Solve system of linear equations — conjugate gradients squared method

attempts to solve the system of linear equations `x`

= cgs(`A`

,`b`

)`A*x = b`

for
`x`

using the Conjugate Gradients Squared Method. When the attempt is
successful, `cgs`

displays a message to confirm convergence. If
`cgs`

fails to converge after the maximum number of iterations or halts
for any reason, it displays a diagnostic message that includes the relative residual
`norm(b-A*x)/norm(b)`

and the iteration number at which the method
stopped.

`[`

returns a flag that specifies whether the algorithm successfully converged. When
`x`

,`flag`

] = cgs(___)`flag = 0`

, convergence was successful. You can use this output syntax
with any of the previous input argument combinations. When you specify the
`flag`

output, `cgs`

does not display any diagnostic
messages.

Convergence of most iterative methods depends on the condition number of the coefficient matrix,

`cond(A)`

. You can use`equilibrate`

to improve the condition number of`A`

, and on its own this makes it easier for most iterative solvers to converge. However, using`equilibrate`

also leads to better quality preconditioner matrices when you subsequently factor the equilibrated matrix`B = R*P*A*C`

.You can use matrix reordering functions such as

`dissect`

and`symrcm`

to permute the rows and columns of the coefficient matrix and minimize the number of nonzeros when the coefficient matrix is factored to generate a preconditioner. This can reduce the memory and time required to subsequently solve the preconditioned linear system.

[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.