networksns.centrality_measures.exponential_quadrature

networksns.centrality_measures.exponential_quadrature(A, u, v, tol=1e-07, maxit=50)

Computes \(q=u^T e^A v\). For the computation the polarization rule and Lanczos iteration are used [1].

Parameters:

  • A (array_like) – sparse/dense symmetric matrix.

  • u (array) – vector.

  • v (array) – vector.

  • tol (float,optional) – tolerance for convergence, relative accuracy, default: 1e-7.

  • maxit (integer, optional) – maximum number of Lanczos iterations, default: 50.

Returns:

q: (float) value of the bilinear form \(u^Te^Av\).

Examples

>>>  from networksns import centrality_measures as cm
>>>  import numpy as np

Create symmetric matrix \(A\)

>>>    A = np.arange(0, 9, 1)
>>>    A = A.reshape(3, 3)
>>>    A = A + A.transpose()
       array([[ 0,  4,  8],
              [ 4,  8, 12],
              [ 8, 12, 16]])

Create vectors \(u\) and \(v\)

>>>    u = np.arange(0, 3)
       array([0, 1, 2])
>>>    v = np.array([2,5,2])
       array([2, 5, 2])

Compute \(q=u^T e^A v\)

>>>    q = cm.exponential_quadrature(A, u, v)

References