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]])