lanczosnosym.m

various matlab programs to slove various problems

function [xk,nres,niter]=lanczosnosym(A,b,x0,m,toll)
r0=b-A*x0; nres0=norm(r0,2); 
if nres0 ~= 0
  V=r0/nres0;  Z=V; gamma(1)=0; beta(1)=0;  k=1; nres=1;
  while k <= m & nres > toll
     vk=V(:,k);  zk=Z(:,k);
     if    k==1,  vk1=0*vk;   zk1=0*zk;
     else,  vk1=V(:,k-1);  zk1=Z(:,k-1);  end
     alpha(k)=zk'*A*vk;
     tildev=A*vk-alpha(k)*vk-beta(k)*vk1;
     tildez=A'*zk-alpha(k)*zk-gamma(k)*zk1;
     gamma(k+1)=sqrt(abs(tildez'*tildev));
     if gamma(k+1) == 0,    k=m+2;
     else
       beta(k+1)=tildez'*tildev/gamma(k+1);
       Z=[Z,tildez/beta(k+1)];
       V=[V,tildev/gamma(k+1)];
     end
     if k~=m+2 
       if k==1
         Tk = alpha;
       else
         Tk=diag(alpha)+diag(beta(2:k),1)+diag(gamma(2:k),-1);
       end
       yk=Tk \ (nres0*[1,0*[1:k-1]]');
       xk=x0+V(:,1:k)*yk;
       nres=abs(gamma(k+1)*[0*[1:k-1],1]*yk)*norm(V(:,k+1),2)/nres0;
       k=k+1;
     end
 end
else
  x=x0;
end
if k==m+2, niter=-k; else, niter=k-1; end
return