octave: Specialized Solvers
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.
DONTPRINTYET See also: 
bicgstab XREFbicgstab, cgs XREFcgs, *noteDONTPRINTYET See also: bicgstab XREFbicgstab, cgs XREFcgs,
gmres XREFgmres, pcg XREFpcg, qmr XREFqmr.
-- : 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.
DONTPRINTYET See also: bicg XREFbicg, cgs XREFcgs, *notegmres:
DONTPRINTYET See also: bicg XREFbicg, cgs XREFcgs, gmres
XREFgmres, pcg XREFpcg, qmr XREFqmr.
-- : 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.
DONTPRINTYET See also: pcg XREFpcg, bicgstab XREFbicgstab, *noteDONTPRINTYET See also: pcg XREFpcg, bicgstab XREFbicgstab,
bicg XREFbicg, gmres XREFgmres, qmr XREFqmr.
-- : 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.
DONTPRINTYET See also: bicg XREFbicg, bicgstab XREFbicgstab, *noteDONTPRINTYET DONTPRINTYET See also: bicg XREFbicg, bicgstab XREFbicgstab,
cgs XREFcgs, pcg XREFpcg, pcr XREFpcr, *noteqmr:
DONTPRINTYET DONTPRINTYET See also: bicg XREFbicg, bicgstab XREFbicgstab,
cgs XREFcgs, pcg XREFpcg, pcr XREFpcr, qmr
XREFqmr.
-- : 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.
DONTPRINTYET See also: bicg XREFbicg, bicgstab XREFbicgstab, *noteDONTPRINTYET See also: bicg XREFbicg, bicgstab XREFbicgstab,
cgs XREFcgs, gmres XREFgmres, pcg XREFpcg.
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