path: root/numpy/linalg/linalg.py

"""Lite version of scipy.linalg.

This module is a lite version of the linalg.py module in SciPy which contains
high-level Python interface to the LAPACK library.  The lite version
only accesses the following LAPACK functions: dgesv, zgesv, dgeev,
zgeev, dgesdd, zgesdd, dgelsd, zgelsd, dsyevd, zheevd, dgetrf, dpotrf.
"""

__all__ = ['solve', 'tensorsolve', 'tensorinv',
           'inv', 'cholesky',
           'eigvals',
           'eigvalsh', 'pinv',
           'det', 'svd',
           'eig', 'eigh','lstsq', 'norm',
           'qr',
           'LinAlgError'
           ]

from numpy.core import array, asarray, zeros, empty, transpose, \
        intc, single, double, csingle, cdouble, inexact, complexfloating, \
        newaxis, ravel, all, Inf, dot, add, multiply, identity, sqrt, \
        maximum, flatnonzero, diagonal, arange, fastCopyAndTranspose, sum, \
        isfinite, size
from numpy.lib import triu
from numpy.linalg import lapack_lite

fortran_int = intc

# Error object
class LinAlgError(Exception):
    pass

def _makearray(a):
    new = asarray(a)
    wrap = getattr(a, "__array_wrap__", new.__array_wrap__)
    return new, wrap

def isComplexType(t):
    return issubclass(t, complexfloating)

_real_types_map = {single : single,
                   double : double,
                   csingle : single,
                   cdouble : double}

_complex_types_map = {single : csingle,
                      double : cdouble,
                      csingle : csingle,
                      cdouble : cdouble}

def _realType(t, default=double):
    return _real_types_map.get(t, default)

def _complexType(t, default=cdouble):
    return _complex_types_map.get(t, default)

def _linalgRealType(t):
    """Cast the type t to either double or cdouble."""
    return double

_complex_types_map = {single : csingle,
                      double : cdouble,
                      csingle : csingle,
                      cdouble : cdouble}

def _commonType(*arrays):
    # in lite version, use higher precision (always double or cdouble)
    result_type = single
    is_complex = False
    for a in arrays:
        if issubclass(a.dtype.type, inexact):
            if isComplexType(a.dtype.type):
                is_complex = True
            rt = _realType(a.dtype.type, default=None)
            if rt is None:
                # unsupported inexact scalar
                raise TypeError("array type %s is unsupported in linalg" %
                        (a.dtype.name,))
        else:
            rt = double
        if rt is double:
            result_type = double
    if is_complex:
        t = cdouble
        result_type = _complex_types_map[result_type]
    else:
        t = double
    return t, result_type

def _castCopyAndTranspose(type, *arrays):
    if len(arrays) == 1:
        return transpose(arrays[0]).astype(type)
    else:
        return [transpose(a).astype(type) for a in arrays]

# _fastCopyAndTranpose is an optimized version of _castCopyAndTranspose.
# It assumes the input is 2D (as all the calls in here are).

_fastCT = fastCopyAndTranspose

def _fastCopyAndTranspose(type, *arrays):
    cast_arrays = ()
    for a in arrays:
        if a.dtype.type is type:
            cast_arrays = cast_arrays + (_fastCT(a),)
        else:
            cast_arrays = cast_arrays + (_fastCT(a.astype(type)),)
    if len(cast_arrays) == 1:
        return cast_arrays[0]
    else:
        return cast_arrays

def _assertRank2(*arrays):
    for a in arrays:
        if len(a.shape) != 2:
            raise LinAlgError, '%d-dimensional array given. Array must be \
            two-dimensional' % len(a.shape)

def _assertSquareness(*arrays):
    for a in arrays:
        if max(a.shape) != min(a.shape):
            raise LinAlgError, 'Array must be square'

def _assertFinite(*arrays):
    for a in arrays:
        if not (isfinite(a).all()):
            raise LinAlgError, "Array must not contain infs or NaNs"

def _assertNonEmpty(*arrays):
    for a in arrays:
        if size(a) == 0:
            raise LinAlgError("Arrays cannot be empty")

# Linear equations

def tensorsolve(a, b, axes=None):
    """Solves the tensor equation a x = b for x

    where it is assumed that all the indices of x are summed over in
    the product.

    a can be N-dimensional.  x will have the dimensions of A subtracted from
    the dimensions of b.
    """
    a = asarray(a)
    b = asarray(b)
    an = a.ndim

    if axes is not None:
        allaxes = range(0, an)
        for k in axes:
            allaxes.remove(k)
            allaxes.insert(an, k)
        a = a.transpose(allaxes)

    oldshape = a.shape[-(an-b.ndim):]
    prod = 1
    for k in oldshape:
        prod *= k

    a = a.reshape(-1, prod)
    b = b.ravel()
    res = solve(a, b)
    res.shape = oldshape
    return res

def solve(a, b):
    """Return the solution of a*x = b
    """
    one_eq = len(b.shape) == 1
    if one_eq:
        b = b[:, newaxis]
    _assertRank2(a, b)
    _assertSquareness(a)
    n_eq = a.shape[0]
    n_rhs = b.shape[1]
    if n_eq != b.shape[0]:
        raise LinAlgError, 'Incompatible dimensions'
    t, result_t = _commonType(a, b)
#    lapack_routine = _findLapackRoutine('gesv', t)
    if isComplexType(t):
        lapack_routine = lapack_lite.zgesv
    else:
        lapack_routine = lapack_lite.dgesv
    a, b = _fastCopyAndTranspose(t, a, b)
    pivots = zeros(n_eq, fortran_int)
    results = lapack_routine(n_eq, n_rhs, a, n_eq, pivots, b, n_eq, 0)
    if results['info'] > 0:
        raise LinAlgError, 'Singular matrix'
    if one_eq:
        return b.ravel().astype(result_t)
    else:
        return b.transpose().astype(result_t)


def tensorinv(a, ind=2):
    """Find the 'inverse' of a N-d array

    ind must be a positive integer specifying
    how many indices at the front of the array are involved
    in the inverse sum.

    the result is ainv with shape a.shape[ind:] + a.shape[:ind]

    tensordot(ainv, a, ind) is an identity operator

    and so is

    tensordot(a, ainv, a.shape-newind)

    Example:

       a = rand(4,6,8,3)
       ainv = tensorinv(a)
       # ainv.shape is (8,3,4,6)
       # suppose b has shape (4,6)
       tensordot(ainv, b) # produces same (8,3)-shaped output as
       tensorsolve(a, b)

       a = rand(24,8,3)
       ainv = tensorinv(a,1)
       # ainv.shape is (8,3,24)
       # suppose b has shape (24,)
       tensordot(ainv, b, 1)  # produces the same (8,3)-shaped output as
       tensorsolve(a, b)

    """
    a = asarray(a)
    oldshape = a.shape
    prod = 1
    if ind > 0:
        invshape = oldshape[ind:] + oldshape[:ind]
        for k in oldshape[ind:]:
            prod *= k
    else:
        raise ValueError, "Invalid ind argument."
    a = a.reshape(prod, -1)
    ia = inv(a)
    return ia.reshape(*invshape)


# Matrix inversion

def inv(a):
    a, wrap = _makearray(a)
    return wrap(solve(a, identity(a.shape[0], dtype=a.dtype)))

# Cholesky decomposition

def cholesky(a):
    _assertRank2(a)
    _assertSquareness(a)
    t, result_t = _commonType(a)
    a = _fastCopyAndTranspose(t, a)
    m = a.shape[0]
    n = a.shape[1]
    if isComplexType(t):
        lapack_routine = lapack_lite.zpotrf
    else:
        lapack_routine = lapack_lite.dpotrf
    results = lapack_routine('L', n, a, m, 0)
    if results['info'] > 0:
        raise LinAlgError, 'Matrix is not positive definite - \
        Cholesky decomposition cannot be computed'
    s = triu(a, k=0).transpose()
    if (s.dtype != result_t):
        return s.astype(result_t)
    return s

# QR decompostion

def qr(a, mode='full'):
    """cacluates A=QR, Q orthonormal, R upper triangular

    mode:  'full' --> (Q,R)
           'r'    --> R
           'economic' --> A2 where the diagonal + upper triangle
                   part of A2 is R. This is faster if you only need R
    """
    _assertRank2(a)
    m, n = a.shape
    t, result_t = _commonType(a)
    a = _fastCopyAndTranspose(t, a)
    mn = min(m, n)
    tau = zeros((mn,), t)
    if isComplexType(t):
        lapack_routine = lapack_lite.zgeqrf
        routine_name = 'zgeqrf'
    else:
        lapack_routine = lapack_lite.dgeqrf
        routine_name = 'dgeqrf'

    # calculate optimal size of work data 'work'
    lwork = 1
    work = zeros((lwork,), t)
    results = lapack_routine(m, n, a, m, tau, work, -1, 0)
    if results['info'] != 0:
        raise LinAlgError, '%s returns %d' % (routine_name, results['info'])

    # do qr decomposition
    lwork = int(abs(work[0]))
    work = zeros((lwork,), t)
    results = lapack_routine(m, n, a, m, tau, work, lwork, 0)

    if results['info'] != 0:
        raise LinAlgError, '%s returns %d' % (routine_name, results['info'])

    #  economic mode. Isn't actually economic.
    if mode[0] == 'e':
        if t != result_t :
            a = a.astype(result_t)
        return a.T

    #  generate r
    r = _fastCopyAndTranspose(result_t, a[:,:mn])
    for i in range(mn):
        r[i,:i].fill(0.0)

    #  'r'-mode, that is, calculate only r
    if mode[0] == 'r':
        return r

    #  from here on: build orthonormal matrix q from a

    if isComplexType(t):
        lapack_routine = lapack_lite.zungqr
        routine_name = 'zungqr'
    else:
        lapack_routine = lapack_lite.dorgqr
        routine_name = 'dorgqr'

    # determine optimal lwork
    lwork = 1
    work = zeros((lwork,), t)
    results = lapack_routine(m, mn, mn, a, m, tau, work, -1, 0)
    if results['info'] != 0:
        raise LinAlgError, '%s returns %d' % (routine_name, results['info'])

    # compute q
    lwork = int(abs(work[0]))
    work = zeros((lwork,), t)
    results = lapack_routine(m, mn, mn, a, m, tau, work, lwork, 0)
    if results['info'] != 0:
        raise LinAlgError, '%s returns %d' % (routine_name, results['info'])

    q = _fastCopyAndTranspose(result_t, a[:mn,:])

    return q, r


# Eigenvalues


def eigvals(a):
    """Compute the eigenvalues of the general 2-d array a.

    A simple interface to the LAPACK routines dgeev and zgeev that sets the
    flags to return only the eigenvalues of general real and complex arrays
    respectively.

    :Parameters:

        a : 2-d array
            A complex or real 2-d array whose eigenvalues and eigenvectors
            will be computed.

    :Returns:

        w : 1-d double or complex array
            The eigenvalues. The eigenvalues are not necessarily ordered, nor
            are they necessarily real for real matrices.

    :SeeAlso:

        - eig : eigenvalues and right eigenvectors of general arrays
        - eigvalsh : eigenvalues of symmetric or Hemitiean arrays.
        - eigh : eigenvalues and eigenvectors of symmetric/Hermitean arrays.

    :Notes:
    -------

        The number w is an eigenvalue of a if there exists a vector v
        satisfying the equation dot(a,v) = w*v. Alternately, if w is a root of
        the characteristic equation det(a - w[i]*I) = 0, where det is the
        determinant and I is the identity matrix.

    """
    _assertRank2(a)
    _assertSquareness(a)
    _assertFinite(a)
    t, result_t = _commonType(a)
    real_t = _linalgRealType(t)
    a = _fastCopyAndTranspose(t, a)
    n = a.shape[0]
    dummy = zeros((1,), t)
    if isComplexType(t):
        lapack_routine = lapack_lite.zgeev
        w = zeros((n,), t)
        rwork = zeros((n,), real_t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine('N', 'N', n, a, n, w,
                                 dummy, 1, dummy, 1, work, -1, rwork, 0)
        lwork = int(abs(work[0]))
        work = zeros((lwork,), t)
        results = lapack_routine('N', 'N', n, a, n, w,
                                 dummy, 1, dummy, 1, work, lwork, rwork, 0)
    else:
        lapack_routine = lapack_lite.dgeev
        wr = zeros((n,), t)
        wi = zeros((n,), t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine('N', 'N', n, a, n, wr, wi,
                                 dummy, 1, dummy, 1, work, -1, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine('N', 'N', n, a, n, wr, wi,
                                 dummy, 1, dummy, 1, work, lwork, 0)
        if all(wi == 0.):
            w = wr
            result_t = _realType(result_t)
        else:
            w = wr+1j*wi
            result_t = _complexType(result_t)
    if results['info'] > 0:
        raise LinAlgError, 'Eigenvalues did not converge'
    return w.astype(result_t)


def eigvalsh(a, UPLO='L'):
    """Compute the eigenvalues of the symmetric or Hermitean 2-d array a.

    A simple interface to the LAPACK routines dsyevd and zheevd that sets the
    flags to return only the eigenvalues of real symmetric and complex
    Hermetian arrays respectively.

    :Parameters:

        a : 2-d array
            A complex or real 2-d array whose eigenvalues and eigenvectors
            will be computed.

        UPLO : string
            Specifies whether the pertinent array date is taken from the upper
            or lower triangular part of a. Possible values are 'L', and 'U' for
            upper and lower respectively. Default is 'L'.

    :Returns:

        w : 1-d double array
            The eigenvalues. The eigenvalues are not necessarily ordered.

    :SeeAlso:

        - eigh : eigenvalues and eigenvectors of symmetric/Hermitean arrays.
        - eigvals : eigenvalues of general real or complex arrays.
        - eig : eigenvalues and eigenvectors of general real or complex arrays.

    :Notes:
    -------

        The number w is an eigenvalue of a if there exists a vector v
        satisfying the equation dot(a,v) = w*v. Alternately, if w is a root of
        the characteristic equation det(a - w[i]*I) = 0, where det is the
        determinant and I is the identity matrix. The eigenvalues of real
        symmetric or complex Hermitean matrices are always real.

    """
    _assertRank2(a)
    _assertSquareness(a)
    t, result_t = _commonType(a)
    real_t = _linalgRealType(t)
    a = _fastCopyAndTranspose(t, a)
    n = a.shape[0]
    liwork = 5*n+3
    iwork = zeros((liwork,), fortran_int)
    if isComplexType(t):
        lapack_routine = lapack_lite.zheevd
        w = zeros((n,), real_t)
        lwork = 1
        work = zeros((lwork,), t)
        lrwork = 1
        rwork = zeros((lrwork,), real_t)
        results = lapack_routine('N', UPLO, n, a, n, w, work, -1,
                                 rwork, -1, iwork, liwork,  0)
        lwork = int(abs(work[0]))
        work = zeros((lwork,), t)
        lrwork = int(rwork[0])
        rwork = zeros((lrwork,), real_t)
        results = lapack_routine('N', UPLO, n, a, n, w, work, lwork,
                                rwork, lrwork, iwork, liwork,  0)
    else:
        lapack_routine = lapack_lite.dsyevd
        w = zeros((n,), t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine('N', UPLO, n, a, n, w, work, -1,
                                 iwork, liwork, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine('N', UPLO, n, a, n, w, work, lwork,
                                 iwork, liwork, 0)
    if results['info'] > 0:
        raise LinAlgError, 'Eigenvalues did not converge'
    return w.astype(result_t)

def _convertarray(a):
    t, result_t = _commonType(a)
    a = _fastCT(a.astype(t))
    return a, t, result_t


# Eigenvectors


def eig(a):
    """Eigenvalues and right eigenvectors of a general matrix.

    A simple interface to the LAPACK routines dgeev and zgeev that compute the
    eigenvalues and eigenvectors of general real and complex arrays
    respectively.

    :Parameters:

        a : 2-d array
            A complex or real 2-d array whose eigenvalues and eigenvectors
            will be computed.

    :Returns:

        w : 1-d double or complex array
            The eigenvalues. The eigenvalues are not necessarily ordered, nor
            are they necessarily real for real matrices.

        v : 2-d double or complex double array.
            The normalized eigenvector corresponding to the eigenvalue w[i] is
            the column v[:,i].

    :SeeAlso:

        - eigvalsh : eigenvalues of symmetric or Hemitiean arrays.
        - eig : eigenvalues and right eigenvectors for non-symmetric arrays
        - eigvals : eigenvalues of non-symmetric array.

    :Notes:
    -------

        The number w is an eigenvalue of a if there exists a vector v
        satisfying the equation dot(a,v) = w*v. Alternately, if w is a root of
        the characteristic equation det(a - w[i]*I) = 0, where det is the
        determinant and I is the identity matrix. The arrays a, w, and v
        satisfy the equation dot(a,v[i]) = w[i]*v[:,i].

        The array v of eigenvectors may not be of maximum rank, that is, some
        of the columns may be dependent, although roundoff error may obscure
        that fact. If the eigenvalues are all different, then theoretically the
        eigenvectors are independent. Likewise, the matrix of eigenvectors is
        unitary if the matrix a is normal, i.e., if dot(a, a.H) = dot(a.H, a).

        The left and right eigenvectors are not necessarily the (Hemitian)
        transposes of each other.

    """
    a, wrap = _makearray(a)
    _assertRank2(a)
    _assertSquareness(a)
    _assertFinite(a)
    a, t, result_t = _convertarray(a) # convert to double or cdouble type
    real_t = _linalgRealType(t)
    n = a.shape[0]
    dummy = zeros((1,), t)
    if isComplexType(t):
        # Complex routines take different arguments
        lapack_routine = lapack_lite.zgeev
        w = zeros((n,), t)
        v = zeros((n, n), t)
        lwork = 1
        work = zeros((lwork,), t)
        rwork = zeros((2*n,), real_t)
        results = lapack_routine('N', 'V', n, a, n, w,
                                 dummy, 1, v, n, work, -1, rwork, 0)
        lwork = int(abs(work[0]))
        work = zeros((lwork,), t)
        results = lapack_routine('N', 'V', n, a, n, w,
                                 dummy, 1, v, n, work, lwork, rwork, 0)
    else:
        lapack_routine = lapack_lite.dgeev
        wr = zeros((n,), t)
        wi = zeros((n,), t)
        vr = zeros((n, n), t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine('N', 'V', n, a, n, wr, wi,
                                  dummy, 1, vr, n, work, -1, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine('N', 'V', n, a, n, wr, wi,
                                  dummy, 1, vr, n, work, lwork, 0)
        if all(wi == 0.0):
            w = wr
            v = vr
            result_t = _realType(result_t)
        else:
            w = wr+1j*wi
            v = array(vr, w.dtype)
            ind = flatnonzero(wi != 0.0)      # indices of complex e-vals
            for i in range(len(ind)/2):
                v[ind[2*i]] = vr[ind[2*i]] + 1j*vr[ind[2*i+1]]
                v[ind[2*i+1]] = vr[ind[2*i]] - 1j*vr[ind[2*i+1]]
            result_t = _complexType(result_t)

    if results['info'] > 0:
        raise LinAlgError, 'Eigenvalues did not converge'
    vt = v.transpose().astype(result_t)
    return w.astype(result_t), wrap(vt)


def eigh(a, UPLO='L'):
    """Compute eigenvalues for a Hermitian-symmetric matrix.

    A simple interface to the LAPACK routines dsyevd and zheevd that compute
    the eigenvalues and eigenvectors of real symmetric and complex Hermitian
    arrays respectively.

    :Parameters:

        a : 2-d array
            A complex Hermitian or symmetric real 2-d array whose eigenvalues
            and eigenvectors will be computed.

        UPLO : string
            Specifies whether the pertinent array date is taken from the upper
            or lower triangular part of a. Possible values are 'L', and 'U'.
            Default is 'L'.

    :Returns:

        w : 1-d double array
            The eigenvalues. The eigenvalues are not necessarily ordered.

        v : 2-d double or complex double array, depending on input array type
            The normalized eigenvector corresponding to the eigenvalue w[i] is
            the column v[:,i].

    :SeeAlso:

        - eigvalsh : eigenvalues of symmetric or Hemitiean arrays.
        - eig : eigenvalues and right eigenvectors for non-symmetric arrays
        - eigvals : eigenvalues of non-symmetric array.

    :Notes:
    -------

        The number w is an eigenvalue of a if there exists a vector v
        satisfying the equation dot(a,v) = w*v. Alternately, if w is a root of
        the characteristic equation det(a - w[i]*I) = 0, where det is the
        determinant and I is the identity matrix. The eigenvalues of real
        symmetric or complex Hermitean matrices are always real. The array v
        of eigenvectors is unitary and a, w, and v satisfy the equation
        dot(a,v[i]) = w[i]*v[:,i].

    """
    a, wrap = _makearray(a)
    _assertRank2(a)
    _assertSquareness(a)
    t, result_t = _commonType(a)
    real_t = _linalgRealType(t)
    a = _fastCopyAndTranspose(t, a)
    n = a.shape[0]
    liwork = 5*n+3
    iwork = zeros((liwork,), fortran_int)
    if isComplexType(t):
        lapack_routine = lapack_lite.zheevd
        w = zeros((n,), real_t)
        lwork = 1
        work = zeros((lwork,), t)
        lrwork = 1
        rwork = zeros((lrwork,), real_t)
        results = lapack_routine('V', UPLO, n, a, n, w, work, -1,
                                 rwork, -1, iwork, liwork,  0)
        lwork = int(abs(work[0]))
        work = zeros((lwork,), t)
        lrwork = int(rwork[0])
        rwork = zeros((lrwork,), real_t)
        results = lapack_routine('V', UPLO, n, a, n, w, work, lwork,
                                 rwork, lrwork, iwork, liwork,  0)
    else:
        lapack_routine = lapack_lite.dsyevd
        w = zeros((n,), t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine('V', UPLO, n, a, n, w, work, -1,
                iwork, liwork, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine('V', UPLO, n, a, n, w, work, lwork,
                iwork, liwork, 0)
    if results['info'] > 0:
        raise LinAlgError, 'Eigenvalues did not converge'
    at = a.transpose().astype(result_t)
    return w.astype(_realType(result_t)), wrap(at)


# Singular value decomposition

def svd(a, full_matrices=1, compute_uv=1):
    """Singular Value Decomposition.

    u,s,vh = svd(a)

    If a is an M x N array, then the svd produces a factoring of the array
    into two unitary (orthogonal) 2-d arrays u (MxM) and vh (NxN) and a
    min(M,N)-length array of singular values such that

                     a == dot(u,dot(S,vh))

    where S is an MxN array of zeros whose main diagonal is s.

    if compute_uv == 0, then return only the singular values
    if full_matrices == 0, then only part of either u or vh is
                           returned so that it is MxN
    """
    a, wrap = _makearray(a)
    _assertRank2(a)
    _assertNonEmpty(a)
    m, n = a.shape
    t, result_t = _commonType(a)
    real_t = _linalgRealType(t)
    a = _fastCopyAndTranspose(t, a)
    s = zeros((min(n, m),), real_t)
    if compute_uv:
        if full_matrices:
            nu = m
            nvt = n
            option = 'A'
        else:
            nu = min(n, m)
            nvt = min(n, m)
            option = 'S'
        u = zeros((nu, m), t)
        vt = zeros((n, nvt), t)
    else:
        option = 'N'
        nu = 1
        nvt = 1
        u = empty((1, 1), t)
        vt = empty((1, 1), t)

    iwork = zeros((8*min(m, n),), fortran_int)
    if isComplexType(t):
        lapack_routine = lapack_lite.zgesdd
        rwork = zeros((5*min(m, n)*min(m, n) + 5*min(m, n),), real_t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                 work, -1, rwork, iwork, 0)
        lwork = int(abs(work[0]))
        work = zeros((lwork,), t)
        results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                 work, lwork, rwork, iwork, 0)
    else:
        lapack_routine = lapack_lite.dgesdd
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                 work, -1, iwork, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                 work, lwork, iwork, 0)
    if results['info'] > 0:
        raise LinAlgError, 'SVD did not converge'
    s = s.astype(_realType(result_t))
    if compute_uv:
        u = u.transpose().astype(result_t)
        vt = vt.transpose().astype(result_t)
        return wrap(u), s, wrap(vt)
    else:
        return s

# Generalized inverse

def pinv(a, rcond=1e-15 ):
    """Return the (Moore-Penrose) pseudo-inverse of a 2-d array

    This method computes the generalized inverse using the
    singular-value decomposition and all singular values larger than
    rcond of the largest.
    """
    a, wrap = _makearray(a)
    _assertNonEmpty(a)
    a = a.conjugate()
    u, s, vt = svd(a, 0)
    m = u.shape[0]
    n = vt.shape[1]
    cutoff = rcond*maximum.reduce(s)
    for i in range(min(n, m)):
        if s[i] > cutoff:
            s[i] = 1./s[i]
        else:
            s[i] = 0.;
    return wrap(dot(transpose(vt),
                       multiply(s[:, newaxis],transpose(u))))

# Determinant

def det(a):
    "The determinant of the 2-d array a"
    a = asarray(a)
    _assertRank2(a)
    _assertSquareness(a)
    t, result_t = _commonType(a)
    a = _fastCopyAndTranspose(t, a)
    n = a.shape[0]
    if isComplexType(t):
        lapack_routine = lapack_lite.zgetrf
    else:
        lapack_routine = lapack_lite.dgetrf
    pivots = zeros((n,), fortran_int)
    results = lapack_routine(n, n, a, n, pivots, 0)
    info = results['info']
    if (info < 0):
        raise TypeError, "Illegal input to Fortran routine"
    elif (info > 0):
        return 0.0
    sign = add.reduce(pivots != arange(1, n+1)) % 2
    return (1.-2.*sign)*multiply.reduce(diagonal(a), axis=-1)

# Linear Least Squares

def lstsq(a, b, rcond=-1):
    """returns x,resids,rank,s
    where x minimizes 2-norm(|b - Ax|)
    resids is the sum square residuals
    rank is the rank of A
    s is the rank of the singular values of A in descending order

    If b is a matrix then x is also a matrix with corresponding columns.
    If the rank of A is less than the number of columns of A or greater than
    the number of rows, then residuals will be returned as an empty array
    otherwise resids = sum((b-dot(A,x)**2).
    Singular values less than s[0]*rcond are treated as zero.
"""
    import math
    a = asarray(a)
    b, wrap = _makearray(b)
    one_eq = len(b.shape) == 1
    if one_eq:
        b = b[:, newaxis]
    _assertRank2(a, b)
    m  = a.shape[0]
    n  = a.shape[1]
    n_rhs = b.shape[1]
    ldb = max(n, m)
    if m != b.shape[0]:
        raise LinAlgError, 'Incompatible dimensions'
    t, result_t = _commonType(a, b)
    real_t = _linalgRealType(t)
    bstar = zeros((ldb, n_rhs), t)
    bstar[:b.shape[0],:n_rhs] = b.copy()
    a, bstar = _fastCopyAndTranspose(t, a, bstar)
    s = zeros((min(m, n),), real_t)
    nlvl = max( 0, int( math.log( float(min(m, n))/2. ) ) + 1 )
    iwork = zeros((3*min(m, n)*nlvl+11*min(m, n),), fortran_int)
    if isComplexType(t):
        lapack_routine = lapack_lite.zgelsd
        lwork = 1
        rwork = zeros((lwork,), real_t)
        work = zeros((lwork,), t)
        results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                 0, work, -1, rwork, iwork, 0)
        lwork = int(abs(work[0]))
        rwork = zeros((lwork,), real_t)
        a_real = zeros((m, n), real_t)
        bstar_real = zeros((ldb, n_rhs,), real_t)
        results = lapack_lite.dgelsd(m, n, n_rhs, a_real, m,
                                     bstar_real, ldb, s, rcond,
                                     0, rwork, -1, iwork, 0)
        lrwork = int(rwork[0])
        work = zeros((lwork,), t)
        rwork = zeros((lrwork,), real_t)
        results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                 0, work, lwork, rwork, iwork, 0)
    else:
        lapack_routine = lapack_lite.dgelsd
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                 0, work, -1, iwork, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                 0, work, lwork, iwork, 0)
    if results['info'] > 0:
        raise LinAlgError, 'SVD did not converge in Linear Least Squares'
    resids = array([], t)
    if one_eq:
        x = array(ravel(bstar)[:n], dtype=result_t, copy=True)
        if results['rank'] == n and m > n:
            resids = array([sum((ravel(bstar)[n:])**2)], dtype=result_t)
    else:
        x = array(transpose(bstar)[:n,:], dtype=result_t, copy=True)
        if results['rank'] == n and m > n:
            resids = sum((transpose(bstar)[n:,:])**2, axis=0).astype(result_t)
    st = s[:min(n, m)].copy().astype(_realType(result_t))
    return wrap(x), resids, results['rank'], st

def norm(x, ord=None):
    """ norm(x, ord=None) -> n

    Matrix or vector norm.

    Inputs:

      x -- a rank-1 (vector) or rank-2 (matrix) array
      ord -- the order of the norm.

     Comments:
       For arrays of any rank, if ord is None:
         calculate the square norm (Euclidean norm for vectors,
         Frobenius norm for matrices)

       For vectors ord can be any real number including Inf or -Inf.
         ord = Inf, computes the maximum of the magnitudes
         ord = -Inf, computes minimum of the magnitudes
         ord is finite, computes sum(abs(x)**ord,axis=0)**(1.0/ord)

       For matrices ord can only be one of the following values:
         ord = 2 computes the largest singular value
         ord = -2 computes the smallest singular value
         ord = 1 computes the largest column sum of absolute values
         ord = -1 computes the smallest column sum of absolute values
         ord = Inf computes the largest row sum of absolute values
         ord = -Inf computes the smallest row sum of absolute values
         ord = 'fro' computes the frobenius norm sqrt(sum(diag(X.H * X),axis=0))

       For values ord < 0, the result is, strictly speaking, not a
       mathematical 'norm', but it may still be useful for numerical purposes.
    """
    x = asarray(x)
    nd = len(x.shape)
    if ord is None: # check the default case first and handle it immediately
        return sqrt(add.reduce((x.conj() * x).ravel().real))

    if nd == 1:
        if ord == Inf:
            return abs(x).max()
        elif ord == -Inf:
            return abs(x).min()
        elif ord == 1:
            return abs(x).sum() # special case for speedup
        elif ord == 2:
            return sqrt(((x.conj()*x).real).sum()) # special case for speedup
        else:
            return ((abs(x)**ord).sum())**(1.0/ord)
    elif nd == 2:
        if ord == 2:
            return svd(x, compute_uv=0).max()
        elif ord == -2:
            return svd(x, compute_uv=0).min()
        elif ord == 1:
            return abs(x).sum(axis=0).max()
        elif ord == Inf:
            return abs(x).sum(axis=1).max()
        elif ord == -1:
            return abs(x).sum(axis=0).min()
        elif ord == -Inf:
            return abs(x).sum(axis=1).min()
        elif ord in ['fro','f']:
            return sqrt(add.reduce((x.conj() * x).real.ravel()))
        else:
            raise ValueError, "Invalid norm order for matrices."
    else:
        raise ValueError, "Improper number of dimensions to norm."