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""" Matrix square root for general matrices and for upper triangular matrices.
This module exists to avoid cyclic imports.
"""
# Local imports
""" Matrix square root of an upper triangular matrix.
This is a helper function for `sqrtm` and `logm`.
Parameters ---------- T : (N, N) array_like upper triangular Matrix whose square root to evaluate blocksize : int, optional If the blocksize is not degenerate with respect to the size of the input array, then use a blocked algorithm. (Default: 64)
Returns ------- sqrtm : (N, N) ndarray Value of the sqrt function at `T`
References ---------- .. [1] Edvin Deadman, Nicholas J. Higham, Rui Ralha (2013) "Blocked Schur Algorithms for Computing the Matrix Square Root, Lecture Notes in Computer Science, 7782. pp. 171-182.
""" T_diag = np.diag(T) keep_it_real = np.isrealobj(T) and np.min(T_diag) >= 0 if not keep_it_real: T_diag = T_diag.astype(complex) R = np.diag(np.sqrt(T_diag))
# Compute the number of blocks to use; use at least one block. n, n = T.shape nblocks = max(n // blocksize, 1)
# Compute the smaller of the two sizes of blocks that # we will actually use, and compute the number of large blocks. bsmall, nlarge = divmod(n, nblocks) blarge = bsmall + 1 nsmall = nblocks - nlarge if nsmall * bsmall + nlarge * blarge != n: raise Exception('internal inconsistency')
# Define the index range covered by each block. start_stop_pairs = [] start = 0 for count, size in ((nsmall, bsmall), (nlarge, blarge)): for i in range(count): start_stop_pairs.append((start, start + size)) start += size
# Within-block interactions. for start, stop in start_stop_pairs: for j in range(start, stop): for i in range(j-1, start-1, -1): s = 0 if j - i > 1: s = R[i, i+1:j].dot(R[i+1:j, j]) denom = R[i, i] + R[j, j] num = T[i, j] - s if denom != 0: R[i, j] = (T[i, j] - s) / denom elif denom == 0 and num == 0: R[i, j] = 0 else: raise SqrtmError('failed to find the matrix square root')
# Between-block interactions. for j in range(nblocks): jstart, jstop = start_stop_pairs[j] for i in range(j-1, -1, -1): istart, istop = start_stop_pairs[i] S = T[istart:istop, jstart:jstop] if j - i > 1: S = S - R[istart:istop, istop:jstart].dot(R[istop:jstart, jstart:jstop])
# Invoke LAPACK. # For more details, see the solve_sylvester implemention # and the fortran dtrsyl and ztrsyl docs. Rii = R[istart:istop, istart:istop] Rjj = R[jstart:jstop, jstart:jstop] if keep_it_real: x, scale, info = dtrsyl(Rii, Rjj, S) else: x, scale, info = ztrsyl(Rii, Rjj, S) R[istart:istop, jstart:jstop] = x * scale
# Return the matrix square root. return R
""" Matrix square root.
Parameters ---------- A : (N, N) array_like Matrix whose square root to evaluate disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True) blocksize : integer, optional If the blocksize is not degenerate with respect to the size of the input array, then use a blocked algorithm. (Default: 64)
Returns ------- sqrtm : (N, N) ndarray Value of the sqrt function at `A`
errest : float (if disp == False)
Frobenius norm of the estimated error, ||err||_F / ||A||_F
References ---------- .. [1] Edvin Deadman, Nicholas J. Higham, Rui Ralha (2013) "Blocked Schur Algorithms for Computing the Matrix Square Root, Lecture Notes in Computer Science, 7782. pp. 171-182.
Examples -------- >>> from scipy.linalg import sqrtm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> r = sqrtm(a) >>> r array([[ 0.75592895, 1.13389342], [ 0.37796447, 1.88982237]]) >>> r.dot(r) array([[ 1., 3.], [ 1., 4.]])
""" A = _asarray_validated(A, check_finite=True, as_inexact=True) if len(A.shape) != 2: raise ValueError("Non-matrix input to matrix function.") if blocksize < 1: raise ValueError("The blocksize should be at least 1.") keep_it_real = np.isrealobj(A) if keep_it_real: T, Z = schur(A) if not np.array_equal(T, np.triu(T)): T, Z = rsf2csf(T, Z) else: T, Z = schur(A, output='complex') failflag = False try: R = _sqrtm_triu(T, blocksize=blocksize) ZH = np.conjugate(Z).T X = Z.dot(R).dot(ZH) except SqrtmError: failflag = True X = np.empty_like(A) X.fill(np.nan)
if disp: if failflag: print("Failed to find a square root.") return X else: try: arg2 = norm(X.dot(X) - A, 'fro')**2 / norm(A, 'fro') except ValueError: # NaNs in matrix arg2 = np.inf
return X, arg2 |