scipy.linalg.expm#

scipy.linalg.expm(A)[source]#

Compute the matrix exponential of an array.

Parameters:

Andarray: Input with last two dimensions are square (..., n, n).

Returns:

eAndarray: The resulting matrix exponential with the same shape of A

Notes

Implements the algorithm given in [1], which is essentially a Pade approximation with a variable order that is decided based on the array data.

For input with size n, the memory usage is in the worst case in the order of 8*(n**2). If the input data is not of single and double precision of real and complex dtypes, it is copied to a new array.

For cases n >= 400, the exact 1-norm computation cost, breaks even with 1-norm estimation and from that point on the estimation scheme given in [2] is used to decide on the approximation order.

References

[1]

Awad H. Al-Mohy and Nicholas J. Higham, (2009), “A New Scaling and Squaring Algorithm for the Matrix Exponential”, SIAM J. Matrix Anal. Appl. 31(3):970-989, DOI:10.1137/09074721X

[2]

Nicholas J. Higham and Francoise Tisseur (2000), “A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra.” SIAM J. Matrix Anal. Appl. 21(4):1185-1201, DOI:10.1137/S0895479899356080

Examples

>>> import numpy as np
>>> from scipy.linalg import expm, sinm, cosm

Matrix version of the formula exp(0) = 1:

>>> expm(np.zeros((3, 2, 2)))
array([[[1., 0.],
        [0., 1.]],

       [[1., 0.],
        [0., 1.]],

       [[1., 0.],
        [0., 1.]]])

Euler’s identity (exp(i*theta) = cos(theta) + i*sin(theta)) applied to a matrix:

>>> a = np.array([[1.0, 2.0], [-1.0, 3.0]])
>>> expm(1j*a)
array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
       [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
>>> cosm(a) + 1j*sinm(a)
array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
       [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])