北师大数值分析作业(一)、(二)、(三)==2390

北师大数值分析作业(一)、(二)、(三) (一)1、设下列各数均为经过四舍五入后得到的近似值，试求各数的绝对误差限和相对误差限。123||||||||24839||||||||=4]0]0-l-l010|||||||-l-ll||||l||||||||lln-1]]||||l1ll1l|ll|0|(2)L-1L-1121]]llln2n2(1)(1)求出A能进行Cholesky分解，即A=LLT(其中L为下三角矩阵)的a||-1|0|-12-100-10-12-1||||「0]10=-1||x32」Lx」4|A=|13求cond(A)及cond(A)w13-17]|29-38|2212212ii2xxii-10．502(1)(1)利用线性插值计算20.3(2)(2)利用二次插值计算20.32、已知函数表为0．520．527xi0．01075yi0．7270．012190．8070．011880．9270．014263、已知函数表为xxii1-7648354、设x,x,,x为n+1个互异节点，l(x)(j=0,1,,n)为这组节点上的n次01njLagrange插值基函数，试证：jjj=0jjj=0P(x)_xnP(x)l(x)=O(x)=nn(x_x)jjn+1jjj=05、已知函数表为xxii2．462592．652712．414503．030356、已知函数表为xxii439253knk=0 (二)计算实习说明书wztu00-0.5-0.-0.284-0.-0.64-0.-0.5-0.-0.4-0.-0.5-0.-0.686-0.1-0.5-0.822-0.-0.5p(x,y)=kcxrysr,s=0ijiji=0j=0ijijijijk(1)算法的设计方案。(2)全部源程序(要求注明主程序和每个子程序的功能)。ijij(4)选择过程的k,值。ijijijijij 程序算法的设计方案(流程图)始利用牛顿法求解非线利用t和u的值进行分片二次插值，得到z的近似值，并输出利用Doolittle法求解线性方程组。插值定位子程序确定插值的位置。利用分片二次插值求t和u对应的z。行曲面拟合，输出过值以及拟合系数顿法和分片二次插值求出z*的值。利用已经求出的拟合系数计算近似表达式并比较逼近的效果。曲面拟合子程序来确定逼近多项式。求逆，主要考虑到求逆的矩阵为方阵，求逆运算和采用高斯消元法原理一样，求逆显得更简洁。开始的，所以在有些地方造成不便，所以有些地度数据方面存在一些问题，主要是直!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!主程序!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!功能：控制整个程序的运行!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!callInitialXY(Xn,Yn)!初始化X,Y向量Ft(i,j)=A(1)!把满足精度的向量u(i,j)=A(2)!分别存在对应向量中write(98,*)"计算结果如下"callCalZ(t(i,j),u(i,j),H,L,Z(i,j))!插值求函数write(98,80)"X(",i,")=",Xn(i),",Y(",j,")=",Yn(j)write8,100)"f(",i,",",j,")=",Z(i,j)callNiheXnYnZ!曲面拟合!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!主程序结束!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!以下为子程序!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!初始化X，Y向量!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!XniD*(i-1)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!非线性方程组!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!一阶导数矩阵矩阵!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!求无穷范数!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!杜利特法解线性方程组!!!!!!!!!!!!!!!!!!!!!!!!!!!iAY=Y+A(k,t)*A(t,j)A(k,j)=A(k,j)-Ywrite(*,*)'错误!!'Y=Y+A(i,t)*A(t,k)A(i,k)=(A(i,k)-Y)/A(k,k)Y=Y+A(i,t)*B(t)B(i)=B(i)-Yj=4-iY=Y+A(j,t)*X(t)X(j)=(B(j)-Y)/A(j,j)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!插值的定位!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!nLi+1!!!!!!!!!!!!!!!!!!!!!!!!!!利用分片二次插值求函数值!!!!!!!!!!!!!!!!!!!!!!!!!!ittoldktolditoldkZ=Z+Lk*Lr*z_old(i,j)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!矩阵的乘法运算!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!A(i,j)=A(i,j)+B(i,k)*C(k,j)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!矩阵的转置运算!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!AT(i,j)=A(j,i)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!矩阵的求逆运算!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!B=AAN(i,j)=AN(i,j)/B(i,i)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!矩阵上三角化程序!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!E=M(j,i)/M(i,i)M(j,i:N)=M(j,i:N)-M(i,i:N)*E!!!!!!!!!!!!!!!!!!!!!!!!!!!!上三角矩阵对角化程序!!!!!!!!!!!!!!!!!!!!!!!!!!!!!E=M(j,i)/M(i,i)M(j,1:N)=M(j,1:N)-M(i,1:N)*E!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!求拟合系数矩阵!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!B(i,j)=Xn(i)**(j-1)G(i,j)=Yn(i)**(j-1)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!最小二乘曲面拟合!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!P(i,j)=P(i,j)+C(r,s)*(Xn(i)**(r-1))*(Yn(j)**(s-1))YXnYnt(i,j)=A(1)uijA(2)HLZijP(i,j)=P(i,j)+CRS(r,s)*(Xn(i)**(r-1))*(Yn(j)**(s-1))write(98,160)"X^(",i,")=",Xn(i),",Y^(",j,")=",Yn(j)writef",i,",",j,")=",Z(i,j)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!重新初始化X,Y向量!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!XniDi!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!==================================X0000000,Y(1)=.5000000f80E+00==================================X0000000,Y(2)=.5500000f927E+00==================================X0000000,Y(3)=.6000000f683E+00==================================X0000000,Y(4)=.6500000f8316E+00==================================X0000000,Y(5)=.7000000f0189556266E-02==================================X0000000,Y(6)=.7500000f7358136380E-01==================================X0000000,Y(7)=.8000000f2750E+00==================================X0000000,Y(8)=.8500000f467E+00==================================X0000000,Y(9)=.9000000f388E+00==================================X0000000,Y(10)=.9500000f110E+00==================================X000000,Y(11)=1.0000000f38E+00==================================X000000,Y(12)=1.0500000f44E+00==================================X=.0000000,Y(13)=1.1000000f884E+00==================================X=.0000000,Y(14)=1.1500000f797E+00==================================X000000,Y(15)=1.2000000f768E+00==================================X000000,Y(16)=1.2500000f090E+00==================================X0000000,Y(17)=1.3000000f876E+00==================================X0000000,Y(18)=1.3500000f186E+00==================================X000000,Y(19)=1.4000000f003E+00==================================X000000,Y(20)=1.4500000f545E+00==================================X0000000,Y(21)=1.5000000f609E+00==================================X0800000,Y(1)=.5000000f510E+00==================================X0800000,Y(2)=.5500000f146E+00==================================X0800000,Y(3)=.6000000f277E+00==================================X0800000,Y(4)=.6500000f154E+00==================================X0800000,Y(5)=.7000000f0285E+00==================================X0800000,Y(6)=.7500000f9926834909E-01==================================X0800000,Y(7)=.8000000f4680402049E-01==================================X0800000,Y(8)=.8500000f6789E+00==================================X0800000,Y(9)=.9000000f309E+00==================================X0800000,Y(10)=.9500000f058E+00==================================X800000,Y(11)=1.0000000f979E+00==================================X800000,Y(12)=1.0500000f22E+00==================================X=.0800000,Y(13)=1.1000000f19E+00==================================X=.0800000,Y(14)=1.1500000f701E+00==================================X800000,Y(15)=1.2000000f962E+00==================================X800000,Y(16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400000,Y(21)=1.5000000f540E+00==================================X7200000,Y(1)=.5000000f266E+01==================================X7200000,Y(2)=.5500000f940E+01==================================X7200000,Y(3)=.6000000f084E+01==================================X7200000,Y(4)=.6500000f2800E+01==================================X7200000,Y(5)=.7000000f2878E+01==================================X7200000,Y(6)=.7500000f2733E+01==================================X7200000,Y(7)=.8000000f3523E+01==================================X7200000,Y(8)=.8500000f5061E+01==================================X7200000,Y(9)=.9000000f0963E+01==================================X7200000,Y(10)=.9500000f261E+00==================================X200000,Y(11)=1.0000000f249E+00==================================X200000,Y(12)=1.0500000f966E+00==================================X=.7200000,Y(13)=1.1000000f504E+00==================================X=.7200000,Y(14)=1.1500000f73E+00==================================X200000,Y(15)=1.2000000f740E+00==================================X200000,Y(16)=1.2500000f033E+00==================================X7200000,Y(17)=1.3000000f8241E+00==================================X7200000,Y(18)=1.3500000f6856518657E-02==================================X200000,Y(19)=1.4000000f6530659133E-01==================================X200000,Y(20)=1.4500000f7819E+00==================================X7200000,Y(21)=1.5000000f752E+00==================================X8000000,Y(1)=.5000000f947E+01==================================X8000000,Y(2)=.5500000f126E+01==================================X8000000,Y(3)=.6000000f686E+01==================================X8000000,Y(4)=.6500000f728E+01==================================X8000000,Y(5)=.7000000f5665E+01==================================X8000000,Y(6)=.7500000f7910E+01==================================X8000000,Y(7)=.8000000f2055E+01==================================X8000000,Y(8)=.8500000f4375E+01==================================X8000000,Y(9)=.9000000f6765E+01==================================X8000000,Y(10)=.9500000f2131E+01==================================X000000,Y(11)=1.0000000f1991E+01==================================X000000,Y(12)=1.0500000f228E+00==================================X=.8000000,Y(13)=1.1000000f111E+00==================================X=.8000000,Y(14)=1.1500000f520E+00==================================X000000,Y(15)=1.2000000f14E+00==================================X000000,Y(16)=1.2500000f898E+00==================================X8000000,Y(17)=1.3000000f181E+00==================================X8000000,Y(18)=1.3500000f2525E+00==================================X000000,Y(19)=1.4000000f5567703264E-01==================================X000000,Y(20)=1.4500000f6985959345E-01==================================X8000000,Y(21)=1.5000000f9709E+00======================================================================================================aE==================================aE==================================aE==================================aE==================================aEC123411356E+01C6844732236E+01C9293729576E+00C557589944E+00C872932266E+00C5)=.674271190394E-01C9190765779E+01C1102653033E+00C714158136E+01C63120138972E+01C729187244E+00C5)=.606161167522E-01C879814138E+00C57997575327E+01C533591393E+00C3)=-.812114292662E-01C02026103127E+00C5)=-.210016719542E-01C257207647E+00C0265519504E+00C07618432390E+01C054404182E+00C2894113382E+00C5)=-.459768761223E-01C459131460E+00C78368833271E+00C2)=-.724135891547E-01C338716861E+00C41338962264E+00C5)=.265110814406E-01C0)=-.559032018033E-01C43198880161E+00C36269574802E+00C3)=.407187801319E-01C4)=.377546298538E-02C5)=-.266778681178E-02=逼近结果如下===========X1000000,Y^(1)=.7000000f918E+00P=.194730352309E+00==================================X1000000,Y^(2)=.9000000f189E+00P=-.183041836493E+00==================================X=.1000000,Y^(3)=1.1000000f5E+00P.445500057353E+00==================================X1000000,Y^(4)=1.3000000f41E+00P-.597558872591E+00==================================X1000000,Y^(5)=1.5000000f01E+00P.646446148983E+00==================================X2000000,Y^(1)=.7000000f8E+00P.405989535099E+00==================================X2000000,Y^(2)=.9000000f159583746E-01P^(2,2)=-.225211143044E-01==================================X=.2000000,Y^(3)=1.1000000f40E+00P-.338224036191E+00==================================X2000000,Y^(4)=1.3000000f22E+00P-.544430464106E+00==================================X2000000,Y^(5)=1.5000000f68E+00P.647348042676E+00==================================X3000000,Y^(1)=.7000000f51E+00P.634787448952E+00==================================X3000000,Y^(2)=.9000000f839E+00P=.158796297231E+00==================================X=.3000000,Y^(3)=1.1000000f71E+00P.207368591334E+00==================================X3000000,Y^(4)=1.3000000f8E+00P.465349933201E+00==================================X3000000,Y^(5)=1.5000000f75E+00P.620257162340E+00==================================X4000000,Y^(1)=.7000000f74E+00P.878969862313E+00==================================X4000000,Y^(2)=.9000000f82E+00P.358646043873E+00==================================X=.4000000,Y^(3)=1.1000000f528211681E-01P^(4,3)=-.552554444560E-01==================================X4000000,Y^(4)=1.3000000f03E+00P-.362671068830E+00==================================X4000000,Y^(5)=1.5000000f55E+00P-.567550595901E+00==================================X5000000,Y^(1)=.7000000f016E+01P=.113662035139E+01==================================X5000000,Y^(2)=.9000000f48E+00P.574975842440E+00==================================X=.5000000,Y^(3)=1.1000000f792E+00P=.115989317915E+00==================================X5000000,Y^(4)=1.3000000f12E+00P.238560420362E+00==================================X5000000,Y^(5)=1.5000000f6E+00P.491420901638E+00==================================X6000000,Y^(1)=.7000000f891E+01P=.140605068646E+01==================================X6000000,Y^(2)=.9000000f63E+00P.805937300660E+00==================================X=.6000000,Y^(3)=1.1000000f45E+00P.304425833078E+00==================================X6000000,Y^(4)=1.3000000f161300996E-01P^(6,4)=-.950089415129E-01==================================X6000000,Y^(5)=1.5000000f46E+00P-.393889825099E+00==================================X7000000,Y^(1)=.7000000f531E+01P=.168579121814E+01==================================X7000000,Y^(2)=.9000000f306E+01P=.104987773701E+01==================================X=.7000000,Y^(3)=1.1000000f40E+00P.508291050844E+00==================================X7000000,Y^(4)=1.3000000f487967065E-01P^(7,4)=.661563657028E-01==================================X7000000,Y^(5)=1.5000000f78E+00P.276822016025E+00==================================X8000000,Y^(1)=.7000000f665E+01P=.197458126370E+01==================================X8000000,Y^(2)=.9000000f765E+01P=.130533200058E+01==================================X=.8000000,Y^(3)=1.1000000f11E+00P.725989322216E+00==================================X8000000,Y^(4)=1.3000000f81E+00P.243260807036E+00==================================X8000000,Y^(5)=1.5000000f709E+00P=-.141938745268E+00==================================xy代法编程，虽然也能解出值来，但是却不能往下做了。后来才弄明白是要将x，。响，降低。 (三)zfxy已知方程组和数表，把每一对待求的x和y值代(3)由(1)和(2)即可以得到数表x、y、f(x,y)，此数表由C程序输出后导入Excel(4)对已求得的数表进行曲线拟合，方法采用教材中的曲线拟合方法，实质上就是再由步骤(4)所确定的函数求得拟合值P(x∗,y∗)，把两组值做一个对比，可以(6)在具体编写程序时没有采用过多的子程序，牛顿法求解非线性方程组、列主元一个子程序进行循环调用。同时步骤(5)中的校验过程均是采用前面的程序计doublex[11],y[21],X[N],dertx[N],F1[N][N],F[N];foriii++)x[i]=0.08*i;for(i=0;i<21;i++)y[i]=0.5+0.05*i;/*牛顿法解非线性方程组*/for(i=0;i<N;i++)X[i]=1;{for(kk=0;kk<M;kk++){F1[0][0]=-0.5*sin(X[0]);F1[0][1]=1;F1[0][2]=1;F1[0][3]=1;FF[1][1]=0.5*cos(X[1]);F1[1][2]=1;F1[1][3]=1;F1[2][0]=0.5;F1[2][1]=1;F1[2][2]=-sin(X[2]);F1[2][3]=1;FFF1;F1[3][3]=cos(X[3]);F[0]=-(0.5*cos(X[0])+X[1]+X[2]+X[3]-x[r]-2.67);F[1]=-(X[0]+0.5*sin(X[1])+X[2]+X[3]-y[s]-1.07);F[2]=-(0.5*X[0]+X[1]+cos(X[2])+X[3]-x[r]-3.74);F[3]=-(X[0]+0.5*X[1]+X[2]+sin(X[3])-y[s]-0.79);wkfor(i=k;i<N;i++)iffabsF[i][k])>max){max=fabs(F1[i][k]);row=i;}if(row!=k){for(i=k;i<N;i++){l1=F1[k][i];F1[k][i]=F1[row][i];F1[row][i]=l1;}l1=F[k];F[k]=F[row];F[row]=l1;}for(i=k+1;i<N;i++){m=F1[i][k]/F1[k][k];for(j=k+1;j<N;j++)F1[i][j]=F1[i][j]-m*F1[k][j];F1[i][k]=0;F[i]=F[i]-m*F[k];}}dertx[N-1]=F[N-1]/F1[N-1][N-1];for(j=k+1;j<N;j++)l1=l1+F1[k][j]*dertx[j];dertx[k]=(F[k]-l1)/F1[k][k];}for(i=0;i<N;i++)X[i]=X[i]+dertx[i];for(i=1;i<N;i++){if(fabs(X[i])>mx)mx=fabs(X[i]);}for(i=1;i<N;i++)}z=f(X[0],X[1]);Z[r][s]=z;}/*曲线拟合*/{for(j=0;j<k1+1;j++)foriii++)B[i][j]=pow(x[i],j);for(j=0;j<k1+1;j++)for(i=0;i<21;i++)G[i][j]=pow(y[i],j);{for(i=0;i<=k1;i++)for(j=0;j<=k1;j++)l=0;l1=l1+B[p][i]*B[p][j];a[i][j]=l1;}for(i=0;i<=k1;i++)ll1=l1+B[p][i]*Z[p][k2];b[i]=l1;}for(i=k;i<k1+1;i++)iffabsaikmax){max=fabs(a[i][k]);row=i;}if(row!=k){for(i=k;i<k1+1;i++){l1=a[k][i];a[k][i]=a[row][i];a[row][i]=l1;}l1=b[k];b[k]=b[row];b[row]=l1;}for(i=k+1;i<k1+1;i++){m=a[i][k]/a[k][k];for(j=k+1;j<k1+1;j++)a[i][j]=a[i][j]-m*a[k][j];a[i][k]=0;b[i]=b[i]-m*b[k];}}x1[k1]=b[k1]/a[k1][k1];for(j=k+1;j<k1+1;j++)l1=l1+a[k][j]*x1[j];x1[k]=(b[k]-l1)/a[k][k];}fori0;i<k1+1;i++)A[i][k2]=x1[i];for(i=0;i<=k1;i++)for(j=0;j<=k1;j++)l=0;l1=l1+G[p][i]*G[p][j];a1[i][j]=l1;}for(i=0;i<=k1;i++)b1[i]=G[k2][i];for(i=k;i<k1+1;i++){max=fabs(a1[i][k]);row=i;}if(row!=k){for(i=k;i<k1+1;i++){l1=a1[k][i];a1[k][i]=a1[row][i];a1[row][i]=l1;}l1=b1[k];b1[k]=b1[row];brow]=l1;}for(i=k+1;i<k1+1;i++){m=a1[i][k]/a1[k][k];for(j=k+1;j<k1+1;j++)a1[i][j]=a1[i][j]-m*a1[k][j];a