tfqmr - Transpose-free quasi-minimal residual method

Syntax

Description

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. 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. See Function Handles in the MATLAB Programming documentation for more information.Parameterizing Functions, in the MATLAB Mathematics documentation, 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:

Flag	Convergence
0	tfqmr converged to the desired tolerance tol within maxit iterations.
1	tfqmr iterated maxit times but did not converge.
2	Preconditioner m was ill-conditioned.
3	tfqmr stagnated. (Two consecutive iterates were the same.)
4	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).

Examples

Example 1

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);

Example 2

This example demonstrates the use of a preconditioner.

1. Load A = west0479, a real 479-by-479 nonsymmetric sparse matrix:

   load west0479;
   A = west0479;

2. Define b so that the true solution is a vector of all ones:

   b = full(sum(A,2));
   

3. Set the tolerance and maximum number of iterations:

   tol = 1e-12; maxit = 20;

4. 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.

5. Plot the behavior of tfqmr:

   semilogy(0:maxit,rv0/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.

6. 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.

   MATLABcannot construct the incomplete LU as it would result in a singular factor, which is useless as a preconditioner.

7. 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).

8. 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:it1,rv1/norm(b),'-o');
xlabel('Iteration number');
ylabel('Relative residual');

See Also

bicg | bicgstab | bicgstabl | cgs | gmres | lsqr | luinc | minres | mldivide | pcg | qmr | symmlq

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