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108 lines
3.3 KiB
108 lines
3.3 KiB
"""
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Matrix square root for general matrices and for upper triangular matrices.
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This module exists to avoid cyclic imports.
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"""
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__all__ = []
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import numpy as np
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# Local imports
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from .lapack import ztrsyl, dtrsyl
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class SqrtmError(np.linalg.LinAlgError):
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pass
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from ._matfuncs_sqrtm_triu import within_block_loop # noqa: E402
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def _sqrtm_triu(T, blocksize=64):
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"""
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Matrix square root of an upper triangular matrix.
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This is a helper function for `sqrtm` and `logm`.
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Parameters
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----------
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T : (N, N) array_like upper triangular
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Matrix whose square root to evaluate
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blocksize : int, optional
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If the blocksize is not degenerate with respect to the
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size of the input array, then use a blocked algorithm. (Default: 64)
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Returns
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-------
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sqrtm : (N, N) ndarray
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Value of the sqrt function at `T`
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References
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----------
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.. [1] Edvin Deadman, Nicholas J. Higham, Rui Ralha (2013)
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"Blocked Schur Algorithms for Computing the Matrix Square Root,
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Lecture Notes in Computer Science, 7782. pp. 171-182.
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"""
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T_diag = np.diag(T)
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keep_it_real = np.isrealobj(T) and np.min(T_diag, initial=0.) >= 0
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# Cast to complex as necessary + ensure double precision
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if not keep_it_real:
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T = np.asarray(T, dtype=np.complex128, order="C")
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T_diag = np.asarray(T_diag, dtype=np.complex128)
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else:
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T = np.asarray(T, dtype=np.float64, order="C")
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T_diag = np.asarray(T_diag, dtype=np.float64)
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R = np.diag(np.sqrt(T_diag))
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# Compute the number of blocks to use; use at least one block.
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n, n = T.shape
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nblocks = max(n // blocksize, 1)
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# Compute the smaller of the two sizes of blocks that
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# we will actually use, and compute the number of large blocks.
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bsmall, nlarge = divmod(n, nblocks)
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blarge = bsmall + 1
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nsmall = nblocks - nlarge
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if nsmall * bsmall + nlarge * blarge != n:
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raise Exception('internal inconsistency')
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# Define the index range covered by each block.
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start_stop_pairs = []
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start = 0
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for count, size in ((nsmall, bsmall), (nlarge, blarge)):
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for i in range(count):
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start_stop_pairs.append((start, start + size))
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start += size
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# Within-block interactions (Cythonized)
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try:
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within_block_loop(R, T, start_stop_pairs, nblocks)
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except RuntimeError as e:
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raise SqrtmError(*e.args) from e
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# Between-block interactions (Cython would give no significant speedup)
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for j in range(nblocks):
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jstart, jstop = start_stop_pairs[j]
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for i in range(j-1, -1, -1):
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istart, istop = start_stop_pairs[i]
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S = T[istart:istop, jstart:jstop]
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if j - i > 1:
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S = S - R[istart:istop, istop:jstart].dot(R[istop:jstart,
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jstart:jstop])
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# Invoke LAPACK.
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# For more details, see the solve_sylvester implementation
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# and the fortran dtrsyl and ztrsyl docs.
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Rii = R[istart:istop, istart:istop]
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Rjj = R[jstart:jstop, jstart:jstop]
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if keep_it_real:
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x, scale, info = dtrsyl(Rii, Rjj, S)
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else:
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x, scale, info = ztrsyl(Rii, Rjj, S)
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R[istart:istop, jstart:jstop] = x * scale
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# Return the matrix square root.
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return R
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