Quasiminimal 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
byn
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(bA*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, 1e6
.
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 





 Preconditioner 
 The method stagnated. (Two consecutive iterates were the same.) 
 One of the scalar quantities calculated during 
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(bA*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(bA*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 = 1e8; 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 5.6e009
This example replaces the matrix A
in the
previous example with a handle to a matrixvector 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 = 1e8; 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:n1) = y(1:n1)  2 * x(2:n); y(2:n) = y(2:n)  x(1:n1); elseif strcmp(transp_flag,'notransp') % y = A*x y = 4 * x; y(2:n) = y(2:n)  2 * x(1:n1); y(1:n1) = y(1:n1)  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 5.6e009
This example demonstrates the use of a preconditioner.
Load A = west0479
, a real 479by479 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 = 1e12; 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 1e12
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',1e5));
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',1e6)); [x1,fl1,rr1,it1,rv1] = qmr(A,b,tol,maxit,L,U);
fl1
is 0 because qmr
drives the relative residual to 4.1410e014
(the value of rr1
). The relative residual is less than the prescribed tolerance of 1e12
at the sixth iteration (the value of it1
) when preconditioned by the incomplete LU factorization with a drop tolerance of 1e6
. The output rv1(1)
is norm(b)
, and the output rv1(7)
is norm(bA*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 quasiminimal residual method for nonHermitian linear systems,” SIAM Journal: Numer. Math. 60, 1991, pp. 315–339.