### 18.5 Specialized Solvers

— Function File: bicgstab (A, b)
— Function File: bicgstab (A, b, tol, maxit, M1, M2, x0)

> This procedure attempts to solve a system of linear equations A*x = b for x. The A must be square, symmetric and positive definite real matrix N*N. The b must be a one column vector with a length of N. The tol specifies the tolerance of the method, the default value is 1e-6. The maxit specifies the maximum number of iterations, the default value is min(20,N). The M1 specifies a preconditioner, can also be a function handler which returns M\X. The M2 combined with M1 defines preconditioner as preconditioner=M1*M2. The x0 is the initial guess, the default value is zeros(N,1).

> The value x is a computed result of this procedure. The value flag can be 0 when we reach tolerance in maxit iterations, 1 when we don't reach tolerance in maxit iterations and 3 when the procedure stagnates. The value relres is a relative residual - norm(b-A*x)/norm(b). The value iter is an iteration number in which x was computed. The value resvec is a vector of relres for each iteration.

— Function File: cgs (A, b)
— Function File: cgs (A, b, tol, maxit, M1, M2, x0)

> This procedure attempts to solve a system of linear equations A*x = b for x. The A must be square, symmetric and positive definite real matrix N*N. The b must be a one column vector with a length of N. The tol specifies the tolerance of the method, default value is 1e-6. The maxit specifies the maximum number of iteration, default value is MIN(20,N). The M1 specifies a preconditioner, can also be a function handler which returns M\X. The M2 combined with M1 defines preconditioner as preconditioner=M1*M2. The x0 is initial guess, default value is zeros(N,1).

— Function File: x = gmres (A, b, m, rtol, maxit, M1, M2, x0)
— Function File: x = gmres (A, b, m, rtol, maxit, P)
— Function File: [x, flag, relres, iter, resvec] = gmres (...)

> Solve `A x = b` using the Preconditioned GMRES iterative method with restart, a.k.a. PGMRES(m).

> - rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used.
> - maxit is the maximum number of outer iterations, if not given or set to [] the default value `min (10, numel (b) / restart)` is used.
> - x0 is the initial guess, if not given or set to [] the default value `zeros(size (b))` is used.
> - m is the restart parameter, if not given or set to [] the default value `numel (b)` is used.

> Argument A can be passed as a matrix, function handle, or inline function `f` such that `f(x) = A*x`.

> The preconditioner P is given as `P = M1 * M2`. Both M1 and M2 can be passed as a matrix, function handle, or inline function `g` such that `g(x) = M1\x` or `g(x) = M2\x`.

> Besides the vector x, additional outputs are:

> - flag indicates the exit status:
>   0 : iteration converged to within the specified tolerance
>   1 : maximum number of iterations exceeded
>   2 : unused, but skipped for compatibility
>   3 : algorithm reached stagnation
> - relres is the final value of the relative residual.
> - iter is a vector containing the number of outer iterations and total iterations performed.
> - resvec is a vector containing the relative residual at each iteration.

> See also: pcg, cgs, bicgstab.