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Quasi-minimal residual method
x = qmr(A,b)
qmr(A,b,tol)
qmr(A,b,tol,maxit)
qmr(A,b,tol,maxit,M)
qmr(A,b,tol,maxit,M1,M2)
qmr(A,b,tol,maxit,M1,M2,x0)
[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) specifies the tolerance of the method. If tol is [], then qmr uses the default, 1e-6.
qmr(A,b,tol,maxit) specifies the maximum number of iterations. If maxit is [], then qmr uses the default, min(n,20).
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(A,b,tol,maxit,M1,M2,x0) specifies the initial guess. If x0 is [], then qmr uses the default, an all zero vector.
[x,flag] = qmr(A,b,...) also returns a convergence flag.
Flag | Convergence |
---|---|
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.
[x,flag,relres] = qmr(A,b,...) also returns the relative residual norm(b-A*x)/norm(b). If flag is 0, relres <= tol.
[x,flag,relres,iter] = qmr(A,b,...) also returns the iteration number at which x was computed, where 0 <= iter <= maxit.
[x,flag,relres,iter,resvec] = qmr(A,b,...) also returns a vector of the residual norms at each iteration, including norm(b-A*x0).
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
x1=run_qmr;
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');
[1] Barrett, R., M. Berry, T. F. Chan, et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.
[2] 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.