scipy.linalg.expm

scipy.linalg.expm(A)

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

>>> 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]])