Symmetric LQ method
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,...)
x = symmlq(A,b)
attempts
to solve the system of linear equations A*x=b
for x
.
The n
byn
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(bA*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, 1e6
.
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 allzero vector.
[x,flag] = symmlq(A,b,...)
also
returns a convergence flag.
Flag  Convergence 

0 

1 

2  Preconditioner 
3 

4  One of the scalar quantities calculated during 
5  Preconditioner 
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(bA*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(bA*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.
n = 100; on = ones(n,1); A = spdiags([2*on 4*on 2*on],1:1,n,n); b = sum(A,2); tol = 1e10; 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 4.3e015
This example replaces the matrix A
in Example
1 with a handle to a matrixvector 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 = 1e8; 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:n1); y(1:n1) = y(1:n1)  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 4.3e015
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 1e006 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,1e6,40); symmlq converged at iteration 39 to a solution with relative residual 1.3e007
[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. 617629.