# scipy/scipy

### Subversion checkout URL

You can clone with
or
.
Fetching contributors…
Cannot retrieve contributors at this time
187 lines (152 sloc) 5.48 KB
 """ Matrix square root for general matrices and for upper triangular matrices. This module exists to avoid cyclic imports. """ from __future__ import division, print_function, absolute_import __all__ = ['sqrtm'] import numpy as np # Local imports from .misc import norm from .lapack import ztrsyl, dtrsyl from .special_matrices import all_mat from .decomp_schur import schur, rsf2csf class SqrtmError(np.linalg.LinAlgError): pass def _has_complex_dtype_char(A): return A.dtype.char in ('F', 'D', 'G') def _sqrtm_triu(T, blocksize=64): """ 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 : 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 `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 = (not _has_complex_dtype_char(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] if not denom: raise SqrtmError('failed to find the matrix square root') R[i,j] = (T[i,j] - s) / denom # 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 def sqrtm(A, disp=True, blocksize=64): """ 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. """ A = np.asarray(A) 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 = not _has_complex_dtype_char(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) R, Z = all_mat(R,Z) X = (Z * R * Z.H) except SqrtmError as e: failflag = True X = np.matrix(np.empty_like(A)) X.fill(np.nan) if disp: nzeig = np.any(np.diag(T) == 0) if nzeig: print("Matrix is singular and may not have a square root.") elif failflag: print("Failed to find a square root.") return X.A else: try: arg2 = norm(X*X - A,'fro')**2 / norm(A,'fro') except ValueError: # NaNs in matrix arg2 = np.inf return X.A, arg2
Something went wrong with that request. Please try again.