MATLAB Function Reference |
Syntax
x = symmlq(A,b) symmlq(A,b,tol) symmlq(A,b,tol,maxit) symmlq(A,b,tol,maxit,M) symmlq(A,b,tol,maxit,M1,M2) symmlq(A,b,tol,maxit,M1,M2,x0) [x,flag] = symmlq(A,b,...) [x,flag,relres] = symmlq(A,b,...) [x,flag,relres,iter] = symmlq(A,b,...) [x,flag,relres,iter,resvec] = symmlq(A,b,...) [x,flag,relres,iter,resvec,resveccg] = symmlq(A,b,...)
Description
x = symmlq(A,b)
attempts to solve the system of linear equations A*x=b
for x
. The n
-by-n
coefficient matrix A
must be symmetric but need not be positive definite. It should also be large and sparse. The column vector b
must have length n
. A
can be a function handle afun
such that afun(x)
returns A*x
. See Function Handles in the MATLAB Programming documentation for more information.
Parameterizing Functions Called by Function Functions, in the MATLAB Mathematics documentation, explains how to provide additional parameters to the function afun
, as well as the preconditioner function mfun
described below, if necessary.
If symmlq
converges, a message to that effect is displayed. If symmlq
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.
symmlq(A,b,tol)
specifies the tolerance of the method. If tol
is []
, then symmlq
uses the default, 1e-6
.
symmlq(A,b,tol,maxit)
specifies the maximum number of iterations. If maxit
is []
, then symmlq
uses the default, min(n,20)
.
symmlq(A,b,tol,maxit,M) and symmlq(A,b,tol,maxit,M1,M2)
use the symmetric positive definite preconditioner M
or M = M1*M2
and effectively solve the system inv(sqrt(M))*A*inv(sqrt(M))*y = inv(sqrt(M))*b
for y
and then return x = inv(sqrt(M))*y
. If M
is []
then symmlq
applies no preconditioner. M
can be a function handle mfun
such that mfun(x)
returns M\x
.
symmlq(A,b,tol,maxit,M1,M2,x0)
specifies the initial guess. If x0
is []
, then symmlq
uses the default, an all-zero vector.
[x,flag] = symmlq(A,b,...)
also returns a convergence flag.
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] = symmlq(A,b,...)
also returns the relative residual norm(b-A*x)/norm(b)
. If flag
is 0
, relres <= tol
.
[x,flag,relres,iter] = symmlq(A,b,...)
also returns the iteration number at which x
was computed, where 0 <= iter <= maxit
.
[x,flag,relres,iter,resvec] = symmlq(A,b,...)
also returns a vector of estimates of the symmlq
residual norms at each iteration, including norm(b-A*x0)
.
[x,flag,relres,iter,resvec,resveccg] = symmlq(A,b,...)
also returns a vector of estimates of the conjugate gradients residual norms at each iteration.
Examples
n = 100; on = ones(n,1); A = spdiags([-2*on 4*on -2*on],-1:1,n,n); b = sum(A,2); tol = 1e-10; maxit = 50; M1 = spdiags(4*on,0,n,n); x = symmlq(A,b,tol,maxit,M1); symmlq converged at iteration 49 to a solution with relative residual 4.3e-015
This example replaces the matrix A
in Example 1 with a handle to a matrix-vector product function afun
. The example is contained in an M-file run_symmlq
that
symmlq
with the function handle @afun
as its first argument.
afun
as a nested function, so that all variables in run_symmlq
are available to afun
.
The following shows the code for run_symmlq
:
function x1 = run_symmlq 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 = symmlq(@afun,b,tol,maxit,M1); function y = afun(x) y = 4 * x; y(2:n) = y(2:n) - 2 * x(1:n-1); y(1:n-1) = y(1:n-1) - 2 * x(2:n); end end
Use a symmetric indefinite matrix that fails with pcg
.
A = diag([20:-1:1,-1:-1:-20]); b = sum(A,2); % The true solution is the vector of all ones. x = pcg(A,b); % Errors out at the first iteration. pcg stopped at iteration 1 without converging to the desired tolerance 1e-006 because a scalar quantity became too small or too large to continue computing. The iterate returned (number 0) has relative residual 1
However, symmlq
can handle the indefinite matrix A
.
x = symmlq(A,b,1e-6,40); symmlq converged at iteration 39 to a solution with relative residual 1.3e-007
See Also
bicg
, bicgstab
, cgs
, lsqr
, gmres
, minres
, pcg
, qmr
function_handle
(@
), mldivide
(\
)
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] Paige, C. C. and M. A. Saunders, "Solution of Sparse Indefinite Systems of Linear Equations." SIAM J. Numer. Anal., Vol.12, 1975, pp. 617-629.
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