inversion_ideas.utils.get_largest_eigenvalue#

inversion_ideas.utils.get_largest_eigenvalue(matrix, *, n_iters=4, seed=None, atol=1e-16)[source]#

Estimate largest absolute eigenvalue for a diagonalizable matrix.

Use the power iteration method to estimate the eigenvector associated with the largest absolute eigenvalue for the given matrix.

Parameters:

matrix(n,n) array or (n,n) sparray or (n,n) LinearOperator: Square matrix or LinearOperator which eigenvalue will be estimated.
n_itersint, optional: Number of iterations used in the power iteration method.
seedint or numpy.random.Generator or None, optional: Random seed used to define a random initial guess for the eigenvector.
atolfloat, optional: Absolute tolerance that will be passed to numpy.isclose() when checking if the norm of the eigenvector is small enought that it can be considered equal to zero. If a null eigenvector is found during the iterations, a ConvergenceError will be raised.

Returns:

float: Estimated largest absolute eigenvalue of the matrix argument.

Raises:

inversion_ideas.errors.ConvergenceError: If the norm of the eigenvector falls below the atol argument, and therefore is considered to be the null vector.

Notes

Given a diagonalizable matrix \(\mathbf{A}\), we can estimate the eigenvector related to its largest absolute eigenvalue \(|\lambda|\) through the power iteration method. In a nutshell, we start with a random unit vector \(\mathbf{b}_0\), and estimate that eigenvector through a recursive relation:

\[\mathbf{b}_{k+1} = \frac{ \mathbf{A}\mathbf{b}_k }{ \lVert \mathbf{A}\mathbf{b}_k \rVert }\]

Once we obtained the \(\mathbf{b}_k\) eigenvector, we can get the absolute value of its corresponding eigenvalue through:

\[|\lambda| = \frac{ \lVert \mathbf{A} \mathbf{b}_k \rVert }{ \lVert \mathbf{b}_k \rVert }\]

If the eigenvector \(\mathbf{b}_k\) is a unit vector (normalized by its magnitude), then we can simplify the last expression to:

\[|\lambda| = \lVert \mathbf{A} \mathbf{b}_k \rVert\]

References

https://en.wikipedia.org/wiki/Power_iteration

Examples

>>> import numpy as np
>>> matrix = np.array(
...     [
...         [1., 0., 0.],
...         [0., 2., 0.],
...         [0., 0., 3.],
...     ]
... )
>>> get_largest_eigenvalue(matrix, seed=42)
3.0