MATLAB Function Reference    

Quasi-Minimal Residual method



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. A can be a function afun such that afun(x) returns A*x and afun(x,'transp') returns A'*x.

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 mfun such that mfun(x) 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.

qmr(afun,b,tol,maxit,m1fun,m2fun,x0,p1,p2,...) passes parameters p1,p2,... to functions afun(x,p1,p2,...) and afun(x,p1,p2,...,'transp') and similarly to the preconditioner functions m1fun and m2fun.

[x,flag] = qmr(A,b,...) also returns a convergence flag.

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).


Example 1.

Alternatively, use this matrix-vector product function

as input to qmr

Example 2.

flag is 1 because qmr does not converge to the default tolerance 1e-6 within the default 20 iterations.

flag1 is 2 because the upper triangular U1 has a zero on its diagonal, and qmr fails in the first iteration when it tries to solve a system such as U1*y = r for y using backslash.

flag2 is 0 because qmr converges to the tolerance of 1.6571e-016 (the value of relres2) at the eighth iteration (the value of iter2) when preconditioned by the incomplete LU factorization with a drop tolerance of 1e-6. resvec2(1) = norm(b) and resvec2(9) = 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).

See Also

bicg, bicgstab, cgs, gmres, lsqr, luinc, minres, pcg, symmlq

@ (function handle), \ (backslash)


[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.

  pwd qr