Quasi-minimal residual method
x = qmr(A,b)
[x,flag] = qmr(A,b,...)
[x,flag,relres] = qmr(A,b,...)
[x,flag,relres,iter] = qmr(A,b,...)
[x,flag,relres,iter,resvec] = qmr(A,b,...)
x = qmr(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 should be large and sparse. The column vector b must have length n. You can specify A as a function handle, afun, such that afun(x,'notransp') returns A*x and afun(x,'transp') returns A'*x.
Parameterizing Functions explains how to provide additional parameters to the function afun, as well as the preconditioner function mfun described below, if necessary.
If qmr converges, a message to that effect is displayed. If qmr fails to converge after the maximum number of iterations or halts for any reason, a warning message is printed displaying the relative residual norm(b-A*x)/norm(b) and the iteration number at which the method stopped or failed.
qmr(A,b,tol,maxit,M) and qmr(A,b,tol,maxit,M1,M2) use preconditioners M or M = M1*M2 and effectively solve the system inv(M)*A*x = inv(M)*b for x. If M is  then qmr applies no preconditioner. M can be a function handle mfun such that mfun(x,'notransp') returns M\x and mfun(x,'transp') returns M'\x.
qmr converged to the desired tolerance tol within maxit iterations.
qmr iterated maxit times but did not converge.
Preconditioner M was ill-conditioned.
The method stagnated. (Two consecutive iterates were the same.)
One of the scalar quantities calculated during qmr became too small or too large to continue computing.
Whenever flag is not 0, the solution x returned is that with minimal norm residual computed over all the iterations. No messages are displayed if the flag output is specified.
This example shows how to use qmr with a matrix input. The code:
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 = qmr(A,b,tol,maxit,M1,M2);
displays the message:
qmr converged at iteration 9 to a solution... with relative residual 5.6e-009
This example replaces the matrix A in the previous example with a handle to a matrix-vector product function afun. The example is contained in a file run_qmr that
Calls qmr with the function handle @afun as its first argument.
Contains afun as a nested function, so that all variables in run_qmr are available to afun.
The following shows the code for run_qmr:
function x1 = run_qmr 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); x1 = qmr(@afun,b,tol,maxit,M1,M2); function y = afun(x,transp_flag) if strcmp(transp_flag,'transp') % y = A'*x y = 4 * x; y(1:n-1) = y(1:n-1) - 2 * x(2:n); y(2:n) = y(2:n) - x(1:n-1); elseif strcmp(transp_flag,'notransp') % y = A*x 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); end end end
When you enter
MATLAB® software displays the message
qmr converged at iteration 9 to a solution with relative residual 5.6e-009
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 qmr to find a solution at the requested tolerance and number of iterations.
[x0,fl0,rr0,it0,rv0] = qmr(A,b,tol,maxit);
fl0 is 1 because qmr 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 qmr.
semilogy(0:maxit,rv0/norm(b),'-o'); xlabel('Iteration number'); ylabel('Relative residual');
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] = qmr(A,b,tol,maxit,L,U);
fl1 is 0 because qmr 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 qmr 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');
 Barrett, R., M. Berry, T. F. Chan, et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.
 Freund, Roland W. and Nöel M. Nachtigal, "QMR: A quasi-minimal residual method for non-Hermitian linear systems," SIAM Journal: Numer. Math. 60, 1991, pp. 315–339.