low_rank_toolbox.krylov.spaces.inverted_krylov_space

Inverted Krylov subspace construction.

Author: Benjamin Carrel, University of Geneva, 2022-2023

Classes

InvertedKrylovSpace(A, X[, invA])

Inverted Krylov space.
class low_rank_toolbox.krylov.spaces.inverted_krylov_space.InvertedKrylovSpace(A, X, invA=None, **extra_args)[source]

Bases: KrylovSpace

Inverted Krylov space.

Constructs the inverted Krylov space:

IK_m(A, X) = span{A^(-1)X, A^(-2)X, …, A^(-m)X}

This class wraps the KrylovSpace class by providing a custom matvec function that computes A^(-1) * v instead of A * v.

How to use

  1. Initialize the inverted Krylov space with matrix A and vector/matrix X.

  2. Augment the basis as needed with the augment_basis method.

  3. Access the basis via the basis or Q attribute.

A

Sparse matrix of shape (n, n)

Type:

spmatrix

X

Initial vector or matrix of shape (n, r)

Type:

ndarray

invA

Function that computes A^(-1) * v for a given vector/matrix v

Type:

callable

Q

Orthonormal basis of the inverted Krylov space

Type:

ndarray

basis

Pointer to Q

Type:

ndarray

Examples

>>> import numpy as np
>>> from scipy.sparse import csr_matrix
>>> from low_rank_toolbox.krylov import InvertedKrylovSpace
>>> # Create a sparse matrix
>>> A = csr_matrix([[4, 1, 0], [1, 3, 1], [0, 1, 2]])
>>> X = np.array([[1.0], [0.0], [0.0]])
>>> # Build inverted Krylov space
>>> IK = InvertedKrylovSpace(A, X)
>>> # Start with A^(-1)*X
>>> IK.Q.shape
(3, 1)
>>> # Augment with A^(-2)*X
>>> IK.augment_basis()
>>> IK.Q.shape
(3, 2)
>>> # Basis is orthonormal
>>> np.allclose(IK.Q.T @ IK.Q, np.eye(2))
True
__init__(A, X, invA=None, **extra_args)[source]
Parameters:

  • A (spmatrix) – The matrix A of the linear system.

  • X (ndarray) – The basis of the Krylov space.

  • invA (callable) – The function that computes the action of A^(-1) on a vector, or a matrix.

Return type:

None

Parameters: