Previous: , Up: Linear Algebra   [Contents][Index]

### 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.

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.

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.

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 (no change between iterations)
• - 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.

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

Solve `A x = b` using the Quasi-Minimal Residual iterative method (without look-ahead).

• - 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 residual norms at each iteration.

References:

1. R. Freund and N. Nachtigal, QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numerische Mathematik, 1991, 60, pp. 315-339.
2. R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhour, R. Pozo, C. Romine, and H. van der Vorst, Templates for the solution of linear systems: Building blocks for iterative methods, SIAM, 2nd ed., 1994.