HishamKamal / idrs_solver

Solving system of linear equations using IDR(s)

IDR(s) Solver in Python

The Induced Dimension Reduction method (IDR(s)) [1] is a short-recurrences Krylov method that solves the system of linear equation, Ax = b. This Python implementation is based on [2]. The interface of the idrs function is compatible with the Krylov methods implemented in Scipy.

  idrs(A, b, x0=None, tol=1e-5, s=4, maxit=None, xtype=None, M=None, callback=None)

Parameters
----------
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`.
s : integer, optional
    specifies the dimension of the shadow space. Normally, a higher
    s gives faster convergence, but also makes the method more expensive.
    Default is 4.
maxiter : integer, optional
    Maximum number of iterations.  Iteration will stop after maxiter
    steps even if the specified tolerance has not been achieved.
xtype : {'f','d','F','D'}
    This parameter is deprecated -- avoid using it.
M : {sparse matrix, dense matrix, LinearOperator}, optional
    Preconditioner for A.  The preconditioner should approximate the
    inverse of A.  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.

Returns
-------
x : array or matrix
    The converged solution.
info : integer
    Provides convergence information:
        - 0  : successful exit
        - >0 : convergence to tolerance not achieved, number of iterations
        - <0 : illegal input or breakdown

References
----------

.. [1] P. Sonneveld and M. B. van Gijzen
         SIAM J. Sci. Comput. Vol. 31, No. 2, pp. 1035--1062, (2008).
.. [2] M. B. van Gijzen and P. Sonneveld
         ACM Trans. Math. Software,, Vol. 38, No. 1, pp. 5:1-5:19, (2011).
.. [3] This file is a translation of the following MATLAB implementation:
        http://ta.twi.tudelft.nl/nw/users/gijzen/idrs.m

License

This software is distributed under the MIT License.