18.5 Specialized Solvers

: x = bicg (A, b, rtol, maxit, M1, M2, x0)

: x = bicg (A, b, rtol, maxit, P)

: [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, qmr.

: x = bicgstab (A, b, rtol, maxit, M1, M2, x0)

: x = bicgstab (A, b, rtol, maxit, P)

: [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, qmr.

: x = cgs (A, b, rtol, maxit, M1, M2, x0)

: x = cgs (A, b, rtol, maxit, P)

: [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, qmr.

: x = gmres (A, b, m, rtol, maxit, M1, M2, x0)

: x = gmres (A, b, m, rtol, maxit, P)

: [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.

See also: bicg, bicgstab, cgs, pcg, pcr, qmr.

: x = qmr (A, b, rtol, maxit, M1, M2, x0)

: x = qmr (A, b, rtol, maxit, P)

: [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.

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