low_rank_toolbox.krylov.spaces.rational_krylov_space

Rational Krylov subspace construction with arbitrary poles.

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

Classes

RationalKrylovSpace(A, X, poles[, inverse_only])

Rational Krylov space.
class low_rank_toolbox.krylov.spaces.rational_krylov_space.RationalKrylovSpace(A, X, poles, inverse_only=False, **extra_args)[source]

Bases: SpaceStructure

Rational Krylov space.

Constructs the rational Krylov space:

RK_m(A, X) = span{X, (A - p_1*I)^{-1}X, …, prod_{i=1}^{m-1}(A - p_i*I)^{-1}X}

where p_1, …, p_{m-1} are specified poles (shifts). This space generalizes the standard Krylov space by using rational functions of A instead of polynomials.

How to use

  1. Initialize with matrix A, vector/matrix X, and list of poles.

  2. Augment the basis as needed with augment_basis.

  3. Access the basis via basis or Q.

A

Sparse matrix of shape (n, n)

Type:

spmatrix

X

Initial vector or matrix of shape (n, r)

Type:

ndarray

poles

Array of poles (shifts) for the rational Krylov space

Type:

ndarray

max_iter

Maximum number of iterations (length of poles list)

Type:

int

m

Current size of the rational Krylov space

Type:

int

Q

Orthonormal basis of shape (n, m*r)

Type:

ndarray

H

Upper triangular matrix from QR factorization

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 RationalKrylovSpace
>>> # 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]])
>>> # Choose poles to emphasize specific spectral regions
>>> # (e.g., near eigenvalues of interest)
>>> poles = [1.0, 2.0, 3.0]
>>> RK = RationalKrylovSpace(A, X, poles)
>>> # Each augmentation uses the next pole
>>> RK.augment_basis()  # Uses pole 1.0
>>> RK.Q.shape
(3, 2)
>>> RK.augment_basis()  # Uses pole 2.0
>>> RK.Q.shape
(3, 3)
>>> # Verify orthogonality
>>> np.allclose(RK.Q.T @ RK.Q, np.eye(3))
True
__init__(A, X, poles, inverse_only=False, **extra_args)[source]

Initialize a rational Krylov space

Parameters:

  • A (spmatrix) – Sparse matrix of shape (n, n)

  • X (ndarray) – Vector or matrix of shape (n, r)

  • poles (list) – Poles (shifts) for the rational Krylov space

  • inverse_only (bool, optional) – If True, solves (A - p*I) * v_new = v_old. If False, solves (A - p*I) * v_new = A * v_old. Default is False. The False case is useful for approximating matrix functions, while True is for solving linear systems.

  • extra_args (dict) – Extra arguments

  • arguments (Extra)

  • ---------------

  • symmetric (bool) – True if A is symmetric, False otherwise (not used for rational Krylov)

Return type:

None

augment_basis()[source]

Augment the basis of the rational Krylov space.

Adds the next block of r basis vectors by solving:

(A - p_{m-1}*I) * v_new = v_old

where p_{m-1} is the next pole in the sequence. The new vectors are then orthogonalized against the existing basis using QR factorization.

Raises:

ValueError – If the space would exceed the problem dimension or if there are not enough poles specified.

property basis: ndarray

The orthonormal basis of the rational Krylov space.

Returns:

Matrix of shape (n, m*r) containing the basis vectors.

Return type:

ndarray

property size: int

The size of the rational Krylov space.

Returns:

The number of basis vectors (m * r).

Return type:

int

Parameters: