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Module « scipy.linalg »

Fonction expm - module scipy.linalg

Signature de la fonction expm

def expm(A) 

Description

help(scipy.linalg.expm)

Compute the matrix exponential of an array.

Parameters
----------
A : ndarray
    Input with last two dimensions are square ``(..., n, n)``.

Returns
-------
eA : ndarray
    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.]],
<BLANKLINE>
       [[1., 0.],
        [0., 1.]],
<BLANKLINE>
       [[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]])

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