— 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.