# scipy.sparse.linalg.expm_multiply¶

scipy.sparse.linalg.expm_multiply(A, B, start=None, stop=None, num=None, endpoint=None)[source]

Compute the action of the matrix exponential of A on B.

Parameters: A : transposable linear operator The operator whose exponential is of interest. B : ndarray The matrix or vector to be multiplied by the matrix exponential of A. start : scalar, optional The starting time point of the sequence. stop : scalar, optional The end time point of the sequence, unless endpoint is set to False. In that case, the sequence consists of all but the last of num + 1 evenly spaced time points, so that stop is excluded. Note that the step size changes when endpoint is False. num : int, optional Number of time points to use. endpoint : bool, optional If True, stop is the last time point. Otherwise, it is not included. expm_A_B : ndarray The result of the action $$e^{t_k A} B$$.

Notes

The optional arguments defining the sequence of evenly spaced time points are compatible with the arguments of numpy.linspace.

The output ndarray shape is somewhat complicated so I explain it here. The ndim of the output could be either 1, 2, or 3. It would be 1 if you are computing the expm action on a single vector at a single time point. It would be 2 if you are computing the expm action on a vector at multiple time points, or if you are computing the expm action on a matrix at a single time point. It would be 3 if you want the action on a matrix with multiple columns at multiple time points. If multiple time points are requested, expm_A_B will always be the action of the expm at the first time point, regardless of whether the action is on a vector or a matrix.

References

  Awad H. Al-Mohy and Nicholas J. Higham (2011) “Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators.” SIAM Journal on Scientific Computing, 33 (2). pp. 488-511. ISSN 1064-8275 http://eprints.ma.man.ac.uk/1591/
  Nicholas J. Higham and Awad H. Al-Mohy (2010) “Computing Matrix Functions.” Acta Numerica, 19. 159-208. ISSN 0962-4929 http://eprints.ma.man.ac.uk/1451/

Examples

>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import expm, expm_multiply
>>> A = csc_matrix([[1, 0], [0, 1]])
>>> A.todense()
matrix([[1, 0],
[0, 1]], dtype=int64)
>>> B = np.array([np.exp(-1.), np.exp(-2.)])
>>> B
array([ 0.36787944,  0.13533528])
>>> expm_multiply(A, B, start=1, stop=2, num=3, endpoint=True)
array([[ 1.        ,  0.36787944],
[ 1.64872127,  0.60653066],
[ 2.71828183,  1.        ]])
>>> expm(A).dot(B)                  # Verify 1st timestep
array([ 1.        ,  0.36787944])
>>> expm(1.5*A).dot(B)              # Verify 2nd timestep
array([ 1.64872127,  0.60653066])
>>> expm(2*A).dot(B)                # Verify 3rd timestep
array([ 2.71828183,  1.        ])


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