qmr

Quasi-minimal residual method

Syntax

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,...)

Description

x = qmr(A,b)attempts to solve the system of linear equationsA*x=bforx. Then-by-ncoefficient matrixAmust be square and should be large and sparse. The column vectorbmust have lengthn. You can specifyAas a function handle,afun, such thatafun(x,'notransp')returnsA*xandafun(x,'transp')returns‘* x.

Parameterizing Functionsexplains how to provide additional parameters to the functionafun, as well as the preconditioner functionmfundescribed below, if necessary.

Ifqmrconverges, a message to that effect is displayed. Ifqmrfails to converge after the maximum number of iterations or halts for any reason, a warning message is printed displaying the relative residualnorm(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. Iftolis[], thenqmruses the default,1e-6.

qmr(A,b,tol,maxit)specifies the maximum number of iterations. Ifmaxitis[], thenqmruses the default,min(n,20).

qmr(A,b,tol,maxit,M)andqmr(A,b,tol,maxit,M1,M2)use preconditionersMorM = M1*M2and effectively solve the systeminv(M)*A*x = inv(M)*bforx. IfMis[]thenqmrapplies no preconditioner.Mcan be a function handlemfunsuch thatmfun(x,'notransp')returnsM\xandmfun(x,'transp')returnsM'\x.

qmr(A,b,tol,maxit,M1,M2,x0)specifies the initial guess. Ifx0is[], thenqmruses the default, an all zero vector.

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

Flag	Convergence
`0`	`qmr`converged to the desired tolerance`tol`within`maxit`iterations.
`1`	`qmr`iterated`maxit`times but did not converge.
`2`	Preconditioner`M`was ill-conditioned.
`3`	The method stagnated. (Two consecutive iterates were the same.)
`4`	One of the scalar quantities calculated during`qmr`became too small or too large to continue computing.

Wheneverflagis not0, the solutionxreturned is that with minimal norm residual computed over all the iterations. No messages are displayed if theflagoutput is specified.

[x,flag,relres] = qmr(A,b,...)also returns the relative residualnorm(b-A*x)/norm(b). Ifflagis0,relres <= tol.

[x,flag,relres,iter] = qmr(A,b,...)also returns the iteration number at whichxwas computed, where0 <= iter <= maxit.

[x,flag,relres,iter,resvec] = qmr(A,b,...)also returns a vector of the residual norms at each iteration, includingnorm(b-A*x0).

Examples

Using qmr with a Matrix Input

This example shows how to useqmrwith 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

Using qmr with a Function Handle

This example replaces the matrixAin the previous example with a handle to a matrix-vector product functionafun. The example is contained in a filerun_qmrthat

Callsqmrwith the function handle@afunas its first argument.
Containsafunas a nested function, so that all variables inrun_qmrare available toafun.

The following shows the code forrun_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

Using qmr with a Preconditioner

This example demonstrates the use of a preconditioner.

LoadA = west0479, a real 479-by-479 nonsymmetric sparse matrix.

loadwest0479; A = west0479;

Definebso 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;

Useqmr在请求的宽容和找到解决办法number of iterations.

[x0,fl0,rr0,it0,rv0] = qmr(A,b,tol,maxit);

fl0is 1 becauseqmrdoes not converge to the requested tolerance1e-12within the requested 20 iterations. The seventeenth iterate is the best approximate solution and is the one returned as indicated byit0 = 17. MATLAB stores the residual history inrv0.

Plot the behavior ofqmr.

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 withilu, since the matrixA是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);

fl1is 0 becauseqmrdrives the relative residual to4.1410e-014(the value ofrr1). The relative residual is less than the prescribed tolerance of1e-12at the sixth iteration (the value ofit1) when preconditioned by the incomplete LU factorization with a drop tolerance of1e-6. The outputrv1(1)isnorm(b), and the outputrv1(7)isnorm(b-A*x2).

You can follow the progress ofqmrby 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');

References

[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.

Extended Capabilities

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

bmust be a full (nonsparse) column vector.

For more information, seeRun MATLAB Functions on a GPU(Parallel Computing Toolbox).

Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

Usage notes and limitations:

IfM1is a function, then it is applied independently to each row.

For more information, seeRun MATLAB Functions with Distributed Arrays(Parallel Computing Toolbox).

qmr

Syntax

Description

Examples

Using qmr with a Matrix Input

Using qmr with a Function Handle

Using qmr with a Preconditioner

References

Extended Capabilities

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

See Also

Topics

Introduced before R2006a

MATLAB Documentation

金宝app

Try MATLAB, Simulink, and Other Products

qmr

Syntax

Description

Examples

Using qmr with a Matrix Input

Using qmr with a Function Handle

Using qmr with a Preconditioner

References

Extended Capabilities

GPU ArraysAccelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Distributed ArraysPartition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

See Also

Topics

Introduced before R2006a

MATLAB Documentation

金宝app

Try MATLAB, Simulink, and Other Products

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.