,,quiver((xx,yyy,Gxx,Gy,,'r')),holldofffDIFFDiifferrenceeanddappproxiimateederrivattive..DIFF(X)),foraveectorrX,is[[X(2))-X(11)XX(3)--X(2))....X(nn)-X((n-1))].DIFF(X)),foramaatrixxX,istthemmatriixoffrowwdifffereencess,[X(22:n,::)-X(1::n-1,,:)]..DIFF(X)),foranNN-DaarrayyX,isttheddiffeerencceallongthefirsstnon--singgletoondiimenssionofXX.DIFF(X,,N)isttheNN-thordeerdiifferrenceealoongttheffirsttnonn-sinnglettondimmensiion((denooteiitbyyDIMM).IIfN>=ssize((X,DIIM),DIFFFtakkessucccesssivediffferenncesalonngthheneextnnon-ssinglletonndimmensiion.（[FX,FY]]=GGRADIIENT((F,HXX,HY)),wheenFis22-D,usesstheespaacinggspeciffiedbyHHXanndHYY.HXXanddHYcaneithherbbesccalarrstoospeecifyythesppacinngbeetweeencooordiinateesorrvecctorsstospeccifythecoorrdinaatesofttheppointts.IfHHXanndHYYareevecctorss,thheirlenggthmmustmatcchthhecoorressponddingdimeensioonoffF.）（QUIVER((X,Y,,U,V))plootsvveloccityvecttorsasaarrowwswiithccompoonentts(uu,v)atthepoinnts((x,y)).TThemmatriicesX,Y,,U,Vmusttalllbethesameesizzeanddconntainncorrresppondiingpposittionandveloocityycommponeents(XaandYYcannalssobeevecctorsstospeccifyaunniforrmgrrid)..QUUIVERRauttomatticalllyscalesthearroowsttofiitwiithinntheegriid.）GRADIENNTAppproxximattegrradieent.[FX,FY]]=GGRADIIENT((F)rreturrnstthennumerricallgraadienntofftheemattrixF.FFXcoorressponddstoodF//dx,thediffferenncesintthex((coluumn)direectioon.FFYcoorressponddstoodF//dy,thediffferenncesinthey(rrow)direectioon.TThesspaciingbbetweeenppointtsinneacchdirrectiioniisasssumeedtoobeone..WheenFisaavecctor,,DF=GRRADIEENT(FF)issthee1-DDgraadiennt.[FX,FY]]=GGRADIIENT((F,H)),whhereHissasscalaar,uusesHasstheespaacinggbettweennpoiintsineeachdireectioon.[FX,FY]]=GGRADIIENT((F,HXX,HY)),whhenFFis2-D,,useesthhesppacinngspeecifiiedbbyHXXanddHY..HXandHYccaneeitheerbeescaalarsstospeccifytheespaacinggbettweenncooordinnatessorvecttorstosspeciifytthecooordinnatessofthepoinnts.IfHXaandHHYarreveectorrs,ttheirrlenngthmusstmaatchthecorrrespoondinngdiimenssionofFF.六．插值:(Innterppolattion))在已知數(shù)據(jù)之間間計(jì)算估計(jì)計(jì)值的過(guò)程程。1．一維插值(1DDIntterpoolatiion)由interp11實(shí)現(xiàn)，用用多項(xiàng)式技技術(shù)計(jì)算插插值點(diǎn)。Yi=inteerp1((x,y,,xi,mmethood)y—函數(shù)值矢矢量，xx—自變量取取值范圍，xi—插值點(diǎn)的自變量矢量，Method——插值方法法選項(xiàng)。MATLAB66.1的4種方法：：*臨近點(diǎn)插值：mmethood=‘nearrest’*線(xiàn)性插值：methhod=‘lineear’*三次樣條插值：：methhod=‘spliine’*立方插值：methhod=‘pchiip’or‘cubiic’選擇插值方法時(shí)時(shí)主要考慮慮因素：運(yùn)算時(shí)間間、占用計(jì)計(jì)算機(jī)內(nèi)存存和插值的的光滑程度度。比較：：運(yùn)算時(shí)間、占用計(jì)算算機(jī)內(nèi)存光滑程度度。*臨近點(diǎn)插值：快少差*線(xiàn)性插值：稍長(zhǎng)較多稍好*三次樣條插值：：最長(zhǎng)較多最好*立方插值：較長(zhǎng)多較好例1：一維插值函數(shù)數(shù)插值方法法的對(duì)比。x=0:10;;y=sin(xx);xi=0:0..25:110;strmod=={'neearesst','linnear'','ssplinne','cubbic'}}%將插值方方法定義為為單元數(shù)組組str1b={{'(a))metthod==nearrest'','((b)mmethood=liinearr',....'(c)meethodd=splline'','((d)mmethood=cuubic''}%將圖標(biāo)定定義為單元元數(shù)組fori=11:4yi=iinterrp1(xx,y,xxi,sttrmodd{i});subplott(2,22,i)plot(x,,y,''ro',xi,,yi,'b')),xlaabel((str11b(i)))endstrmod='neearesst''llineaar''ssplinne''ccubicc'例2：三次樣條插值x0=0:100;y0=sin((x0);;x=0:.255:10;;y=splinne(x00,y0,,x);plot(x00,y0,,'or'',x,yy,'k'')與interp11結(jié)果一樣樣2．二維插值(2DDIntterpoolatiion)用于圖形圖象處處理和三維維曲線(xiàn)擬合合等領(lǐng)域，由由inteerp2實(shí)實(shí)現(xiàn)，一般般格式為：：ZI=inteerp2((X,Y,,Z,XII,YI,,methhod)X,,Y—自變量組組成的數(shù)組組，尺寸相相同。XI,YYI—插值點(diǎn)的的自變量數(shù)數(shù)組Methhod—插值方法法選項(xiàng)，4種*臨近點(diǎn)插值：mmethood=‘nearrest’*線(xiàn)性插值：methhod=‘lineear’該方法是是inteerp2函函數(shù)的缺省省方法*三次樣條插值：：methhod=‘spliine’*立方插值：methhod=‘pchiip’or‘cubiic’例：二維插值4種種方法的對(duì)對(duì)比。[x,y,z]]=peaaks(77);ffigurre(1)),mmesh((x,y,,z)[xi,yi]]=messhgriid(-33:0.22:3,--3:0..2:3));z1=inteerp2((x,y,,z,xii,yi,,'neaarestt');z2=inteerp2((x,y,,z,xii,yi,,'linnear'');z3=inteerp2((x,y,,z,xii,yi,,'splline'');z4=inteerp2((x,y,,z,xii,yi,,'cubbic'));figure((2),subbplott(2,22,1)mesh(xxi,yii,z1))title(''nearrest'')subplott(2,22,2)mesh(xii,yi,,z2)title(('linnear'')subplott(2,22,3)mesh(xxi,yii,z3))title(''spiiine'))subplott(2,22,4)mesh(xii,yi,,z4)title(('cubbic'))3．多維插值:((3DIInterrpolaationn)包括三維插值函函數(shù)intterp33和多維插插值函數(shù)iinterrpn,函數(shù)調(diào)用用格式與一一、二維插插值基本相相同。VI=inteerp3((X,Y,,Z,V,,XI,YYI,ZII,metthod))其中：X,YY,Z—自變量組組成的數(shù)組組；V—三維函數(shù)數(shù)組XI,YI,ZZI—插值點(diǎn)的的自變量數(shù)數(shù)組Method--插值方法法選項(xiàng)。（FLOWAsimppleffuncttionof33varriablles.FLOOW,aafunnctioonoffthrreevvariaabless,isstheespeeedpprofiileoofasubbmerggedjjetwwithiinainfiiniteetannk.FLOWWisuseffulffordemmonsttratiingSSLICEEanddINTTERP33.）(SLICE(XX,Y,ZZ,V,XXI,YII,ZI))draawsssliceesthhrougghthhevoolumeeVaalonggtheesurrfaceedeffineddbythearraaysXXI,YII,ZI..)(SHADINGGFLAATseetstthesshadiingoofthhecuurrenntgrraphtofflat..SHAADINGGINTTERPsetsstheeshaadinggtointeerpollatedd.SHAADINGGFACCETEDDsettsthheshhadinngtoofaccetedd,whhichisttheddefauult.))例：三維插值實(shí)實(shí)例。[x,y,z,,v]=fflow((10);;%三變量無(wú)無(wú)限大容器器淹沒(méi)射流流場(chǎng)的速度度剖面圖figure((1),slice(xx,y,zz,v,[[699.5],,2,[--2..2])%在X=6,,9.5,,Y=22,z==-2,0.2五處取切切面[xi,yi,,zi]==meshhgridd(.1::.25::10,--3:.225:3,,-3:..25:33);vi=inteerp3((x,y,,z,v,,xi,yyi,zii);%全場(chǎng)沿坐坐標(biāo)面插值值figure((2),slice(xxi,yii,zi,,vi,[[699.5],,2,[--2..2])%切面插值值shadinggflaatFLOW