# scipy/scipy

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 """SVD decomposition functions.""" import numpy from numpy import asarray_chkfinite, zeros, r_, diag from scipy.linalg import calc_lwork # Local imports. from misc import LinAlgError, _datacopied from lapack import get_lapack_funcs __all__ = ['svd', 'svdvals', 'diagsvd', 'orth'] def svd(a, full_matrices=True, compute_uv=True, overwrite_a=False): """ Singular Value Decomposition. Factorizes the matrix a into two unitary matrices U and Vh, and a 1-D array s of singular values (real, non-negative) such that ``a == U*S*Vh``, where S is a suitably shaped matrix of zeros with main diagonal s. Parameters ---------- a : ndarray Matrix to decompose, of shape ``(M,N)``. full_matrices : bool, optional If True, `U` and `Vh` are of shape ``(M,M)``, ``(N,N)``. If False, the shapes are ``(M,K)`` and ``(K,N)``, where ``K = min(M,N)``. compute_uv : bool, optional Whether to compute also `U` and `Vh` in addition to `s`. Default is True. overwrite_a : bool, optional Whether to overwrite `a`; may improve performance. Default is False. Returns ------- U : ndarray Unitary matrix having left singular vectors as columns. Of shape ``(M,M)`` or ``(M,K)``, depending on `full_matrices`. s : ndarray The singular values, sorted in non-increasing order. Of shape (K,), with ``K = min(M, N)``. Vh : ndarray Unitary matrix having right singular vectors as rows. Of shape ``(N,N)`` or ``(K,N)`` depending on `full_matrices`. For ``compute_uv = False``, only `s` is returned. Raises ------ LinAlgError If SVD computation does not converge. See also -------- svdvals : Compute singular values of a matrix. diagsvd : Construct the Sigma matrix, given the vector s. Examples -------- >>> from scipy import linalg >>> a = np.random.randn(9, 6) + 1.j*np.random.randn(9, 6) >>> U, s, Vh = linalg.svd(a) >>> U.shape, Vh.shape, s.shape ((9, 9), (6, 6), (6,)) >>> U, s, Vh = linalg.svd(a, full_matrices=False) >>> U.shape, Vh.shape, s.shape ((9, 6), (6, 6), (6,)) >>> S = linalg.diagsvd(s, 6, 6) >>> np.allclose(a, np.dot(U, np.dot(S, Vh))) True >>> s2 = linalg.svd(a, compute_uv=False) >>> np.allclose(s, s2) True """ a1 = asarray_chkfinite(a) if len(a1.shape) != 2: raise ValueError('expected matrix') m,n = a1.shape overwrite_a = overwrite_a or (_datacopied(a1, a)) gesdd, = get_lapack_funcs(('gesdd',), (a1,)) if gesdd.module_name[:7] == 'flapack': lwork = calc_lwork.gesdd(gesdd.prefix, m, n, compute_uv)[1] u,s,v,info = gesdd(a1,compute_uv = compute_uv, lwork = lwork, full_matrices=full_matrices, overwrite_a = overwrite_a) else: # 'clapack' raise NotImplementedError('calling gesdd from %s' % gesdd.module_name) if info > 0: raise LinAlgError("SVD did not converge") if info < 0: raise ValueError('illegal value in %d-th argument of internal gesdd' % -info) if compute_uv: return u, s, v else: return s def svdvals(a, overwrite_a=False): """ Compute singular values of a matrix. Parameters ---------- a : ndarray Matrix to decompose, of shape ``(M, N)``. overwrite_a : bool, optional Whether to overwrite `a`; may improve performance. Default is False. Returns ------- s : ndarray The singular values, sorted in decreasing order. Of shape ``(K,)``, with``K = min(M, N)``. Raises ------ LinAlgError If SVD computation does not converge. See also -------- svd : Compute the full singular value decomposition of a matrix. diagsvd : Construct the Sigma matrix, given the vector s. """ return svd(a, compute_uv=0, overwrite_a=overwrite_a) def diagsvd(s, M, N): """ Construct the sigma matrix in SVD from singular values and size M, N. Parameters ---------- s : array_like, shape (M,) or (N,) Singular values M : int Size of the matrix whose singular values are `s`. N : int Size of the matrix whose singular values are `s`. Returns ------- S : array, shape (M, N) The S-matrix in the singular value decomposition """ part = diag(s) typ = part.dtype.char MorN = len(s) if MorN == M: return r_['-1', part, zeros((M, N-M), typ)] elif MorN == N: return r_[part, zeros((M-N,N), typ)] else: raise ValueError("Length of s must be M or N.") # Orthonormal decomposition def orth(A): """Construct an orthonormal basis for the range of A using SVD Parameters ---------- A : array, shape (M, N) Returns ------- Q : array, shape (M, K) Orthonormal basis for the range of A. K = effective rank of A, as determined by automatic cutoff See also -------- svd : Singular value decomposition of a matrix """ u, s, vh = svd(A) M, N = A.shape eps = numpy.finfo(float).eps tol = max(M,N) * numpy.amax(s) * eps num = numpy.sum(s > tol, dtype=int) Q = u[:,:num] return Q
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