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714 lines
22 KiB
Python
714 lines
22 KiB
Python
"""Compute the action of the matrix exponential.
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"""
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import numpy as np
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import scipy.linalg
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import scipy.sparse.linalg
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from scipy.sparse.linalg import aslinearoperator
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from scipy.sparse.sputils import is_pydata_spmatrix
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__all__ = ['expm_multiply']
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def _exact_inf_norm(A):
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# A compatibility function which should eventually disappear.
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if scipy.sparse.isspmatrix(A):
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return max(abs(A).sum(axis=1).flat)
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elif is_pydata_spmatrix(A):
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return max(abs(A).sum(axis=1))
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else:
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return np.linalg.norm(A, np.inf)
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def _exact_1_norm(A):
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# A compatibility function which should eventually disappear.
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if scipy.sparse.isspmatrix(A):
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return max(abs(A).sum(axis=0).flat)
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elif is_pydata_spmatrix(A):
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return max(abs(A).sum(axis=0))
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else:
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return np.linalg.norm(A, 1)
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def _trace(A):
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# A compatibility function which should eventually disappear.
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if scipy.sparse.isspmatrix(A):
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return A.diagonal().sum()
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elif is_pydata_spmatrix(A):
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return A.to_scipy_sparse().diagonal().sum()
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else:
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return np.trace(A)
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def _ident_like(A):
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# A compatibility function which should eventually disappear.
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if scipy.sparse.isspmatrix(A):
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return scipy.sparse.construct.eye(A.shape[0], A.shape[1],
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dtype=A.dtype, format=A.format)
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elif is_pydata_spmatrix(A):
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import sparse
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return sparse.eye(A.shape[0], A.shape[1], dtype=A.dtype)
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else:
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return np.eye(A.shape[0], A.shape[1], dtype=A.dtype)
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def expm_multiply(A, B, start=None, stop=None, num=None, endpoint=None):
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"""
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Compute the action of the matrix exponential of A on B.
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Parameters
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----------
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A : transposable linear operator
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The operator whose exponential is of interest.
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B : ndarray
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The matrix or vector to be multiplied by the matrix exponential of A.
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start : scalar, optional
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The starting time point of the sequence.
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stop : scalar, optional
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The end time point of the sequence, unless `endpoint` is set to False.
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In that case, the sequence consists of all but the last of ``num + 1``
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evenly spaced time points, so that `stop` is excluded.
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Note that the step size changes when `endpoint` is False.
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num : int, optional
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Number of time points to use.
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endpoint : bool, optional
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If True, `stop` is the last time point. Otherwise, it is not included.
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Returns
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-------
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expm_A_B : ndarray
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The result of the action :math:`e^{t_k A} B`.
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Notes
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-----
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The optional arguments defining the sequence of evenly spaced time points
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are compatible with the arguments of `numpy.linspace`.
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The output ndarray shape is somewhat complicated so I explain it here.
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The ndim of the output could be either 1, 2, or 3.
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It would be 1 if you are computing the expm action on a single vector
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at a single time point.
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It would be 2 if you are computing the expm action on a vector
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at multiple time points, or if you are computing the expm action
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on a matrix at a single time point.
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It would be 3 if you want the action on a matrix with multiple
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columns at multiple time points.
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If multiple time points are requested, expm_A_B[0] will always
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be the action of the expm at the first time point,
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regardless of whether the action is on a vector or a matrix.
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References
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----------
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.. [1] Awad H. Al-Mohy and Nicholas J. Higham (2011)
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"Computing the Action of the Matrix Exponential,
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with an Application to Exponential Integrators."
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SIAM Journal on Scientific Computing,
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33 (2). pp. 488-511. ISSN 1064-8275
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http://eprints.ma.man.ac.uk/1591/
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.. [2] Nicholas J. Higham and Awad H. Al-Mohy (2010)
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"Computing Matrix Functions."
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Acta Numerica,
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19. 159-208. ISSN 0962-4929
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http://eprints.ma.man.ac.uk/1451/
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Examples
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--------
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>>> from scipy.sparse import csc_matrix
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>>> from scipy.sparse.linalg import expm, expm_multiply
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>>> A = csc_matrix([[1, 0], [0, 1]])
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>>> A.todense()
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matrix([[1, 0],
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[0, 1]], dtype=int64)
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>>> B = np.array([np.exp(-1.), np.exp(-2.)])
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>>> B
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array([ 0.36787944, 0.13533528])
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>>> expm_multiply(A, B, start=1, stop=2, num=3, endpoint=True)
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array([[ 1. , 0.36787944],
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[ 1.64872127, 0.60653066],
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[ 2.71828183, 1. ]])
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>>> expm(A).dot(B) # Verify 1st timestep
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array([ 1. , 0.36787944])
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>>> expm(1.5*A).dot(B) # Verify 2nd timestep
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array([ 1.64872127, 0.60653066])
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>>> expm(2*A).dot(B) # Verify 3rd timestep
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array([ 2.71828183, 1. ])
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"""
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if all(arg is None for arg in (start, stop, num, endpoint)):
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X = _expm_multiply_simple(A, B)
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else:
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X, status = _expm_multiply_interval(A, B, start, stop, num, endpoint)
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return X
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def _expm_multiply_simple(A, B, t=1.0, balance=False):
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"""
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Compute the action of the matrix exponential at a single time point.
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Parameters
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----------
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A : transposable linear operator
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The operator whose exponential is of interest.
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B : ndarray
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The matrix to be multiplied by the matrix exponential of A.
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t : float
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A time point.
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balance : bool
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Indicates whether or not to apply balancing.
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Returns
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-------
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F : ndarray
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:math:`e^{t A} B`
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Notes
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-----
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This is algorithm (3.2) in Al-Mohy and Higham (2011).
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"""
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if balance:
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raise NotImplementedError
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if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
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raise ValueError('expected A to be like a square matrix')
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if A.shape[1] != B.shape[0]:
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raise ValueError('shapes of matrices A {} and B {} are incompatible'
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.format(A.shape, B.shape))
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ident = _ident_like(A)
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n = A.shape[0]
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if len(B.shape) == 1:
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n0 = 1
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elif len(B.shape) == 2:
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n0 = B.shape[1]
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else:
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raise ValueError('expected B to be like a matrix or a vector')
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u_d = 2**-53
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tol = u_d
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mu = _trace(A) / float(n)
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A = A - mu * ident
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A_1_norm = _exact_1_norm(A)
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if t*A_1_norm == 0:
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m_star, s = 0, 1
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else:
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ell = 2
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norm_info = LazyOperatorNormInfo(t*A, A_1_norm=t*A_1_norm, ell=ell)
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m_star, s = _fragment_3_1(norm_info, n0, tol, ell=ell)
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return _expm_multiply_simple_core(A, B, t, mu, m_star, s, tol, balance)
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def _expm_multiply_simple_core(A, B, t, mu, m_star, s, tol=None, balance=False):
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"""
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A helper function.
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"""
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if balance:
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raise NotImplementedError
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if tol is None:
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u_d = 2 ** -53
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tol = u_d
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F = B
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eta = np.exp(t*mu / float(s))
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for i in range(s):
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c1 = _exact_inf_norm(B)
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for j in range(m_star):
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coeff = t / float(s*(j+1))
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B = coeff * A.dot(B)
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c2 = _exact_inf_norm(B)
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F = F + B
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if c1 + c2 <= tol * _exact_inf_norm(F):
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break
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c1 = c2
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F = eta * F
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B = F
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return F
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# This table helps to compute bounds.
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# They seem to have been difficult to calculate, involving symbolic
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# manipulation of equations, followed by numerical root finding.
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_theta = {
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# The first 30 values are from table A.3 of Computing Matrix Functions.
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1: 2.29e-16,
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2: 2.58e-8,
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3: 1.39e-5,
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4: 3.40e-4,
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5: 2.40e-3,
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6: 9.07e-3,
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7: 2.38e-2,
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8: 5.00e-2,
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9: 8.96e-2,
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10: 1.44e-1,
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# 11
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11: 2.14e-1,
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12: 3.00e-1,
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13: 4.00e-1,
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14: 5.14e-1,
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15: 6.41e-1,
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16: 7.81e-1,
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17: 9.31e-1,
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18: 1.09,
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19: 1.26,
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20: 1.44,
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# 21
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21: 1.62,
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22: 1.82,
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23: 2.01,
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24: 2.22,
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25: 2.43,
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26: 2.64,
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27: 2.86,
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28: 3.08,
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29: 3.31,
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30: 3.54,
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# The rest are from table 3.1 of
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# Computing the Action of the Matrix Exponential.
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35: 4.7,
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40: 6.0,
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45: 7.2,
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50: 8.5,
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55: 9.9,
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}
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def _onenormest_matrix_power(A, p,
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t=2, itmax=5, compute_v=False, compute_w=False):
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"""
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Efficiently estimate the 1-norm of A^p.
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Parameters
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----------
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A : ndarray
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Matrix whose 1-norm of a power is to be computed.
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p : int
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Non-negative integer power.
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t : int, optional
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A positive parameter controlling the tradeoff between
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accuracy versus time and memory usage.
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Larger values take longer and use more memory
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but give more accurate output.
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itmax : int, optional
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Use at most this many iterations.
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compute_v : bool, optional
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Request a norm-maximizing linear operator input vector if True.
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compute_w : bool, optional
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Request a norm-maximizing linear operator output vector if True.
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Returns
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-------
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est : float
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An underestimate of the 1-norm of the sparse matrix.
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v : ndarray, optional
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The vector such that ||Av||_1 == est*||v||_1.
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It can be thought of as an input to the linear operator
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that gives an output with particularly large norm.
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w : ndarray, optional
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The vector Av which has relatively large 1-norm.
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It can be thought of as an output of the linear operator
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that is relatively large in norm compared to the input.
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"""
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#XXX Eventually turn this into an API function in the _onenormest module,
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#XXX and remove its underscore,
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#XXX but wait until expm_multiply goes into scipy.
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return scipy.sparse.linalg.onenormest(aslinearoperator(A) ** p)
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class LazyOperatorNormInfo:
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"""
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Information about an operator is lazily computed.
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The information includes the exact 1-norm of the operator,
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in addition to estimates of 1-norms of powers of the operator.
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This uses the notation of Computing the Action (2011).
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This class is specialized enough to probably not be of general interest
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outside of this module.
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"""
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def __init__(self, A, A_1_norm=None, ell=2, scale=1):
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"""
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Provide the operator and some norm-related information.
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Parameters
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----------
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A : linear operator
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The operator of interest.
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A_1_norm : float, optional
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The exact 1-norm of A.
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ell : int, optional
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A technical parameter controlling norm estimation quality.
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scale : int, optional
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If specified, return the norms of scale*A instead of A.
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"""
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self._A = A
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self._A_1_norm = A_1_norm
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self._ell = ell
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self._d = {}
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self._scale = scale
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def set_scale(self,scale):
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"""
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Set the scale parameter.
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"""
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self._scale = scale
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def onenorm(self):
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"""
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Compute the exact 1-norm.
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"""
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if self._A_1_norm is None:
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self._A_1_norm = _exact_1_norm(self._A)
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return self._scale*self._A_1_norm
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def d(self, p):
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"""
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Lazily estimate d_p(A) ~= || A^p ||^(1/p) where ||.|| is the 1-norm.
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"""
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if p not in self._d:
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est = _onenormest_matrix_power(self._A, p, self._ell)
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self._d[p] = est ** (1.0 / p)
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return self._scale*self._d[p]
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def alpha(self, p):
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"""
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Lazily compute max(d(p), d(p+1)).
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"""
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return max(self.d(p), self.d(p+1))
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def _compute_cost_div_m(m, p, norm_info):
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"""
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A helper function for computing bounds.
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This is equation (3.10).
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It measures cost in terms of the number of required matrix products.
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Parameters
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----------
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m : int
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A valid key of _theta.
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p : int
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A matrix power.
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norm_info : LazyOperatorNormInfo
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Information about 1-norms of related operators.
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Returns
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-------
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cost_div_m : int
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Required number of matrix products divided by m.
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"""
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return int(np.ceil(norm_info.alpha(p) / _theta[m]))
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def _compute_p_max(m_max):
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"""
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Compute the largest positive integer p such that p*(p-1) <= m_max + 1.
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Do this in a slightly dumb way, but safe and not too slow.
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Parameters
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----------
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m_max : int
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A count related to bounds.
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"""
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sqrt_m_max = np.sqrt(m_max)
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p_low = int(np.floor(sqrt_m_max))
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p_high = int(np.ceil(sqrt_m_max + 1))
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return max(p for p in range(p_low, p_high+1) if p*(p-1) <= m_max + 1)
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def _fragment_3_1(norm_info, n0, tol, m_max=55, ell=2):
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"""
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A helper function for the _expm_multiply_* functions.
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Parameters
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----------
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norm_info : LazyOperatorNormInfo
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Information about norms of certain linear operators of interest.
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n0 : int
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Number of columns in the _expm_multiply_* B matrix.
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tol : float
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Expected to be
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:math:`2^{-24}` for single precision or
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:math:`2^{-53}` for double precision.
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m_max : int
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A value related to a bound.
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ell : int
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The number of columns used in the 1-norm approximation.
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This is usually taken to be small, maybe between 1 and 5.
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Returns
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-------
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best_m : int
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Related to bounds for error control.
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best_s : int
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Amount of scaling.
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Notes
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-----
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This is code fragment (3.1) in Al-Mohy and Higham (2011).
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The discussion of default values for m_max and ell
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is given between the definitions of equation (3.11)
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and the definition of equation (3.12).
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"""
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if ell < 1:
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raise ValueError('expected ell to be a positive integer')
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best_m = None
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best_s = None
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if _condition_3_13(norm_info.onenorm(), n0, m_max, ell):
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for m, theta in _theta.items():
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s = int(np.ceil(norm_info.onenorm() / theta))
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if best_m is None or m * s < best_m * best_s:
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best_m = m
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best_s = s
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else:
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# Equation (3.11).
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for p in range(2, _compute_p_max(m_max) + 1):
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for m in range(p*(p-1)-1, m_max+1):
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if m in _theta:
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s = _compute_cost_div_m(m, p, norm_info)
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if best_m is None or m * s < best_m * best_s:
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best_m = m
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best_s = s
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best_s = max(best_s, 1)
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return best_m, best_s
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def _condition_3_13(A_1_norm, n0, m_max, ell):
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"""
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A helper function for the _expm_multiply_* functions.
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Parameters
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----------
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A_1_norm : float
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The precomputed 1-norm of A.
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n0 : int
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Number of columns in the _expm_multiply_* B matrix.
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m_max : int
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A value related to a bound.
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ell : int
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The number of columns used in the 1-norm approximation.
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This is usually taken to be small, maybe between 1 and 5.
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Returns
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-------
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value : bool
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Indicates whether or not the condition has been met.
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Notes
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-----
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This is condition (3.13) in Al-Mohy and Higham (2011).
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"""
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# This is the rhs of equation (3.12).
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p_max = _compute_p_max(m_max)
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a = 2 * ell * p_max * (p_max + 3)
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# Evaluate the condition (3.13).
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b = _theta[m_max] / float(n0 * m_max)
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return A_1_norm <= a * b
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def _expm_multiply_interval(A, B, start=None, stop=None,
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num=None, endpoint=None, balance=False, status_only=False):
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"""
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Compute the action of the matrix exponential at multiple time points.
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Parameters
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----------
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A : transposable linear operator
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The operator whose exponential is of interest.
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|
B : ndarray
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|
The matrix to be multiplied by the matrix exponential of A.
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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.
|
|
balance : bool
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|
Indicates whether or not to apply balancing.
|
|
status_only : bool
|
|
A flag that is set to True for some debugging and testing operations.
|
|
|
|
Returns
|
|
-------
|
|
F : ndarray
|
|
:math:`e^{t_k A} B`
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|
status : int
|
|
An integer status for testing and debugging.
|
|
|
|
Notes
|
|
-----
|
|
This is algorithm (5.2) in Al-Mohy and Higham (2011).
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|
|
|
There seems to be a typo, where line 15 of the algorithm should be
|
|
moved to line 6.5 (between lines 6 and 7).
|
|
|
|
"""
|
|
if balance:
|
|
raise NotImplementedError
|
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if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
|
|
raise ValueError('expected A to be like a square matrix')
|
|
if A.shape[1] != B.shape[0]:
|
|
raise ValueError('shapes of matrices A {} and B {} are incompatible'
|
|
.format(A.shape, B.shape))
|
|
ident = _ident_like(A)
|
|
n = A.shape[0]
|
|
if len(B.shape) == 1:
|
|
n0 = 1
|
|
elif len(B.shape) == 2:
|
|
n0 = B.shape[1]
|
|
else:
|
|
raise ValueError('expected B to be like a matrix or a vector')
|
|
u_d = 2**-53
|
|
tol = u_d
|
|
mu = _trace(A) / float(n)
|
|
|
|
# Get the linspace samples, attempting to preserve the linspace defaults.
|
|
linspace_kwargs = {'retstep': True}
|
|
if num is not None:
|
|
linspace_kwargs['num'] = num
|
|
if endpoint is not None:
|
|
linspace_kwargs['endpoint'] = endpoint
|
|
samples, step = np.linspace(start, stop, **linspace_kwargs)
|
|
|
|
# Convert the linspace output to the notation used by the publication.
|
|
nsamples = len(samples)
|
|
if nsamples < 2:
|
|
raise ValueError('at least two time points are required')
|
|
q = nsamples - 1
|
|
h = step
|
|
t_0 = samples[0]
|
|
t_q = samples[q]
|
|
|
|
# Define the output ndarray.
|
|
# Use an ndim=3 shape, such that the last two indices
|
|
# are the ones that may be involved in level 3 BLAS operations.
|
|
X_shape = (nsamples,) + B.shape
|
|
X = np.empty(X_shape, dtype=np.result_type(A.dtype, B.dtype, float))
|
|
t = t_q - t_0
|
|
A = A - mu * ident
|
|
A_1_norm = _exact_1_norm(A)
|
|
ell = 2
|
|
norm_info = LazyOperatorNormInfo(t*A, A_1_norm=t*A_1_norm, ell=ell)
|
|
if t*A_1_norm == 0:
|
|
m_star, s = 0, 1
|
|
else:
|
|
m_star, s = _fragment_3_1(norm_info, n0, tol, ell=ell)
|
|
|
|
# Compute the expm action up to the initial time point.
|
|
X[0] = _expm_multiply_simple_core(A, B, t_0, mu, m_star, s)
|
|
|
|
# Compute the expm action at the rest of the time points.
|
|
if q <= s:
|
|
if status_only:
|
|
return 0
|
|
else:
|
|
return _expm_multiply_interval_core_0(A, X,
|
|
h, mu, q, norm_info, tol, ell,n0)
|
|
elif not (q % s):
|
|
if status_only:
|
|
return 1
|
|
else:
|
|
return _expm_multiply_interval_core_1(A, X,
|
|
h, mu, m_star, s, q, tol)
|
|
elif (q % s):
|
|
if status_only:
|
|
return 2
|
|
else:
|
|
return _expm_multiply_interval_core_2(A, X,
|
|
h, mu, m_star, s, q, tol)
|
|
else:
|
|
raise Exception('internal error')
|
|
|
|
|
|
def _expm_multiply_interval_core_0(A, X, h, mu, q, norm_info, tol, ell, n0):
|
|
"""
|
|
A helper function, for the case q <= s.
|
|
"""
|
|
|
|
# Compute the new values of m_star and s which should be applied
|
|
# over intervals of size t/q
|
|
if norm_info.onenorm() == 0:
|
|
m_star, s = 0, 1
|
|
else:
|
|
norm_info.set_scale(1./q)
|
|
m_star, s = _fragment_3_1(norm_info, n0, tol, ell=ell)
|
|
norm_info.set_scale(1)
|
|
|
|
for k in range(q):
|
|
X[k+1] = _expm_multiply_simple_core(A, X[k], h, mu, m_star, s)
|
|
return X, 0
|
|
|
|
|
|
def _expm_multiply_interval_core_1(A, X, h, mu, m_star, s, q, tol):
|
|
"""
|
|
A helper function, for the case q > s and q % s == 0.
|
|
"""
|
|
d = q // s
|
|
input_shape = X.shape[1:]
|
|
K_shape = (m_star + 1, ) + input_shape
|
|
K = np.empty(K_shape, dtype=X.dtype)
|
|
for i in range(s):
|
|
Z = X[i*d]
|
|
K[0] = Z
|
|
high_p = 0
|
|
for k in range(1, d+1):
|
|
F = K[0]
|
|
c1 = _exact_inf_norm(F)
|
|
for p in range(1, m_star+1):
|
|
if p > high_p:
|
|
K[p] = h * A.dot(K[p-1]) / float(p)
|
|
coeff = float(pow(k, p))
|
|
F = F + coeff * K[p]
|
|
inf_norm_K_p_1 = _exact_inf_norm(K[p])
|
|
c2 = coeff * inf_norm_K_p_1
|
|
if c1 + c2 <= tol * _exact_inf_norm(F):
|
|
break
|
|
c1 = c2
|
|
X[k + i*d] = np.exp(k*h*mu) * F
|
|
return X, 1
|
|
|
|
|
|
def _expm_multiply_interval_core_2(A, X, h, mu, m_star, s, q, tol):
|
|
"""
|
|
A helper function, for the case q > s and q % s > 0.
|
|
"""
|
|
d = q // s
|
|
j = q // d
|
|
r = q - d * j
|
|
input_shape = X.shape[1:]
|
|
K_shape = (m_star + 1, ) + input_shape
|
|
K = np.empty(K_shape, dtype=X.dtype)
|
|
for i in range(j + 1):
|
|
Z = X[i*d]
|
|
K[0] = Z
|
|
high_p = 0
|
|
if i < j:
|
|
effective_d = d
|
|
else:
|
|
effective_d = r
|
|
for k in range(1, effective_d+1):
|
|
F = K[0]
|
|
c1 = _exact_inf_norm(F)
|
|
for p in range(1, m_star+1):
|
|
if p == high_p + 1:
|
|
K[p] = h * A.dot(K[p-1]) / float(p)
|
|
high_p = p
|
|
coeff = float(pow(k, p))
|
|
F = F + coeff * K[p]
|
|
inf_norm_K_p_1 = _exact_inf_norm(K[p])
|
|
c2 = coeff * inf_norm_K_p_1
|
|
if c1 + c2 <= tol * _exact_inf_norm(F):
|
|
break
|
|
c1 = c2
|
|
X[k + i*d] = np.exp(k*h*mu) * F
|
|
return X, 2
|