Symmetric LQ method
x = symmlq(A,b)
[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,...)
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. You can specify A as a function handle, afun, such that afun(x) 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 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,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 converged to the desired tolerance tol within maxit iterations.
symmlq iterated maxit times but did not converge.
Preconditioner M was ill-conditioned.
symmlq stagnated. (Two consecutive iterates were the same.)
One of the scalar quantities calculated during symmlq became too small or too large to continue computing.
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.
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 the function 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
MATLAB® software displays the message
symmlq converged at iteration 49 to a solution with relative residual 4.3e-015
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
 Barrett, R., M. Berry, T. F. Chan, et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.
 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.