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symmlq - Symmetric LQ method

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.

Parametrizing 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 = in(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.

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

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

Example 1

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

Example 2

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

Calls symmlq with the function handle @afun as its first argument.
Contains 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

When you enter

x1=run_symmlq;

MATLAB software displays the message

symmlq converged at iteration 49 to a solution with relative 
residual 4.3e-015

Example 3

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

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.