大规模矩阵问题的Krylov子空间方法综述课件.ppt

1、大规模矩阵问题的大规模矩阵问题的Krylov 子空间方法综述子空间方法综述K(L)X(0)X(1)KLX(0)X(1)TW AV1(0)()TTyW AVW r(0)()0TWrAVy(1)(0)1(0)()TTxxV W AVW r(1)()min( )(0)xzzx K1/2( )( (),)zA xz xz(0)(1)()min( )z xxzK1/22( )(,)zbAzbAz bAz(0)(0)(0)2(0)(0)( ,),AmA rspan rArA rrm -1，Kdim( , )min , mA vmK(, )mA vK(1)1/ 2( , )vvv v( )( )(1)(2)

2、( )12 (2.1)jjjjjjjwAvh vh vh v( )( )(,), 1,2,jiijhAvvij( )( ) 1/21,(,)jjjjhww( )(1)1,jjjjwvh mH()()1()mmTmmmmmAVV HweVHTmmmV AVH( )(1)(2)( 1)( )121,(,), 1,2, .jiiijjjijhAvh vh vhvvijTW AV1(0)()TTyW AVW r(0)()0TWrAVy(1)(0)1(0)()TTxxV W AVW r()(0)()mmmxxV ymH()()1()mmTmmmmmAVV HweVHTmmmV AVH( )(1)(2)(

3、 1)( )121,(,), 1,2, .jiiijjjijhAvh vh vhvvij1222311mmmmmmT(0)(1)()min( )z xxzK1/22( )(,)zbAzbAz bAzmH()()1()mmTmmmmmAVV HweVHTmmmV AVH( )(1)(2)( 1)( )121,(,), 1,2, .jiiijjjijhAvh vh vhvvij(0)( )(1)( )(1)( )(1)( )122222( )()mmmmmmmmmzb AzrAV yvAV yVeH yeH ymH()()1()mmTmmmmmAVV HweVHTmmmV AVH()(1)1,mm

4、mmwhv1,0mmh( , )mA vK1,0mmh(,)iiyimiiH yyimiiH yy(1)()()1,1,()()mmTmTimmmmmmiiiiAV yV yhveyheyimixV yTMLL(0)(0)(0)2(0)(0)2(0)(0)(0)(0)(0)( ,),A(,),mTTTTmA rspan rArA rrArspan rA rArArm -1m -1，KL(1)(1)( )vq A v( )q (1)1niiivp(1)(1)111( )()()()nrniiiiiiiiiiii rvq A vqpqpqp ( )q 1( , ) ,mmA Vspan V AVA

5、VKfunction x=cg(A,b,tol)t=cputime;n = numel(b);x0=rand(n,1)r=b-A*x0;s=b-A*x0;x=x0;j=1;while (norm(r)tol) count(j)=j-1; res(j)=norm(r); alpha(j)=dot(r,r)/dot(s,(A*r);b1=dot(r,r); (续续)x=x+(alpha(j)*s;r=r-(alpha(j)*(A*s);b2=dot(r,r);beta(j+1)=b2/b1;s=r+(beta(j+1)*s; j=j+1;end t=cputime-tplot(count,res),xlabel(迭代步迭代步),ylabel(残量残量),title(残量随迭代步的变化残量随迭代步的变化),set(findobj(gca,Type,line,Color,0 0 1),.Color,red,.LineWidth,2)