Module Contents


_fgmres(matvec,v0,m,atol,lpsolve=None,rpsolve=None,cs=tuple,outer_v=tuple,prepend_outer_v=False) FGMRES Arnoldi process, with optional projection or augmentation
gcrotmk(A,b,x0=None,tol=1e-05,maxiter=1000,M=None,callback=None,m=20,k=None,CU=None,discard_C=False,truncate=”oldest”) Solve a matrix equation using flexible GCROT(m,k) algorithm.
_fgmres(matvec, v0, m, atol, lpsolve=None, rpsolve=None, cs=tuple, outer_v=tuple, prepend_outer_v=False)

FGMRES Arnoldi process, with optional projection or augmentation

matvec : callable
Operation A*x
v0 : ndarray
Initial vector, normalized to nrm2(v0) == 1
m : int
Number of GMRES rounds
atol : float
Absolute tolerance for early exit
lpsolve : callable
Left preconditioner L
rpsolve : callable
Right preconditioner R
CU : list of (ndarray, ndarray)
Columns of matrices C and U in GCROT
outer_v : list of ndarrays
Augmentation vectors in LGMRES
prepend_outer_v : bool, optional
Whether augmentation vectors come before or after Krylov iterates
If nans encountered
Q, R : ndarray
QR decomposition of the upper Hessenberg H=QR
B : ndarray
Projections corresponding to matrix C
vs : list of ndarray
Columns of matrix V
zs : list of ndarray
Columns of matrix Z
y : ndarray
Solution to ||H y - e_1||_2 = min!
gcrotmk(A, b, x0=None, tol=1e-05, maxiter=1000, M=None, callback=None, m=20, k=None, CU=None, discard_C=False, truncate="oldest")

Solve a matrix equation using flexible GCROT(m,k) algorithm.

A : {sparse matrix, dense matrix, LinearOperator}
The real or complex N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
x0 : {array, matrix}
Starting guess for the solution.
tol : float, optional
Tolerance to achieve. The algorithm terminates when either the relative or the absolute residual is below tol.
maxiter : int, optional
Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.
M : {sparse matrix, dense matrix, LinearOperator}, optional
Preconditioner for A. The preconditioner should approximate the inverse of A. gcrotmk is a ‘flexible’ algorithm and the preconditioner can vary from iteration to iteration. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance.
callback : function, optional
User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.
m : int, optional
Number of inner FGMRES iterations per each outer iteration. Default: 20
k : int, optional
Number of vectors to carry between inner FGMRES iterations. According to [2], good values are around m. Default: m
CU : list of tuples, optional
List of tuples (c, u) which contain the columns of the matrices C and U in the GCROT(m,k) algorithm. For details, see [2]. The list given and vectors contained in it are modified in-place. If not given, start from empty matrices. The c elements in the tuples can be None, in which case the vectors are recomputed via c = A u on start and orthogonalized as described in [3].
discard_C : bool, optional
Discard the C-vectors at the end. Useful if recycling Krylov subspaces for different linear systems.
truncate : {‘oldest’, ‘smallest’}, optional
Truncation scheme to use. Drop: oldest vectors, or vectors with smallest singular values using the scheme discussed in [1,2]. See [2] for detailed comparison. Default: ‘oldest’
x : array or matrix
The solution found.
info : int

Provides convergence information:

  • 0 : successful exit
  • >0 : convergence to tolerance not achieved, number of iterations
[1]E. de Sturler, ‘’Truncation strategies for optimal Krylov subspace methods’‘, SIAM J. Numer. Anal. 36, 864 (1999).
[2](1, 2, 3) J.E. Hicken and D.W. Zingg, ‘’A simplified and flexible variant of GCROT for solving nonsymmetric linear systems’‘, SIAM J. Sci. Comput. 32, 172 (2010).
[3]M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti, ‘’Recycling Krylov subspaces for sequences of linear systems’‘, SIAM J. Sci. Comput. 28, 1651 (2006).