stag.spectrum Namespace Reference

Functions
Tuple[np.ndarray, np.ndarray]	compute_eigensystem (scipy.sparse.spmatrix mat, int num, str which='SM')
	Compute the eigenvalues and eigenvectors of a given matrix.
np.ndarray	compute_eigenvalues (scipy.sparse.spmatrix mat, int num, str which='SM')
	Compute the eigenvalues of a given matrix.
np.ndarray	compute_eigenvectors (scipy.sparse.spmatrix mat, int num, str which='SM')
	Compute the eigenvectors of a given matrix.
np.ndarray	power_method (scipy.sparse.spmatrix mat, int num_iterations=None, np.ndarray initial_vector=None)
	Apply the power method to compute the dominant eigenvector of a matrix.
float	rayleigh_quotient (scipy.sparse.spmatrix mat, np.ndarray vec)
	Compute the Rayleigh quotient of the given vector and matrix.

Function Documentation

compute_eigensystem()

Tuple[np.ndarray, np.ndarray] stag.spectrum.compute_eigensystem	(	scipy.sparse.spmatrix	mat,
		int	num,
		str	which = 'SM'
	)

Compute the eigenvalues and eigenvectors of a given matrix.

By default, this will compute the eigenvalues of smallest magnitude. This default can be overridden by the whichparameter which takes a string and should take one of the following values.

• SM will return the eigenvalues with smallest magnitude
• LM will return the eigenvalues with largest magnitude

The following example demonstrates how to compute the 3 largest eigenvectors and eigenvalues of a cycle graph.

import stag.graph
import stag.spectrum
 
myGraph = stag.graph.cycle_graph(10)
lap = myGraph.normalised_laplacian()
eigenvalues, eigenvectors = stag.spectrum.compute_eigensystem(
    lap, 3, 'LM')

Parameters

mat	the matrix on which to operate
num	the number of eigenvalues and eigenvectors to compute
which	(optional) a string indicating which eigenvectors to calculate

Returns

a tuple containing the computed eigenvalues and eigenvectors

compute_eigenvalues()

np.ndarray stag.spectrum.compute_eigenvalues	(	scipy.sparse.spmatrix	mat,
		int	num,
		str	which = 'SM'
	)

Compute the eigenvalues of a given matrix.

By default, this will compute the eigenvalues of smallest magnitude. This default can be overridden by the which parameter which takes a string and should be one of the following.

• SM will return the eigenvalues with smallest magnitude
• LM will return the eigenvalues with largest magnitude

If you would like to calculate the eigenvectors and eigenvalues together, then you should instead use stag.spectrum.compute_eigensystem.

Parameters

mat	the matrix on which to operate
num	the number of eigenvalues to compute
which	(optional) a string indicating which eigenvalues to calculate

Returns

a numpy array containing the computed eigenvalues

compute_eigenvectors()

np.ndarray stag.spectrum.compute_eigenvectors	(	scipy.sparse.spmatrix	mat,
		int	num,
		str	which = 'SM'
	)

Compute the eigenvectors of a given matrix.

By default, this will compute the eigenvectors corresponding to the eigenvalues of smallest magnitude. This default can be overridden by the which parameter which takes a string and should be one of the following.

• SM will return the vectors corresponding to eigenvalues with smallest magnitude
• LM will return the vectors corresponding to eigenvalues with largest magnitude

If you would like to calculate the eigenvectors and eigenvalues together, then you should instead use stag.spectrum.compute_eigensystem.

Parameters

mat	the matrix on which to operate
num	the number of eigenvectors to compute
which	(optional) a string indicating which eigenvectors to calculate

Returns

a numpy array containing the computed eigenvectors as columns

power_method()

np.ndarray stag.spectrum.power_method	(	scipy.sparse.spmatrix	mat,
		int	num_iterations = None,
		np.ndarray	initial_vector = None
	)

Apply the power method to compute the dominant eigenvector of a matrix.

Given a matrix \(M\), an initial vector \(v_0\), and a number of iterations \(t\), the power method calculates the vector

\[ v_t = M^t v_0, \]

which is close to the eigenvector of \(M\) with largest eigenvalue.

The running time of the power method is \(O(t \cdot \mathrm{nnz}(M))\), where \(\mathrm{nnz}(M)\) is the number of non-zero elements in the matrix \(M\).

Parameters

mat	the matrix \(M\) on which to operate.
num_iterations	(optional) the number of iterations of the power method to apply. It this argument is omitted, \(O(\log(n))\) iterations are used which results in a vector whose Rayleigh quotient is a \((1 - \epsilon)\) approximation of the dominant eigenvalue.
initial_vector	(optional) the initial vector to use for the power iteration. If this argument is omitted, a random unit vector will be used.

Returns

the vector \(v_t\) computed by repeated multiplication with \(M\).

rayleigh_quotient()

float stag.spectrum.rayleigh_quotient	(	scipy.sparse.spmatrix	mat,
		np.ndarray	vec
	)

Compute the Rayleigh quotient of the given vector and matrix.

Given a matrix \(M\), the Rayleigh quotient of vector \(v\) is

\[ R(M, v) = \frac{v^\top M v}{v^\top v}. \]

Parameters

mat	a sparse matrix \(M \in \mathbb{R}^{n \times n}\).
vec	a vector \(v \in \mathbb{R}^n\).

Returns

the Rayleigh quotient \(R(M, v)\).