cgs (MATLAB Functions)

Conjugate gradients squared method

Syntax

x = cgs(A,b)
cgs(A,b,tol)
cgs(A,b,tol,maxit)
cgs(A,b,tol,maxit,M)
cgs(A,b,tol,maxit,M1,M2)
cgs(A,b,tol,maxit,M1,M2,x0)
[x,flag] = cgs(A,b,...)
[x,flag,relres] = cgs(A,b,...)
[x,flag,relres,iter] = cgs(A,b,...)
[x,flag,relres,iter,resvec] = cgs(A,b,...)

Description

x = cgs(A,b) attempts to solve the system of linear equations A*x = b for x. The n-by-n coefficient matrix A must be square and should 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 cgs converges, a message to that effect is displayed. If cgs 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.

cgs(A,b,tol) specifies the tolerance of the method, tol. If tol is [], then cgs uses the default, 1e-6.

cgs(A,b,tol,maxit) specifies the maximum number of iterations, maxit. If maxit is [] then cgs uses the default, min(n,20).

cgs(A,b,tol,maxit,M) and cgs(A,b,tol,maxit,M1,M2) use the preconditioner M or M = M1*M2 and effectively solve the system inv(M)*A*x = inv(M)*b for x. If M is [] then cgs applies no preconditioner. M can be a function handle mfun such that mfun(x) returns M\x.

cgs(A,b,tol,maxit,M1,M2,x0) specifies the initial guess x0. If x0 is [], then cgs uses the default, an all-zero vector.

[x,flag] = cgs(A,b,...) returns a solution x and a flag that describes the convergence of cgs.

Flag
Convergence

0
cgs converged to the desired tolerance tol within maxit iterations.

1
cgs iterated maxit times but did not converge.

2
Preconditioner M was ill-conditioned.

3
cgs stagnated. (Two consecutive iterates were the same.)

4
One of the scalar quantities calculated during cgs became too small or too large to continue computing.

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

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] = cgs(A,b,...) also returns the relative residual norm(b-A*x)/norm(b). If flag is 0, then relres <= tol.

[x,flag,relres,iter] = cgs(A,b,...) also returns the iteration number at which x was computed, where 0 <= iter <= maxit.

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

Examples

Example 1.

A = gallery('wilk',21);
b = sum(A,2);
tol = 1e-12;  maxit = 15; 
M1 = diag([10:-1:1 1 1:10]);
x = cgs(A,b,tol,maxit,M1);

displays the message

cgs converged at iteration 13 to a solution with relative residual 
1.3e-016

Example 2.

This example replaces the matrix A in Example 1 with a handle to a matrix-vector product function afun, and the preconditioner M1 with a handle to a backsolve function mfun. The example is contained in an M-file run_cgs that

Calls cgs with the function handle @afun as its first argument.
Contains afun as a nested function, so that all variables in run_cgs are available to afun and myfun.

The following shows the code for run_cgs:

function x1 = run_cgs
n = 21;
A = gallery('wilk',n);
b = sum(A,2);
tol = 1e-12;  maxit = 15; 
x1 = cgs(@afun,b,tol,maxit,@mfun);
 
    function y = afun(x)
        y = [0; x(1:n-1)] + ...
              [((n-1)/2:-1:0)'; (1:(n-1)/2)'].*x + ...
              [x(2:n); 0];
    end
 
    function y = mfun(r)
        y = r ./ [((n-1)/2:-1:1)'; 1; (1:(n-1)/2)'];
    end
end

When you enter

```
x1 = run_cgs
```

MATLAB returns

cgs converged at iteration 13 to a solution with relative residual 
1.3e-016

Example 3.

load west0479
A = west0479
b = sum(A,2)
[x,flag] = cgs(A,b)

flag is 1 because cgs does not converge to the default tolerance 1e-6 within the default 20 iterations.

[L1,U1] = luinc(A,1e-5)
[x1,flag1] = cgs(A,b,1e-6,20,L1,U1)

flag1 is 2 because the upper triangular U1 has a zero on its diagonal, and cgs fails in the first iteration when it tries to solve a system such as U1*y = r for y with backslash.

[L2,U2] = luinc(A,1e-6)
[x2,flag2,relres2,iter2,resvec2] = cgs(A,b,1e-15,10,L2,U2)

flag2 is 0 because cgs converges to the tolerance of 6.344e-16 (the value of relres2) at the fifth iteration (the value of iter2) when preconditioned by the incomplete LU factorization with a drop tolerance of 1e-6. resvec2(1) = norm(b) and resvec2(6) = norm(b-A*x2). You can follow the progress of cgs by plotting the relative residuals at each iteration starting from the initial estimate (iterate number 0) with

semilogy(0:iter2,resvec2/norm(b),'-o')
xlabel('iteration number')
ylabel('relative residual')

See Also

bicg, bicgstab, gmres, lsqr, luinc, minres, pcg, qmr, symmlq

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] Sonneveld, Peter, "CGS: A fast Lanczos-type solver for nonsymmetric linear systems", SIAM J. Sci. Stat. Comput., January 1989, Vol. 10, No. 1, pp. 36-52.

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