数值分析计算实习题(三).doc

北航数值分析计算实习题目三SY1004114 全昌彪一、算法设计方案1、解非线性方程组首先将x，y当作已知的常数，求解四个未知数t，u，w，u。利用Newton法（简单迭代法不收敛）求解非线性方程组，得到与x，y对应的向量t，u。求解步骤：1）、选取初始向量t,u,v,w=1，1，1，1；2）、计算和；3）、解关于的线性方程组（调用Doolittle分解法求解此线性方程组）；4）、若，则取；否则转5；5）、计算；题目中Newton法迭代公式为：其中2、分片二次代数插值解题思路：由1得到的x，y和t，u的映射表，f(t(x，y)，u(x，y)，即求得f（x，y）。但由于得到的t，u不可能正好是题目提供的二维数表中的值，需要用相关规则对插值节点加以规范。利用(x，y)以及对应的f（x，y），就可能通过二元拉格朗日插值多项式得到f（x，y）的表达式。插值节点：1）、根据计算得到的t、u值，选取插值节点；选择标准如下：假设对（t，u），这里用（x，y）代替：设： a)、若满足： 则应选择 为插值节点 b)、若满足：或，则取或；或，则取或；2）、双元二次插值子程序相应的插值多项式为：其中3、最小二乘法曲面拟合设在三维直角坐标系中给定（m+1）*(n+1)个点（即三维坐标）在本题中即为。选定M+1个x的函数以及N+1个y的函数。本题中，于是得到乘积型基函数构成的曲面，随着k值的不断增大，精度会越来越大，题目要求精度为，此时的k即为要求的最小值。解题思路：1）、求解矩阵A固定，以为基函数对数据作最小二乘拟合，得到n+1条拟合曲线其中是法方程的解，而，求解n+1线性方程组，得到矩阵A。2）、求解矩阵G3）、系数矩阵C4、子程序说明子程序名称功能subroutine f_fit(t1,t2,c,sigma)最小二乘法曲面拟合子程序，可给出拟合精度sigmasubroutine f_pxy(c,t1,t2,x,y,p_xy)以x,y的幂函数为基，得到拟合系数矩阵Csubroutine f_zxy(z)分片插值子函数，利用已知的（x,y）,得到z(x,y)subroutine f_zut(u,t,p)分片插值子函数，利用求取的（u,t）,得到z(u,t)subroutine DLU(a,b,x)Doolittle分解求线性方程组子函数subroutine f_newton_iteration(x,y,u,t)Newton迭代法解非线性方程组子程序5、主程序main功能说明主程序对xi,yi赋值，通过调用子程序对非线性方程组求解，得到相应的数据（t,u,v,w），通过调用插值子程序，得到对应的z=f(x,y),并以文件的形式进行输出。通过调用拟合子程序对拟合系数矩阵及拟合精度的求解，结果以文件形式输出。二、fortran源程序!/曲面拟合子函数，并给出拟合精度/subroutine f_fit(t1,t2,c,sigma)use imslimplicit noneinteger i,j,t1,t2parameter n1=11parameter n2=21dimension b(n1,t1),b_trans(t1,n1),b_trans_b(t1,t1),b_inverse(t1,t1)dimension g(n2,t2),g_trans(t2,n2),g_trans_g(t2,t2),g_inverse(t2,t2)double precision b,g,b_trans,g_trans,b_trans_b,g_trans_g,b_inverse,g_inversedimension temp_1(t1,n1),temp_2(t1,n2),temp_3(t1,t2),&c(t1,t2),u(n1,n2),p_xy(n1,n2),x(n1),y(n2)double precision temp_1,temp_2,temp_3,c,u,p_xy,sigma, x,y!/初始化x,y/do i=1,n1 x(i)=0.08*(i-1)end do do j=1,n2 y(j)=0.5+0.05*(j-1)end do!/据题意，求出矩阵b,g/do i=1,n1 do j=1,t1 b(i,j)=x(i)*(j-1) end doend dodo i=1,n2 do j=1,t2 g(i,j)=y(i)*(j-1) end doend do!/确定b,g的转置b_trans和g_trans/do i=1,n1 do j=1,t1 b_trans(j,i)=b(i,j) end doend dodo i=1,n2 do j=1,t2 g_trans(j,i)=g(i,j) end doend do!/求解b_trans_b和g_trans_g的逆矩阵b_inverse和g_inverse/b_trans_b=matmul(b_trans,b)!matmul为矩阵相乘函数，库函数g_trans_g=matmul(g_trans,g)b_inverse=.i.b_trans_bg_inverse=.i.g_trans_g !/求解c矩阵/call f_zxy(u)temp_1=matmul(b_inverse,b_trans)temp_2=matmul(temp_1,u)temp_3=matmul(temp_2,g)c=matmul(temp_3,g_inverse)!/求拟合精度误差/sigma=0do i=1,n1 do j=1,n2 call f_pxy(c,t1,t2,x(i),y(j),p_xy(i,j) sigma=sigma+(u(i,j)-p_xy(i,j)*2 end doend dowrite (*,*) sigmaend subroutine f_fit!/子函数f_pxy给定拟合曲面的近似表达式/subroutine f_pxy(c,t1,t2,x,y,p_xy)implicit noneinteger t1,t2dimension c(t1,t2)double precision p_xy,temp_4,x,y,cinteger i,j,ktemp_4=0.0d0do i=1,t1 do j=1,t2 temp_4=temp_4+c(i,j)*(x*(i-1)*(y*(j-1) end doend dop_xy=temp_4 !拟合曲面end subroutine f_pxy!/分片插值子函数，得出z=z(x,y)/subroutine f_zxy(z)dimension x(11),y(21),u(11,21),t(11,21),z(11,21)double precision x,y,u,t,zinteger i,jdo i=1,11 do j=1,21 x(i)=0.08*(i-1) y(j)=0.5+0.05*(j-1)!调用牛顿迭代法求非线性方程子函数 call f_newton_iteration(x(i),y(j),u(i,j),t(i,j) call f_zut(u(i,j),t(i,j),z(i,j) end doend doend subroutine f_zxy!/ Doolittle分解子函数求解线性方程组/subroutine DLU(a,b,x)integer,parameter : kkk=4real(kind=8) mreal(kind=8),dimension(kkk,kkk),intent(in): areal(kind=8),dimension(kkk,kkk):l=0,u=0real(kind=8),dimension(kkk): yreal(kind=8),dimension(kkk),intent(out):xreal(kind=8),dimension(kkk),intent(in):binteger i,j,kdo k=1,kkk,1 do j=k,kkk,1 m=0 do t=1,k-1,1 m=l(k,t)*u(t,j)+m end do u(k,j)=a(k,j)-m end do do i=k+1,kkk,1 m=0 do t=1,k-1,1 m=m+l(i,t)*u(t,k) end do l(i,k)=(a(i,k)-m)/u(k,k) end do l(k,k)=1end do!/解方程/y(1)=b(1)do i=1,kkk,1 m=0 do t=1,i-1,1 m=m+l(i,t)*y(t) end do y(i)=b(i)-mend dox(kkk)=y(kkk)/u(kkk,kkk)do i=kkk-1,1,-1 m=0 do t=i+1,kkk,1 m=m+u(i,t)*x(t) end do x(i)=(y(i)-m)/u(i,i)end doend subroutine DLU!/牛顿迭代法子程序/subroutine f_newton_iteration(x,y,u,t)parameter n=4dimension f(n),aa(n,n),f_delta(n)double precision f,aa,f_deltadouble precision t,u,v,w,x,ydouble precision epsion,s1,s2integer i,j!迭代初始值t=1.0u=1.0v=1.0w=1.0epsion=1.0do while(epsion.ge.1e-12)f(1)=-(0.5*cos(t)+u+v+w-x-2.67)f(2)=-(t+0.5*sin(u)+v+w-y-1.07)f(3)=-(0.5*t+u+cos(v)+w-x-3.74)f(4)=-(t+0.5*u+v+sin(w)-y-0.79)aa(1,1)=-0.5*sin(t)aa(1,2)=1aa(1,3)=1aa(1,4)=1aa(2,1)=1aa(2,2)=0.5*cos(u)aa(2,3)=1aa(2,4)=1aa(3,1)=0.5aa(3,2)=1aa(3,3)=-sin(v)aa(3,4)=1aa(4,1)=1aa(4,2)=0.5aa(4,3)=1aa(4,4)=cos(w)call DLU(aa,f,f_delta)s1=f_delta(1)*f_delta(1)+f_delta(2)*f_delta(2)+f_delta(3)*f_delta(3)+f_delta(4)*f_delta(4)s1=sqrt(s1)s2=t*t+u*u+v*v+w*ws2=sqrt(s2)epsion=s1/s2t=t+f_delta(1)u=u+f_delta(2)v=v+f_delta(3)w=w+f_delta(4)end doend subroutine f_newton_iteration!/分片插值子函数，得出z=z(u,t)/subroutine f_zut(u,t,p)implicit noneparameter n=6dimension x(n),y(n)double precision x,y,h,h1double precision u,tinteger i,j,ii,jj !ii,jj 插值节点integer k,r,ti dimension lx(n),ly(n),z(n,n) double precision p,lx,ly,z !p,lx,ly,z 插值多项式中的项x(1)=0y(1)=0h=0.4h1=0.2do i=2,nx(i)=x(1)+(i-1)*hy(i)=y(1)+(i-1)*h1end doii=0jj=0p=0!/给定z(n,n)/z(:,1)=(/-0.5,-0.34,0.14,0.94,2.06,3.5/)z(:,2)=(/-0.42,-0.5,-0.26,0.3,1.18,2.38/)z(:,3)=(/-0.18,-0.5,-0.5,-0.18,0.46,1.42/)z(:,4)=(/0.22,-0.34,-0.58,-0.5,-0.1,0.62/)z(:,5)=(/0.78,-0.02,-0.5,-0.66,-0.5,-0.02/)z(:,6)=(/1.5,0.46,-0.26,-0.66,-0.74,-0.5/)do i=3,n-2 if (u.gt.(x(i)-h/2).and.(u.le.(x(i)+h/2) then ii=i goto 10 else if (u.le.(x(3)-h/2) then ii=2 else ii=n-1 end ifend do10 do j=3,n-2 if (t.gt.(y(j)-h1/2).and.(t.le.(y(j)+h1/2) then jj=j goto 20 else if (t.le.(y(3)-h1/2) then jj=2 else jj=n-1 end if end do20 do k=ii-1,ii+1 do r=jj-1,jj+1 lx(k)=1 ly(r)=1 do ti=ii-1,ii+1 if (ti.ne.k) then lx(k)=lx(k)*(u-x(ti)/(x(k)-x(ti) end if end do do ti=jj-1,jj+1 if (ti.ne.r) then ly(r)=ly(r)*(t-y(ti)/(y(r)-y(ti) end if end do p=p+lx(k)*ly(r)*z(k,r) end do end doend subroutine f_zut!/主函数main/program main implicit nonedouble precision vector_x(0:10),vector_y(0:20)double precision xx(0:7),yy(0:4)double precision tu(0:230,0:1),ut(0:1),zt(0:230),c(0:10,0:10),tu2(45,0:1),zt2(0:39),tt(45)double precision z,sigmainteger i,j,m,n,l,t2,n2n=10m=20do i=0,10 vector_x(i)=0.08*iend dodo j=0,20 vector_y(j)=0.5+0.05*jend doopen(11,file=xytuz.txt)write(11,100)x,y,t,u,zdo i=0,10 do j=0,20 call f_newton_iteration(vector_x(i),vector_y(j),&tu(i*21+j,1),tu(i*21+j,0) call f_zut(tu(i*21+j,1),tu(i*21+j,0),zt(i*21+j) write(11,200) vector_x(i),vector_y(j),&tu(i*21+j,1),tu(i*21+j,0),zt(i*21+j) end doend doclose(11)open(22,file=f(x,y).txt)write(22,400)xi,yi,f(xi,yi)do i=0,n do j=0,m write(22,300) vector_x(i),vector_y(j),zt(i*21+j) end doend doclose(22)do l=1,6 call f_fit(l,l,c,sigma)end doopen(33,file=matrix_c.txt) do i=0,5 write(33,(20e20.12)(c(i,j),j=0,5) end doclose(33)t2=8n2=5do i=1,t2 xx(i-1)=0.1*iend dodo j=1,n2 yy(j-1)=0.5+0.2*jend dodo i=1,t2 do j=1,n2 call f_newton_iteration(xx(i-1),yy(j-1),tu2(i-1)*5+j,1), tu2(i-1)*5+j,0) end doend dodo i=1,40 call f_zut(tu2(i,1),tu2(i,0),zt2(i-1)end dodo i=1,t2 do j=1,n2 call f_pxy(c,6,6,xx(i-1),yy(j-1),tt(i-1)*5+j) end doend doopen(44,file=f2(x,y).txt)write(44,600)x*i,y*i,f(xi,yi),p(xi,yi)do i=1,t2 do j=1,n2 write(44,500) xx(i-1),yy(j-1),zt2(i-1)*5+j-1),tt(i-1)*5+j) end doend doclose(44)! /输出格式控制/100 format(2a10,3a15)200 format(2e12.4,3e20.12)300 format(2e12.4,e20.12)400 format(2a10,a18)500 format(2e20.7,2e20.12)600 format(2a18,2a18)end program main 三、结果汇总1、数表：xi,yi、f(xi,yi) xi yi f(xi,yi)0.0000E+00 0.5000E+00 0.446504069241E+00 0.0000E+00 0.5500E+00 0.324683310597E+00 0.0000E+00 0.6000E+00 0.210159730631E+00 0.0000E+00 0.6500E+00 0.103043643055E+00 0.0000E+00 0.7000E+00 0.340192524000E-02 0.0000E+00 0.7500E+00 -0.887357886680E-01 0.0000E+00 0.8000E+00 -0.173371611924E+00 0.0000E+00 0.8500E+00 -0.250534594048E+00 0.0000E+00 0.9000E+00 -0.320276491493E+00 0.0000E+00 0.9500E+00 -0.382668052743E+00 0.0000E+00 0.1000E+01 -0.437795745638E+00 0.0000E+00 0.1050E+01 -0.485758921243E+00 0.0000E+00 0.1100E+01 -0.526667236102E+00 0.0000E+00 0.1150E+01 -0.560638463024E+00 0.0000E+00 0.1200E+01 -0.587796524654E+00 0.0000E+00 0.1250E+01 -0.608269768328E+00 0.0000E+00 0.1300E+01 -0.622189446194E+00 0.0000E+00 0.1350E+01 -0.629688384281E+00 0.0000E+00 0.1400E+01 -0.630899769389E+00 0.0000E+00 0.1450E+01 -0.625956164380E+00 0.0000E+00 0.1500E+01 -0.614988550047E+00 0.8000E-01 0.5000E+00 0.638015234587E+00 0.8000E-01 0.5500E+00 0.506611795876E+00 0.8000E-01 0.6000E+00 0.382176408075E+00 0.8000E-01 0.6500E+00 0.264863525560E+00 0.8000E-01 0.7000E+00 0.154780230633E+00 0.8000E-01 0.7500E+00 0.519927097057E-01 0.8000E-01 0.8000E+00 -0.434680179481E-01 0.8000E-01 0.8500E+00 -0.131601038093E+00 0.8000E-01 0.9000E+00 -0.212431072587E+00 0.8000E-01 0.9500E+00 -0.286004537561E+00 0.8000E-01 0.1000E+01 -0.352386059629E+00 0.8000E-01 0.1050E+01 -0.411655437858E+00 0.8000E-01 0.1100E+01 -0.463904893928E+00 0.8000E-01 0.1150E+01 -0.509236708669E+00 0.8000E-01 0.1200E+01 -0.547761104059E+00 0.8000E-01 0.1250E+01 -0.579594377202E+00 0.8000E-01 0.1300E+01 -0.604857251213E+00 0.8000E-01 0.1350E+01 -0.623673426166E+00 0.8000E-01 0.1400E+01 -0.636168257103E+00 0.8000E-01 0.1450E+01 -0.642467668455E+00 0.8000E-01 0.1500E+01 -0.642697116743E+00 0.1600E+00 0.5000E+00 0.840081396883E+00 0.1600E+00 0.5500E+00 0.699764166271E+00 0.1600E+00 0.6000E+00 0.566061444836E+00 0.1600E+00 0.6500E+00 0.439171614444E+00 0.1600E+00 0.7000E+00 0.319242167288E+00 0.1600E+00 0.7500E+00 0.206376218657E+00 0.1600E+00 0.8000E+00 0.100638546919E+00 0.1600E+00 0.8500E+00 0.206075984654E-02 0.1600E+00 0.9000E+00 -0.893540080136E-01 0.1600E+00 0.9500E+00 -0.173626954709E+00 0.1600E+00 0.1000E+01 -0.250799943998E+00 0.1600E+00 0.1050E+01 -0.320932252750E+00 0.1600E+00 0.1100E+01 -0.384097719188E+00 0.1600E+00 0.1150E+01 -0.440382160815E+00 0.1600E+00 0.1200E+01 -0.489881139368E+00 0.1600E+00 0.1250E+01 -0.532697954784E+00 0.1600E+00 0.1300E+01 -0.568941871351E+00 0.1600E+00 0.1350E+01 -0.598726545062E+00 0.1600E+00 0.1400E+01 -0.622168637157E+00 0.1600E+00 0.1450E+01 -0.639386546614E+00 0.1600E+00 0.1500E+01 -0.650499364497E+00 0.2400E+00 0.5000E+00 0.105151509248E+01 0.2400E+00 0.5500E+00 0.902927427685E+00 0.2400E+00 0.6000E+00 0.760580262813E+00 0.2400E+00 0.6500E+00 0.624715195878E+00 0.2400E+00 0.7000E+00 0.495519776775E+00 0.2400E+00 0.7500E+00 0.373134067685E+00 0.2400E+00 0.8000E+00 0.257656776698E+00 0.2400E+00 0.8500E+00 0.149150588766E+00 0.2400E+00 0.9000E+00 0.476470043700E-01 0.2400E+00 0.9500E+00 -0.468493081747E-01 0.2400E+00 0.1000E+01 -0.134356747670E+00 0.2400E+00 0.1050E+01 -0.214913334079E+00 0.2400E+00 0.1100E+01 -0.288573687009E+00 0.2400E+00 0.1150E+01 -0.355406352189E+00 0.2400E+00 0.1200E+01 -0.415491385199E+00 0.2400E+00 0.1250E+01 -0.468918240380E+00 0.2400E+00 0.1300E+01 -0.515783875727E+00 0.2400E+00 0.1350E+01 -0.556191070573E+00 0.2400E+00 0.1400E+01 -0.590246929357E+00 0.2400E+00 0.1450E+01 -0.618061557834E+00 0.2400E+00 0.1500E+01 -0.639746851961E+00 0.3200E+00 0.5000E+00 0.127124675843E+01 0.3200E+00 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