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### 18.5 Specialized Solvers

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

> Solve `A x = b` using the Bi-conjugate gradient iterative method.

> - rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used.
> - maxit the maximum number of outer iterations, if not given or set to [] the default value `min (20, numel (b))` is used.
> - x0 the initial guess, if not given or set to [] the default value `zeros (size (b))` is used.

> A can be passed as a matrix or as a function handle or inline function `f` such that `f(x, "notransp") = A*x` and `f(x, "transp") = A'*x`.

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

> If called with more than one output parameter

> - flag indicates the exit status:
>   - 0: iteration converged to the within the chosen tolerance
>   - 1: the maximum number of iterations was reached before convergence
>   - 3: the algorithm reached stagnation
>   (the value 2 is unused but skipped for compatibility).
> - relres is the final value of the relative residual.
> - iter is the number of iterations performed.
> - resvec is a vector containing the relative residual at each iteration.

> See also: bicgstab, cgs, gmres, pcg.

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

> Solve `A x = b` using the stabilizied Bi-conjugate gradient iterative method.

> - rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used.
> - maxit the maximum number of outer iterations, if not given or set to [] the default value `min (20, numel (b))` is used.
> - x0 the initial guess, if not given or set to [] the default value `zeros (size (b))` is used.

> A can be passed as a matrix or as a 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 or as a function handle or inline function `g` such that `g(x) = M1 \ x` or `g(x) = M2 \ x`.

> If called with more than one output parameter

> - flag indicates the exit status:
>   - 0: iteration converged to the within the chosen tolerance
>   - 1: the maximum number of iterations was reached before convergence
>   - 3: the algorithm reached stagnation
>   (the value 2 is unused but skipped for compatibility).
> - relres is the final value of the relative residual.
> - iter is the number of iterations performed.
> - resvec is a vector containing the relative residual at each iteration.

> See also: bicg, cgs, gmres, pcg.

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

> Solve `A x = b`, where A is a square matrix, using the Conjugate Gradients Squared method.

> - rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used.
> - maxit the maximum number of outer iterations, if not given or set to [] the default value `min (20, numel (b))` is used.
> - x0 the initial guess, if not given or set to [] the default value `zeros (size (b))` is used.

> A can be passed as a matrix or as a 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 or as a function handle or inline function `g` such that `g(x) = M1 \ x` or `g(x) = M2 \ x`.

> If called with more than one output parameter

> - flag indicates the exit status:
>   - 0: iteration converged to the within the chosen tolerance
>   - 1: the maximum number of iterations was reached before convergence
>   - 3: the algorithm reached stagnation
>   (the value 2 is unused but skipped for compatibility).
> - relres is the final value of the relative residual.
> - iter is the number of iterations performed.
> - resvec is a vector containing the relative residual at each iteration.

> See also: pcg, bicgstab, bicg, gmres.

— 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: bicg, bicgstab, cgs, pcg.