pyyeti.expmint.expmint_pow

pyyeti.expmint.expmint_pow(A, h)[source]

Compute the matrix exponential and its integrals using the power series expansion.

Parameters:

  • A ((M, M) array_like or sparse matrix) – 2D Array or Matrix (sparse or dense) to be exponentiated

  • h (scalar) – Time step

Returns:

  • E ((M, M) ndarray) – Matrix exponential of A*h: exp(A*h)

  • I ((M, M) ndarray) – Integral of exp(A*t) dt from 0 to h

  • I2 ((M, M) ndarray) – Integral of exp(A*t)*t dt from 0 to h

Notes

This routine is a simple brute-force alternative to the more elegant and optimized expmint(). The power series expansions for these matrices are (I = identity):

E = I + A*h + (A*h)**2/2! + (A*h)**3/3! + ...
I1 = h*(I + A*h/2 + (A*h)**2/3! + (A*h)**3/4! + ...)
I2 = h*h*(I/2 + A*h/3 + (A*h)**2/(4*2!) + (A*h)**3/(5*3!) + ...)

Examples

>>> from pyyeti import expmint
>>> import numpy as np
>>> import scipy.linalg as la
>>> np.set_printoptions(4)
>>> a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> e, i, i2 = expmint.expmint_pow(a, .05)
>>> e
array([[ 1.0996,  0.1599,  0.2202],
       [ 0.3099,  1.3849,  0.46  ],
       [ 0.5202,  0.61  ,  1.6998]])

>>> i
array([[ 0.052 ,  0.0034,  0.0048],
       [ 0.0067,  0.0583,  0.01  ],
       [ 0.0114,  0.0133,  0.0651]])

>>> i2
array([[ 0.0013,  0.0001,  0.0002],
       [ 0.0002,  0.0015,  0.0003],
       [ 0.0004,  0.0005,  0.0018]])