工業(yè)大學(xué)

計(jì)算機(jī)與信息學(xué)院

實(shí)驗(yàn)報(bào)告課程：計(jì)算方法專業(yè)班級(jí):號(hào):名:回苗eAef1FI1抑牢匪鞋呈環(huán)綁囂雜卑來(lái)率期￥F關(guān)氷融SE舉界E遜則繰則鼎醸退I磔丨叵1占I其實(shí)都不難按照程序流程圖就可以完成了實(shí)驗(yàn)一插值與擬合一.實(shí)驗(yàn)?zāi)康拿鞔_插值多項(xiàng)式和分段插值多項(xiàng)式各自的優(yōu)缺點(diǎn)；編程實(shí)現(xiàn)三次樣條插值算法,分析實(shí)驗(yàn)結(jié)果體會(huì)高次插值產(chǎn)生的龍格現(xiàn)象；理解最小二乘擬合,并編程實(shí)現(xiàn)線由以合,掌握非線性擬合轉(zhuǎn)化為線性擬合的方法運(yùn)用常用的插值和擬合方法解決實(shí)際問(wèn)題。二實(shí)驗(yàn)容⑴對(duì)于f(x)"/(l+x*x)實(shí)現(xiàn)三次樣條插值⑵實(shí)現(xiàn)最小二乘法的直線擬合數(shù)據(jù)如下：6165123150123141187126172125148三.基本原理(計(jì)算公式)(1)三次樣條插值在每個(gè)節(jié)點(diǎn)上具有2階導(dǎo)數(shù)。(2)最小二乘法擬合直線為y=a+bx，而a,b有如下等式(N為給出的數(shù)據(jù)點(diǎn)的總個(gè)數(shù))血+吃兀=X；"工兀+吃彳=藝否X出■誹裁吐刼r檢日日is対誌:開(kāi)始x[NJ,xOW三次樣條播值法最小二乘法直線擬合：輸入數(shù)據(jù)后，按照公式計(jì)算a,b。用得到的擬合直線計(jì)算預(yù)測(cè)點(diǎn)的近似函數(shù)值。五、輸入與輸出三次樣條插值輸入：區(qū)間長(zhǎng)度，n+1個(gè)數(shù)據(jù)點(diǎn)，預(yù)測(cè)點(diǎn)輸出：預(yù)測(cè)點(diǎn)的近似函數(shù)值，精確值,及誤差最小二乘法直線擬合輸入：n個(gè)數(shù)據(jù)點(diǎn)，預(yù)測(cè)點(diǎn)輸出：預(yù)測(cè)點(diǎn)的近似函數(shù)值六.結(jié)果討論和分析廣E:\計(jì)算萬(wàn)法儼少二病去直線擬合）?€xe代碼三次樣條插值#include<iostream>#include<fstream>#defineN10usingnamespacestd;doubleu0(doublex){return(x-l)*(x-l)*(2*x+l);}doubleul(doublex){returnx*x*(3-2*x);}doublev0(doublex){returnx*(x-l)*(x-l);}doublevl(doublex){returnx*x*(x-l);}doubles3(doublex,doubley,doubleyl.doubleredoublem^doubleh){returnu0(x)*y+ul(x)*yl+h*v0(x)*m+h*vl(x)*ml;}doublef(doublex){return1/(1+x*x);}intmain(){ifstreamfin;cout<<'erroropeninginputstream"<<endl;system(npause");return0;}doublex[N+l],y[N+l],m[N+l],A[N],B[N],C[N];doubleh[N];doublea[N],b[N];doublefO,fn;doubletemp;inti;for(i=0;i<=N;i++){fin>>x[i]>>y[i];}fin>>fO>>fn;h[0]=x[l]-x[0];for(i=l;i<N;i++){h[i]=x[i+l]-x[i];a[i]=h[i-l]/(h[i-l]+h[i]);b[i]=3*((l-a[i])*(y[i]-y[i-l])/h[i-l]+a[i]*(y[i+l]-y[i])/h[i]);m[l]=b[l]-(l-a[l])*fO;m[N-l]=b[N-l]-a[N-l]*fn;for(i=2;i<N-l;i++){m[i]=b[i];}for(i=l;i<N;i++){B[i]=2;C[i]=a[i];}for(i=2;i<N;i++){A[i]=l-a[i];}C[1]=C[1]/B[1];m[l]=m[l]/B[l];doublet;for(i=2;i!=N-2;i++){t=B[i]-C[i-l]*A[i];C[i]=C[i]/t;m[i]=(m[i]-m[i-l]*A[i])/t;)m[N-l]=(m[N-l]-m[N-2]*A[N-l])/(B[N-l]-C[N-2]*A[N-l]);for(i=N—2;ivoT-imnTm=lc日*)cos.AA?p_ease(郵ASBM計(jì)/AXoAAn埋?AAXzAA?a)?AAendbwhi-e(cinv:>temsdoub-ettHtemp;if(tempcx@=temp:>xz=(coutAA=盛?甜曲涉?AA^AA?■甜圧諒Blsl?AAend_;continue)for?lx-u5++)(if(tempcxebrea?T)tempH?mpix〒s、h〒一「tempuaaempyvlLy目旦一’lLmsh〒一D8C7A=Efiftrit=DT甘?AAttAA-f=AAfat)AA-AAfAA=湎?謎人Af(tt)—tempcAend-;systemupausevfin.coseo;returnoj最小二乘法的直線擬合#inelude<iostream>#include<fstream>#definen5usingnamespacestd;doublesum(doublex[]jntk){inti;doublesum=0;for(i=0;ivk;i++)sum=sum+x[i];returnsum;}doublesum2(doublex[]Jntk){inti;doublesum=0;for(i=0;ivk;i++)sum=sum+x[i]*x[i];returnsum;}doublesumxy(doublex[],doubley[]rintk){inti;doublesum=0;for(i=0;i<k;i++)sum=sum+x[i]*y[i];returnsum;}intmain(){ifstreamfin;fin.openCE:\\t?txtJ;if(!fin){cout<<"erroropeninginputstreamM<<endl;systemC*pause");return0;}doublex[n],y[n]ab;doublex0,y0;inti;for(i=0;i<n;i++){fin>>x[i]>>y[i];}b=(n*sumxy(x7/n)-sum(x/n)*sum(yrn))/(n*sum2(xrn)-sum(x/n)*sum(x/n));a=(sum(y/n)-b*sum(x/n))/n;cout<<-最小二乘法直線擬合得到a:-<<a<<\b:M<<b<<\擬合直線為y=,,<<a<<n+,,<<b<<"xMv<endl;COUtvc1■請(qǐng)輸入插值節(jié)點(diǎn)X:'；while(cin>>xO){yO=a+b*xO;cout<v■當(dāng)x=11v<x0<<\y=n<<y0<<endl;COUtcc1■請(qǐng)輸入插值節(jié)點(diǎn)X:'；}system(Mpause");fin.close();return0;}實(shí)驗(yàn)二數(shù)值積分一.實(shí)驗(yàn)?zāi)康?1)熟悉復(fù)化梯形方法、復(fù)化Simpson方法、梯形遞推算法、龍貝格算i去；⑵能編程實(shí)現(xiàn)龍貝格算法和中點(diǎn)加速；理解并掌握自適應(yīng)算法和收斂加速算法的基本思想；分析實(shí)驗(yàn)結(jié)果體會(huì)各種方法的精確度,建立計(jì)算機(jī)求解走積分問(wèn)題的感性認(rèn)識(shí)二、實(shí)驗(yàn)容(1)用龍貝格算i去計(jì)算竺dx⑵用中點(diǎn)加速方法計(jì)算J的一階導(dǎo)數(shù)

三.基本原理(計(jì)算公式)(1)龍貝格算法Th—1梯形遞推公式r+玄竹)加權(quán)平均公式:4Tln~Tn4-1(2)中點(diǎn)加速中點(diǎn)公式:G(h)=(f(a+h)-f(a-h))/2/h加權(quán)平均:Gl(h)=4P(h/2)/3?G(h)/3G2(h)=16*Gl(h/2)/15-Gl(h)/15G3(h)=64*G2(h/2)/63-G2(h)/63圖2.2梯形遞推算法流程圖圖2.3龍貝格算法流程圖圖2.2梯形遞推算法流程圖圖2.3龍貝格算法流程圖中點(diǎn)加速：輸入i后根據(jù)公式計(jì)算融值五.輸入與輸出用龍貝格算法計(jì)算(竺Jx輸入：積分區(qū)間,誤差限輸出：序列Tn,Sn,Cn,Rn及積分結(jié)果用中點(diǎn)加速方法計(jì)算J的一階導(dǎo)數(shù)輸入：求導(dǎo)節(jié)點(diǎn)，步長(zhǎng)輸出：求得的導(dǎo)數(shù)值，精確值六、結(jié)果討論和分析■rrxitwmwvpsttM幅exe吐兼嚴(yán)諱木壬產(chǎn)浮—曰_?0|零血譴蟄耐0?、?38注卩石汀莒掰湃alx浮—至al霧商自7.38U06、泓詆仝8?4049卞—010片X4MT.加于巴62冷—琴如聲商鐫^?￥54?5982、-3-紡苣冏5?浮—琴4(0曹商兇54二982、%Dlm-N6?一.y4p2aQHOTH6MT.歩*4、法—曰一1零壓芻B&403E辿0崗3T淳—孚40零血自403?￡、祈湘Rl?79625e—01QF?Mt.-!0102、x^—琴4D蹲血恭環(huán)血扌2980?9廠?>』尿gr浮—多4D零血兇29396、BHI4?8658elaLl建nc-udeciostreamv#indudecfstreamv#indudeccmarhvusingnamespacestd」doub-ef(doub-ex)(if(XHHO)「eiurnLreiumsin(x)、x;)intmain。亠ifstreamfin;fin?open(mA\t?txtwif(!fin){cout<<"erroropeninginputstream,,<<endl;system(,'pause");return0;}doubleafb/ertlrt2rslrS2/cl/c2jlj2;doublex「h,s;fin>>a>>b>>e;cout<<-積分區(qū)間為["<va<<","<<b<<"],要求精度為"<<e<<endl;cout<<'kT2S2C2R2"<<endl;h=b-a;tl=(f(a)+f(b))*h/2;coutvvOvv”"<<tl<<endl;intk;for(k=l;k<=10;k++/h=h/2/tl=t2isl=s2){s=0;x=a+h/2;do{s=s+f(x);x=x+h;}while(x<b);t2=tl/2+h*s/2;s2=t2+(t2-tl)/3;if(k==l){cout<<k<<"'<<t2<<""<<s2<<endl;continue;}c2=s2+(s2-sl)/15;if(k==2){cout<<k<<"'<<t2<<-"<<s2<<"'<<c2<<endl;cl=c2;continue;}r2=c2+(c2-cl)/63;cout<<k<<""<<t2<<""<<s2<<""<<c2<<""<<r2<<endl;if(k==3){rl=r2;cl=c2;continue;}if(fabs(r2-rl)<e){cout<<"數(shù)值積分結(jié)果為'<<r2<<endl;break;rl=r2;cl=c2;}systemC1'pause");return0;}中點(diǎn)加速算法#include<iostream>#include<fstream>#includevcmath>usingnamespacestd;doublef(doublex){returnexp(x);}doublefl(doublex){returnexp(x);}doubleg(doublex,doubleh){return(f(x+h)-f(x-h))/2/h;}doublegl(doublex.doubleh){SEM(1X4>\3?)U①do?u匸c匸lueMSJ-ou-elulu-亠勵(lì)9、(wx)epe9、(0Arx)0ptz9Ernal5①-qnopx①-qnop酉①-qnop亠lx17(lfx)gLAT、(0AVX)TP9Hu」a①」5①-qnopx①-qnop)%①-qnop亠w(wx)6CA、(0、qx)pt7E3?O①sop?u匸Mosned-E①-SAS=pu①VV26)2丄emVVkM黑八VV(we)m6VV共?檢即盒—直x?咲姻呂匪甘八VVS口vvkfl^K迺檢助直—敘XJVVqvviJZ*^八vvevvix淚?vvlno。(JZAAeAAU匸)①-ZM2:6①-qnop(ou」nl①」MQsned-E①-SAS=pu①Vv.lueMS-ndu一6UC①do」ou①?v:>lno:>return0;實(shí)驗(yàn)三非線性方程求根迭代法一.實(shí)驗(yàn)?zāi)康氖煜し蔷€性方程求根簡(jiǎn)單迭代法，牛頓迭代及牛頓下山法能編程實(shí)現(xiàn)牛頓下山法⑶認(rèn)識(shí)選擇迭代格式的重要性(4)對(duì)迭代速度建立感性的認(rèn)識(shí)；分析實(shí)驗(yàn)結(jié)果體會(huì)初值對(duì)迭代的影響二實(shí)驗(yàn)容用牛頓下山法解方程x3-x-l=0(初值為0.6)三、基本原1(計(jì)算公式)求非線性方程組的解是科學(xué)計(jì)算常遇到的問(wèn)題，有很多實(shí)際背景.各種算法層出不窮,其中迭代是主流算法。只有建立有效的迭代格式，迭代數(shù)列才可以收斂于所求的根。因此設(shè)計(jì)算法之前，對(duì)于一般迭代進(jìn)行收斂性的判斷是至關(guān)重要的。牛頓法也叫切線法，是迭代算法中典型方法,只要初值選取適當(dāng),在單根附近,牛頓法收斂速度很快,初值對(duì)于牛頓迭代至關(guān)重要。當(dāng)初值選取不當(dāng)可以采用牛頓下山算法進(jìn)行糾正?！愕喝?1=0(?)x=0(x)o/(x)=O牛頓公式：畑=Xk/E)廣E)牛頓下山公式：％嚴(yán)心一2光牛下山因子;1=1,*,*,*,…下山條件i/'EJivi/E)i圖3.2牛頓下山算法流程五、輸入與輸出輸入：初值，誤差限，迭代最大次數(shù)，下山最大次數(shù)輸出：近似根各步下山因子六.結(jié)果討論和分析” E:\計(jì)算方法\牛頓下LUexe迭代初值英x0：0旳4誤蔥限姿"隔，最人下山迖礬玲最人迭代次數(shù)為5一?卞山汝魏込下山因子乩迭代洛諏下山次裁尹下山因子込9迭住詐舉：，下山癱存?下口因子嘰25?迭代獺68倍了魔:侶霊爲(wèi)尊g繃214863,卞山?吹舉叱卞山因辛1.36680衛(wèi)山次數(shù)込衛(wèi)山1.32628，—_!］「；、i.32472■衛(wèi)山欽數(shù)沖1.32472—邃代取茲遊;略真為丄?良如2…請(qǐng)扮任意璉繼線???0U0066666xlxlxlxlxlxl：0：0lfe0旳?03125,迭彳涉遨：0F山伙茲汨'下山因子f迭代伏數(shù)M小25,卞925..…7&2?下山？17.9941114063,xl36681,xl32628.xl32472Axl代碼牛頓下山法#include<iostream>#include<fstream>#include<cmath>usingnamespacestd;doublef(doublex){//函數(shù)returnx*x*x-x-l;}doublefl(doublex){〃一階導(dǎo)數(shù)return3*x*x-l;}intmainQ{ifstreamfin;fin.open(nE\\Ltxv)；ifcfiscoutAcmfroropeninginpuistream-AAend-jsystemspause」；return0;doub-exoxLefeempjintN'Mxi;finvvxovvevVMVV28c7A.^iB^=AAX0AA^l^/A^AA=h^&^§=AA^AA=.?^亠衣為勞逬AANAAend-;X1HPfor(kupkcNQ?fi?fl(xo)lro>r++)(To-jytempHf(xo)、f1(X0)；do(xluxo-temp/j;coufAxpnA人X0AA?xl「AX1AA?nqlj為磐「仝人人JTE岡出?AAl/jA~濟(jì)左為聲1Akccendkj42;if(ivHM)breakj)whi_e(fabs(f(xl))VHfabs(f(xo)))；if(iv=M)(coutAAX0-AAend_;bre*if(fabs(xllxo)A￡COU7AAAxleAendrX0HX1;if(kVHNxourAA■卑簷?AAend一；if(fl(xoT=o)coutAA0=AAendkfin.doseo;systemupausgreturnoj醤E9I,醤皿3熟悉求解線性方程組的有關(guān)理論^方法;能編程實(shí)現(xiàn)高斯-塞德?tīng)柕ā⒘兄髟咚瓜シ?、LU分解法通過(guò)測(cè)試，進(jìn)一步了解各種方法的優(yōu)缺點(diǎn)根據(jù)不同類型的方程組，選擇合適的數(shù)值方法二實(shí)驗(yàn)容:10*1—“2—2*3=7.2⑴用高斯-塞德?tīng)柕ㄇ螅?小+10工2-2?=8.3~x\~xi+5心=4.2(lox】-x2-2x3=7.2-x,+10x2-2x3=8.3-Xj—x2+5x3=4.2⑶LU分解法求解方程組三.基本原理(計(jì)算公式)線性方程組大致分迭代法和直接法。只有收斂條件滿足時(shí),才可以進(jìn)行迭代。雅可比及高斯■塞德?tīng)柺亲罨镜膬深惖椒?，最大區(qū)別是迭代過(guò)程中是否引用新值逬行剩下的計(jì)算。消元是最簡(jiǎn)單的直接法，并且也十分有效的，列主元高斯消去法對(duì)求解一般的線性方程組都適用,同時(shí)可以用來(lái)求矩陣對(duì)應(yīng)的行列式。約當(dāng)消去實(shí)質(zhì)是經(jīng)過(guò)初等行變換將系數(shù)矩陣化為單位陣，主要用來(lái)求矩陣的逆。在使用直接法,要注意從空間、時(shí)間兩方面對(duì)算法進(jìn)行優(yōu)化。ii=1,2,高斯■塞德?tīng)柕和?

i—1 /I5產(chǎn)j 一XauXJ；=1 J-r+1列主元高斯i肖去法：列主元IalkI=maxIaik1工0;卜嚴(yán)=即一叫硝 冒-ix*消元［b；z=bf-叫磧' 回代七=——粹—— (I=“，?.?,1)對(duì)應(yīng)于矩陣分解A=IXA方程tli(25)即Ak=&亦即L(Ux)Ly=d和Ur=y兩個(gè)方程紐來(lái)求解?先解ty=</即/iy\=d、ay(-I4-Ayz= , /=2,3,???,得?129?(33)Vi=d\/Z(33)尸=Cdi?a<y^i)/li9 i=2,3,???,//而Ux=yW(2S)的求解公式己由丫30/給出?這樣?方程紐(25)的求解可!/I納為如下三步:步1 按武(32丿計(jì)算厶一m-*厶一…一￡/.<IA;步2 按式(33丿順序求卩一“一…一”；步3按式『30丿逆序求Xu-*x-1—…—R?輸入aJjku荷出解狀i=l…■山圖4.3列主元的髙斯消去流程圖輸入aJjku荷出解狀i=l…■山圖4.3列主元的髙斯消去流程圖11 :?: >leuk+1兀uyf輸出迭代先疑 |輸出幾圖4.1G-S迭代算法流程圖LU分解法：依次求得L、U、y禾[]x五、輸入與輸出用高斯-塞德?tīng)柕ㄝ斎耄合禂?shù)矩陣A,最大迭代次數(shù)N,初始向呈,誤差限e輸出：解向呈列主元高斯;肖去法輸入：系數(shù)矩陣A輸出：解向呈⑶LU分解法輸入：系數(shù)矩陣A輸出：解向呈六.結(jié)果討論和分析代碼高斯塞德?tīng)柕?include<iostream>#include<fstream>#include<cmath>#definen3usingnamespacestd;voidshow(doublea[n+l][n+l]rdoubleb[n+l]){intij;cout<<"原方程為：，'vvendl;for(i=1;i<=n;i++){for(j=1;j<n;j++)if(a[i][j+1]<0)cout<<a[i][j]<<"*x"<<j;elsecout<<a[i][j]<<"*xn<<j<<"+";cout<<a[i][j]<<"*x"<<j<<"="<<b[i]<<endl;}}intmain(){ifstreamfin;fin.openCE:\\t?txt");cout<<"ErroropeninginputstreamM<<endl;system^pause”)；return0;}doublex[n+l]ry[n+l]ra[n+1][n+l]rb[n+l]rertempfmax;inti,j,k,N;fin>>e>>N;for(i=l;i<=n;i++){fin>>x[i];y[i]=x[i]；}cout<<"初始向星為:?；for(i=l;i<n;i++)cout<<x[i]<<"rcoutvvx[i]vvendl;for(i=l;i<=n;i++){for(j=l;j<=n;j++)fin>>a[i][j];fin>>b[i];}show(a/b);k=0;while(true){for(i=l;i<=n;i++){temp=O;for(j=l;j<=n;j++)if(j!=i)temp=temp+a[i][j]*y[j];y[i]=(b[i]-temp)/a[i][i];}max=fabs(y[l]-x[l]);for(i=2;i<=n;i++)if(max<fabs(y[i]-x[i]))max=fabs(y[i]-刈i]);if(max<e){cout<<?高斯客德?tīng)柕蟮迷匠痰慕鉃?<<endl;for(i=l;i<=n;i++)cout<<"x"<<i<<"為?<<y[i]cout<<endl;break;}if(k==N){cout<<?迭代失敗?<<endl;break;else{k++;for(i=l;i<=n;i++)x[i]=y[i];}}fin.closeO；systemCpauseJ;return0;}高斯消去#include<iostream>#include<fstream>#include<cmath>#definen3usingnamespacestd;voidshow(doublea[n+l][n+l]rdoubleb[n+l]){intij;cout<<"原方程為：Ivendl;for(i=1;i<=n;i++){for(j=1;j<n；j++)if(a[i][j+l]v0)cout<<a[i][j]<<"*xn<<j;elsecout<<a[i][j]<<"*xn<<j<<"+";cout<<a[i][j]<<"*x"<<j<<"="<<b[i]<<endl;}}intmain(){ifstreamfin;fin.open("E:\Xt.txt");if(!fin){cout<<"erroropeninginputstream1^<endl;system(,,pauseM);return0;}doublea[n+1][n+l]rb[n+l]rdft;inti,j,k,l;for(i=l;i<=n;i++){for(j=l;j<=n;j++)fin>>a[i][j];fin>>b[i];}show(a/b);k=l;do{d=a[k][k];l=k;i=k+l;do{if(i>n)break;if(fabs(a[i][k])>fabs(d)){d=a[i][k];l=i；}if(i==n)break;i++；}while(true);if(d==O){cout<<"奇異"<<endl;systemC1pause");fin.closeO;return0;for(j=k;j<=n;j++){a[l][j]=a[k][j];a[k][j]=t;}t=b[k];b[k]=b[l];b[l]=t;}for(j=k+l;j<=n;j++){a[k][j]=a[k][j]/a[k][k];}b[k]=b[k]/a[k][k];for(i=k+l;i<=n;i++){for(j=k+l;j<=n;j++){a[i][j]=a[i][j]-a[i][k]*a[k][j];}for(i=k+l;i<=n;i++)b[i]=b[i]-a[i][k]*b[k];if(k==n)break;k++;}while(true);for(i=n-l;i>=l;i—){t=0；for(j=i+l;j<=n;j++){t=t+a[i][j]*b[j];}b[i]=b[i]-t;}cout<<"列主元的高斯消去法求得原方程的解為:";for(i=l;i<=n;i++)cout<<"x"<<i<<"為"<<b[i]<<'cout<<endl;system(Mpause");fin.close();return0;}LU分解#include<iostream>#include<fstream>#definen3usingnamespacestd;voidshow(doublea[n+l][n+l]rdoubleb[n+l]){intij;cout<<"原方程為「vvendl;for(i=1;i<=n;i++){for(j=1;j<n;j++)if(a[i][j+1]<0)cout<<a[i][j]<<"*x"<<j;elsecout<<a[i][j]<<"*x"<<j<<"+";cout<<a[i][j]<<"*x"<<j<<"="<<b[i]<<endl;})intmain(){ifstreamfin;fin.open("E:\\t.txtn);if(!fin){cout<<"erroropeninginputstream1^<endl;system(Mpause");return0;doublea[n+1][n+l]rb[n+l]ru[n+1][n+1]J[n+1][n+l]rx[n+l],y[n+1];doublet;inti,j,k;for(i=l;iv=n;i++){for(j=l;j<=n;j++)fin>>a[i][j];fin>>b[i];)show(azb);for(i=l;i<=n;i++){for(j=l;j<i;j++){t=0;for(k=l;k<j;k++)t=t+l[i][k]*u[k][j];l[i][j]=(a[i][j]-t)/u[j][j];}for(j=i;j<=n;j++){t=0;for(k=l;k<i;k++)t=t+l[i][k]*u[k][j];u[i][j]=a[i][j]-t;for(i=l;i<=n;i++){t=0;for(j=l;j<i;j++)t=t+l[i][j]*y[j];y[i]=b[i]-t;}for(i=n;i>=l;i—){t=0;for(j=i+l;j<=n;j++)t=t+u[i][j]*x[j];x[i]=(y[i]-t)/u[i][i];}cout<<'LU分解法求得原方程的解為：-;for(i=l;i<=n;i++)cout<<"x"<<i<<"為"<<x[i]<<"cout<<endl;system("pause");fin.close();return0;}貓五數(shù)值微分一.實(shí)驗(yàn)?zāi)康?/p>

(1)熟悉數(shù)值微分中Euler法，改進(jìn)Euler法,Rung-Kutta方法；⑵能編程實(shí)現(xiàn)亞當(dāng)姆斯方法，Rung-Kutta方法;⑶通過(guò)實(shí)驗(yàn)結(jié)果分析各個(gè)算法的優(yōu)缺點(diǎn)；(4)明確步長(zhǎng)對(duì)算法的影響并理解變步長(zhǎng)的Rung-Kutta方法二實(shí)驗(yàn)容0VX<1,2xy=y—° y0VX<1.y(°)=i取h=o.l時(shí)用亞當(dāng)姆斯方法rRung-Kutta方法求其數(shù)值解并與精確解進(jìn)行比較。三.基本原理(計(jì)算公式)在許多科學(xué)技術(shù)問(wèn)題中，建立的模型常常以常微分方程的形式表示。然而，除了少數(shù)特殊類型的常微分方程能用解析方法求其精確解外，要給出一般方程解析解的表達(dá)式是困難的。所以只能用近似方法求其數(shù)值解，在實(shí)際工作中常用計(jì)算機(jī)求常微分方程的數(shù)值解。所謂常微分方程的數(shù)值解即對(duì)于常微分方程初值問(wèn)題=/U，>)計(jì)算出在一系列節(jié)點(diǎn)aU(Xo)=兒=Xo<XK...<Xn=b處的未知函數(shù)y(x)ifi｛以值yo,yi,...yn,即找到一系列離散點(diǎn)(xo,yo)(XI,yi)(X2$2)...(Xn,yn)近似滿足常微分方程。數(shù)值解法的基本思想用差商代替導(dǎo)數(shù),實(shí)現(xiàn)連續(xù)問(wèn)題離散化，選取不同的差商代替導(dǎo)數(shù)可以得到不同公式。改逬歐拉公式是常用方法之一,包括預(yù)測(cè)和校正兩步。先用歐拉公式進(jìn)行預(yù)報(bào)，再將預(yù)報(bào)值代入梯形公式進(jìn)行校正,從而達(dá)到二階精度。通過(guò)龍格-庫(kù)塔法我們可以獲得更高精度。經(jīng)典龍格-庫(kù)塔法即在區(qū)間［Xn,Xn+l］取四點(diǎn)，并對(duì)這四點(diǎn)的斜率進(jìn)行加權(quán)平均作為平均斜率，通過(guò)泰勒公式尋找使局部截?cái)嗾`差為0(h5)(即4階精度)的參數(shù)滿足條件。四階(經(jīng)典)龍格-庫(kù)塔公式兒+軟KI+2K2+2K3+K4)/(X”，兒)/K+P兒+壬＜)/(“”+%兒+今瓦)f(x?+h,yn+hK})亞當(dāng)姆斯預(yù)報(bào)■校正系統(tǒng)：預(yù)報(bào)歹”+】=X+￡（55無(wú)一59y'”_i+37幾_2—9北_3）羅”+1=/（召+1，孑”+1）校正，”+】=九+備陽(yáng)”+1+i9y?-5_/i+y._2）/-+1=/（召+】，兒+1）IMI111圖5.3經(jīng)典龍格庫(kù)塔算法五、輸入與輸出輸入：求解區(qū)間,初值，數(shù)值解個(gè)數(shù)輸出：數(shù)值解六.結(jié)果討論和分析?TF|卜@hnr](Ti90EE99 \OSS88Z66G3COO)寸I98*rg080seorH EZ6II04?HSZ?H審??93IIIIIIIrKLGO亍樂(lè)不汗&\長(zhǎng)樂(lè)寸苗^*、8語(yǔ)刪加啊刪岫你、漫?出臨兆兆毗:兆毗吐9腳H尺二總卡囑■HVZ■W沁冒S99E? ? ?COTrK□OIllia:oo)ogoszi8ZNSiGCP卜S1GC99TH1ceiinSii■■■■?■ ■■4tHTZcIlli即徹湘I圳刪刪醐瞌瀘曲賦:發(fā)陋班?刼的臉蠟鞘兆陳汽&J匚$&fUO站哋叟3南更番JnZEXm很皿七\(yùn)心■Psz0N。9KZ-HCO審I8 -ebbsCM6Ze9P<占、0… ■9ZE0了、刁7TT\才、丁、了、TT*、TT、.-、■<IIAfAAAAAA亠尸半心龍永空乙總臣譽(yù)松總什］卜j、?ixdi'cvu 冬匸nt<1吋&喃md(44吋4球rrtirq述韓率臟韓袈枠珅粋珅粋.涇旺旺氏旺坯旺旺旺圧掘*rrr殳心於心眷辿線t?gM七怎-ZZI寸0CO6ZT8PHb寺S9R■w■?.=匸■HflllHwj4011411114miJRnt41111401]+U1LR粹 ?_懾前苗薛苗苗苗理drHZe審G9OC06埜#破■^(sssssscsiiss二車x/aHHI*iiiinnInn.{><-fe-XXXXXXXXXX興ciue①」1SOV①pnpw#s#include<fstream>#include<cmath>usingnamespacestd;doublef(doublex,doubley){returny-2*x/y;//函數(shù)}doublefO(doub