Transpose-free quasi-minimal residual method
x = tfqmr(A,b)
x = tfqmr(afun,b)
x = tfqmr(a,b,tol)
x = tfqmr(a,b,tol,maxit)
x = tfqmr(a,b,tol,maxit,m)
x = tfqmr(a,b,tol,maxit,m1,m2,x0)
[x,flag] = tfqmr(A,B,...)
[x,flag,relres] = tfqmr(A,b,...)
[x,flag,relres,iter,resvec] = tfqmr(A,b,...)
x = tfqmr(A,b) attempts to solve the system of linear equations A*x=b for x. The n-by-n coefficient matrix A must be square and the right-hand side column vector b must have length n.
x = tfqmr(afun,b) accepts a function handle, afun, instead of the matrix A. The function, afun(x), accepts a vector input x and returns the matrix-vector product A*x. In all of the following syntaxes, you can replace A by afun. Parameterizing Functions explains how to provide additional parameters to the function afun.
x = tfqmr(a,b,tol) specifies the tolerance of the method. If tol is  then tfqmr uses the default, 1e-6.
x = tfqmr(a,b,tol,maxit) specifies the maximum number of iterations. If maxit is  then tfqmr uses the default, min(N,20).
x = tfqmr(a,b,tol,maxit,m) and x = tfqmr(a,b,tol,maxit,m1,m2) use preconditioners m or m=m1*m2 and effectively solve the system A*inv(M)*x = B for x. If M is  then a preconditioner is not applied. M may be a function handle mfun such that mfun(x) returns m\x.
x = tfqmr(a,b,tol,maxit,m1,m2,x0) specifies the initial guess. If x0 is  then tfqmr uses the default, an all zero vector.
[x,flag] = tfqmr(A,B,...) also returns a convergence flag:
tfqmr converged to the desired tolerance tol within maxit iterations.
tfqmr iterated maxit times but did not converge.
Preconditioner m was ill-conditioned.
tfqmr stagnated. (Two consecutive iterates were the same.)
One of the scalar quantities calculated during tfqmr became too small or too large to continue computing.
[x,flag,relres] = tfqmr(A,b,...) also returns the relative residual norm(b-A*x)/norm(b). If flag is 0, then relres <= tol.
[x,flag,relres,y]y(A,b,...) also returns the iteration number at which x was computed: 0 <= iter <= maxit.
[x,flag,relres,iter,resvec] = tfqmr(A,b,...) also returns a vector of the residual norms at each iteration, including norm(b-A*x0).
This example shows how to use tfqmr with a matrix input and with a function input.
n = 100; on = ones(n,1); A = spdiags([-2*on 4*on -on],-1:1,n,n); b = sum(A,2); tol = 1e-8; maxit = 15; M1 = spdiags([on/(-2) on],-1:0,n,n); M2 = spdiags([4*on -on],0:1,n,n); x = tfqmr(A,b,tol,maxit,M1,M2,);
You can also use a matrix-vector product function as input:
function y = afun(x,n) y = 4 * x; y(2:n) = y(2:n) - 2 * x(1:n-1); y(1:n-1) = y(1:n-1) - x(2:n); x1 = tfqmr(@(x)afun(x,n),b,tol,maxit,M1,M2);
If applyOp is a function suitable for use with qmr, it may be used with tfqmr by wrapping it in an anonymous function:
x1 = tfqmr(@(x)applyOp(x,'notransp'),b,tol,maxit,M1,M2);
This example demonstrates the use of a preconditioner.
Load A = west0479, a real 479-by-479 nonsymmetric sparse matrix.
load west0479; A = west0479;
Define b so that the true solution is a vector of all ones.
b = full(sum(A,2));
Set the tolerance and maximum number of iterations.
tol = 1e-12; maxit = 20;
Use tfqmr to find a solution at the requested tolerance and number of iterations.
[x0,fl0,rr0,it0,rv0] = tfqmr(A,b,tol,maxit);
fl0 is 1 because tfqmr does not converge to the requested tolerance 1e-12 within the requested 20 iterations. The seventeenth iterate is the best approximate solution and is the one returned as indicated by it0 = 17. MATLAB® stores the residual history in rv0.
Plot the behavior of tfqmr.
semilogy(0:maxit,rv0(1:maxit+1)/norm(b),'-o'); xlabel('Iteration number'); ylabel('Relative residual');
Note that like bicgstab, tfqmr keeps track of half iterations. The plot shows that the solution does not converge. You can use a preconditioner to improve the outcome.
Create the preconditioner with ilu, since the matrix A is nonsymmetric.
[L,U] = ilu(A,struct('type','ilutp','droptol',1e-5));
Error using ilu There is a pivot equal to zero. Consider decreasing the drop tolerance or consider using the 'udiag' option.
MATLAB cannot construct the incomplete LU as it would result in a singular factor, which is useless as a preconditioner.
You can try again with a reduced drop tolerance, as indicated by the error message.
[L,U] = ilu(A,struct('type','ilutp','droptol',1e-6)); [x1,fl1,rr1,it1,rv1] = tfqmr(A,b,tol,maxit,L,U);
fl1 is 0 because tfqmr drives the relative residual to 4.1410e-014 (the value of rr1). The relative residual is less than the prescribed tolerance of 1e-12 at the sixth iteration (the value of it1) when preconditioned by the incomplete LU factorization with a drop tolerance of 1e-6. The output rv1(1) is norm(b), and the output rv1(7) is norm(b-A*x2).
You can follow the progress of tfqmr by plotting the relative residuals at each iteration starting from the initial estimate (iterate number 0).
semilogy(0:0.5:it1,rv1/norm(b),'-o'); xlabel('Iteration number'); ylabel('Relative residual');