def eig(a):
"""
Compute eigenvalues and right eigenvectors of an array.
Parameters
----------
a : array_like, shape (M, M)
A complex or real 2-D array.
Returns
-------
w : ndarray, shape (M,)
The eigenvalues, each repeated according to its multiplicity.
The eigenvalues are not necessarily ordered, nor are they
necessarily real for real matrices.
v : ndarray, shape (M, M)
The normalized eigenvector corresponding to the eigenvalue ``w[i]`` is
the column ``v[:,i]``.
Raises
------
LinAlgError
If the eigenvalue computation does not converge.
See Also
--------
eigvalsh : eigenvalues of symmetric or Hemitiean arrays.
eig : eigenvalues and right eigenvectors for non-symmetric arrays
eigvals : eigenvalues of non-symmetric array.
Notes
-----
This is a simple interface to the LAPACK routines dgeev and zgeev
that compute the eigenvalues and eigenvectors of general real and
complex arrays respectively.
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 (Hermitian)
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 assert_array_compare(comparison, x, y, err_msg='', verbose=True,
header=''):
from numpy.core import array, isnan, isinf, isna, any, all, inf
x = array(x, copy=False, subok=True)
y = array(y, copy=False, subok=True)
def isnumber(x):
return x.dtype.char in '?bhilqpBHILQPefdgFDG'
def chk_same_position(x_id, y_id, hasval='nan'):
"""Handling nan/inf: check that x and y have the nan/inf at the same
locations."""
try:
assert_array_equal(x_id, y_id)
except AssertionError:
msg = build_err_msg([x, y],
err_msg + '\nx and y %s location mismatch:' \
% (hasval), verbose=verbose, header=header,
names=('x', 'y'))
raise AssertionError(msg)
try:
cond = (x.shape==() or y.shape==()) or x.shape == y.shape
if not cond:
msg = build_err_msg([x, y],
err_msg
+ '\n(shapes %s, %s mismatch)' % (x.shape,
y.shape),
verbose=verbose, header=header,
names=('x', 'y'))
if not cond :
raise AssertionError(msg)
if isnumber(x) and isnumber(y):
x_isna, y_isna = isna(x), isna(y)
x_isnan, y_isnan = isnan(x), isnan(y)
x_isinf, y_isinf = isinf(x), isinf(y)
# Remove any NAs from the isnan and isinf arrays
if x.ndim == 0:
if x_isna:
x_isnan = False
x_isinf = False
else:
x_isnan[x_isna] = False
x_isinf[x_isna] = False
if y.ndim == 0:
if y_isna:
y_isnan = False
y_isinf = False
else:
y_isnan[y_isna] = False
y_isinf[y_isna] = False
# Validate that the special values are in the same place
if any(x_isnan) or any(y_isnan):
chk_same_position(x_isnan, y_isnan, hasval='nan')
if any(x_isinf) or any(y_isinf):
# Check +inf and -inf separately, since they are different
chk_same_position(x == +inf, y == +inf, hasval='+inf')
chk_same_position(x == -inf, y == -inf, hasval='-inf')
if any(x_isna) or any(y_isna):
chk_same_position(x_isna, y_isna, hasval='NA')
# Combine all the special values
x_id, y_id = x_isnan, y_isnan
x_id |= x_isinf
y_id |= y_isinf
x_id |= x_isna
y_id |= y_isna
# Only do the comparison if actual values are left
if all(x_id):
return
if any(x_id):
val = comparison(x[~x_id], y[~y_id])
else:
val = comparison(x, y)
# field-NA isn't supported yet, so skip struct dtypes for this
elif (not x.dtype.names and not y.dtype.names) and \
(any(isna(x)) or any(isna(y))):
x_isna, y_isna = isna(x), isna(y)
if any(x_isna) or any(y_isna):
chk_same_position(x_isna, y_isna, hasval='NA')
if all(x_isna):
return
val = comparison(x[~x_isna], y[~y_isna])
else:
val = comparison(x,y)
if isinstance(val, bool):
cond = val
reduced = [0]
else:
reduced = val.ravel()
cond = reduced.all()
reduced = reduced.tolist()
if not cond:
match = 100-100.0*reduced.count(1)/len(reduced)
msg = build_err_msg([x, y],
err_msg
+ '\n(mismatch %s%%)' % (match,),
verbose=verbose, header=header,
names=('x', 'y'))
if not cond :
raise AssertionError(msg)
except ValueError, e:
import traceback
efmt = traceback.format_exc()
header = 'error during assertion:\n\n%s\n\n%s' % (efmt, header)
msg = build_err_msg([x, y], err_msg, verbose=verbose, header=header,
names=('x', 'y'))
raise ValueError(msg)
def eigvals(a):
"""
Compute the eigenvalues of a general matrix.
Parameters
----------
a : array_like, shape (M, M)
A complex or real matrix whose eigenvalues and eigenvectors
will be computed.
Returns
-------
w : ndarray, shape (M,)
The eigenvalues, each repeated according to its multiplicity.
They are not necessarily ordered, nor are they necessarily
real for real matrices.
Raises
------
LinAlgError
If the eigenvalue computation does not converge.
See Also
--------
eig : eigenvalues and right eigenvectors of general arrays
eigvalsh : eigenvalues of symmetric or Hemitiean arrays.
eigh : eigenvalues and eigenvectors of symmetric/Hermitean arrays.
Notes
-----
This is 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.
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.
"""
a, wrap = _makearray(a)
_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 eigvals(a):
"""
Compute the eigenvalues of a general matrix.
Parameters
----------
a : array_like, shape (M, M)
A complex or real matrix whose eigenvalues and eigenvectors
will be computed.
Returns
-------
w : ndarray, shape (M,)
The eigenvalues, each repeated according to its multiplicity.
They are not necessarily ordered, nor are they necessarily
real for real matrices.
Raises
------
LinAlgError
If the eigenvalue computation does not converge.
See Also
--------
eig : eigenvalues and right eigenvectors of general arrays
eigvalsh : eigenvalues of symmetric or Hemitiean arrays.
eigh : eigenvalues and eigenvectors of symmetric/Hermitean arrays.
Notes
-----
This is 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.
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.
"""
a, wrap = _makearray(a)
_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 eig(a):
"""
Compute eigenvalues and right eigenvectors of an array.
Parameters
----------
a : array_like, shape (M, M)
A complex or real 2-D array.
Returns
-------
w : ndarray, shape (M,)
The eigenvalues, each repeated according to its multiplicity.
The eigenvalues are not necessarily ordered, nor are they
necessarily real for real matrices.
v : ndarray, shape (M, M)
The normalized eigenvector corresponding to the eigenvalue ``w[i]`` is
the column ``v[:,i]``.
Raises
------
LinAlgError
If the eigenvalue computation does not converge.
See Also
--------
eigvalsh : eigenvalues of symmetric or Hemitiean arrays.
eig : eigenvalues and right eigenvectors for non-symmetric arrays
eigvals : eigenvalues of non-symmetric array.
Notes
-----
This is a simple interface to the LAPACK routines dgeev and zgeev
that compute the eigenvalues and eigenvectors of general real and
complex arrays respectively.
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 (Hermitian)
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 assert_array_compare(comparison, x, y, err_msg="", verbose=True, header=""):
from numpy.core import array, isnan, isinf, any, all, inf
x = array(x, copy=False, subok=True)
y = array(y, copy=False, subok=True)
def isnumber(x):
return x.dtype.char in "?bhilqpBHILQPefdgFDG"
def chk_same_position(x_id, y_id, hasval="nan"):
"""Handling nan/inf: check that x and y have the nan/inf at the same
locations."""
try:
assert_array_equal(x_id, y_id)
except AssertionError:
msg = build_err_msg(
[x, y],
err_msg + "\nx and y %s location mismatch:" % (hasval),
verbose=verbose,
header=header,
names=("x", "y"),
)
raise AssertionError(msg)
try:
cond = (x.shape == () or y.shape == ()) or x.shape == y.shape
if not cond:
msg = build_err_msg(
[x, y],
err_msg + "\n(shapes %s, %s mismatch)" % (x.shape, y.shape),
verbose=verbose,
header=header,
names=("x", "y"),
)
if not cond:
raise AssertionError(msg)
if isnumber(x) and isnumber(y):
x_isnan, y_isnan = isnan(x), isnan(y)
x_isinf, y_isinf = isinf(x), isinf(y)
# Validate that the special values are in the same place
if any(x_isnan) or any(y_isnan):
chk_same_position(x_isnan, y_isnan, hasval="nan")
if any(x_isinf) or any(y_isinf):
# Check +inf and -inf separately, since they are different
chk_same_position(x == +inf, y == +inf, hasval="+inf")
chk_same_position(x == -inf, y == -inf, hasval="-inf")
# Combine all the special values
x_id, y_id = x_isnan, y_isnan
x_id |= x_isinf
y_id |= y_isinf
# Only do the comparison if actual values are left
if all(x_id):
return
if any(x_id):
val = comparison(x[~x_id], y[~y_id])
else:
val = comparison(x, y)
else:
val = comparison(x, y)
if isinstance(val, bool):
cond = val
reduced = [0]
else:
reduced = val.ravel()
cond = reduced.all()
reduced = reduced.tolist()
if not cond:
match = 100 - 100.0 * reduced.count(1) / len(reduced)
msg = build_err_msg(
[x, y], err_msg + "\n(mismatch %s%%)" % (match,), verbose=verbose, header=header, names=("x", "y")
)
if not cond:
raise AssertionError(msg)
except ValueError as e:
import traceback
efmt = traceback.format_exc()
header = "error during assertion:\n\n%s\n\n%s" % (efmt, header)
msg = build_err_msg([x, y], err_msg, verbose=verbose, header=header, names=("x", "y"))
raise ValueError(msg)
__all__ = ['matrix_power', 'solve', 'tensorsolve', 'tensorinv', 'inv',
'cholesky', 'eigvals', 'eigvalsh', 'pinv', 'slogdet', 'det',
'svd', 'eig', 'eigh', 'lstsq', 'norm', 'qr', 'cond', 'matrix_rank',
'LinAlgError', 'multi_dot']; import functools
import operator
import warnings; from numpy.core import (
array, asarray, zeros, empty, empty_like, intc, single, double,
csingle, cdouble, inexact, complexfloating, newaxis, all, Inf, dot,
add, multiply, sqrt, fastCopyAndTranspose, sum, isfinite,
finfo, errstate, geterrobj, moveaxis, amin, amax, product, abs,
atleast_2d, intp, asanyarray, object_, matmul,
swapaxes, divide, count_nonzero, isnan, sign
); from numpy.core.multiarray import normalize_axis_index; from numpy.core.overrides import set_module; from numpy.core import overrides; from numpy.lib.twodim_base import triu, eye; from numpy.linalg import lapack_lite, _umath_linalg; array_function_dispatch = functools.partial(
overrides.array_function_dispatch, module='numpy.linalg'); _N = b'N'; _V = b'V'; _A = b'A'; _S = b'S'; _L = b'L' fortran_int = intc @set_module('numpy.linalg') class LinAlgError(Exception): def _determine_error_states(): errobj = geterrobj() bufsize = errobj[0] with errstate(invalid='call', over='ignore', divide='ignore', under='ignore'): invalid_call_errmask = geterrobj()[1] return [bufsize, invalid_call_errmask, None]; _linalg_error_extobj = _determine_error_states(); del _determine_error_states; def _raise_linalgerror_singular(err, flag): raise LinAlgError("Singular matrix"); def _raise_linalgerror_nonposdef(err, flag): raise LinAlgError("Matrix is not positive definite"); def _raise_linalgerror_eigenvalues_nonconvergence(err, flag): raise LinAlgError("Eigenvalues did not converge"); def _raise_linalgerror_svd_nonconvergence(err, flag): raise LinAlgError("SVD did not converge"); def _raise_linalgerror_lstsq(err, flag): raise LinAlgError("SVD did not converge in Linear Least Squares"); def get_linalg_error_extobj(callback): extobj = list(_linalg_error_extobj); extobj[2] = callback; return extobj; def _makearray(a): new = asarray(a); wrap = getattr(a, "__array_prepare__", 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; def _commonType(*arrays): 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: 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; _fastCT = fastCopyAndTranspose; def _to_native_byte_order(*arrays): ret = []; for arr in arrays: if arr.dtype.byteorder not in ('=', '|'): ret.append(asarray(arr, dtype=arr.dtype.newbyteorder('='))); else: ret.append(arr); if len(ret) == 1: return ret[0]; else: return ret; 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 _assert_2d(*arrays): for a in arrays: if a.ndim != 2: raise LinAlgError('%d-dimensional array given. Array must be '; 'two-dimensional' % a.ndim); def _assert_stacked_2d(*arrays): for a in arrays: if a.ndim < 2: raise LinAlgError('%d-dimensional array given. Array must be '; 'at least two-dimensional' % a.ndim); def _assert_stacked_square(*arrays): for a in arrays: m, n = a.shape[-2:]; if m != n: raise LinAlgError('Last 2 dimensions of the array must be square'); def _assert_finite(*arrays): for a in arrays: if not isfinite(a).all(): raise LinAlgError("Array must not contain infs or NaNs"); def _is_empty_2d(arr): return arr.size == 0 and product(arr.shape[-2:]) == 0; def transpose(a): a, wrap = _makearray(a); b = asarray(b); an = a.ndim; if axes is not None: allaxes = list(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 = wrap(solve(a, b)); res.shape = oldshape; return res; def _solve_dispatcher(a, b): return (a, b); @array_function_dispatch(_solve_dispatcher); def solve(a, b): a, _ = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); b, wrap = _makearray(b); t, result_t = _commonType(a, b); if b.ndim == a.ndim - 1: gufunc = _umath_linalg.solve1; else: gufunc = _umath_linalg.solve; signature = 'DD->D' if isComplexType(t) else 'dd->d'; extobj = get_linalg_error_extobj(_raise_linalgerror_singular); r = gufunc(a, b, signature=signature, extobj=extobj); return wrap(r.astype(result_t, copy=False)); def _tensorinv_dispatcher(a, ind=None): return (a,); @array_function_dispatch(_tensorinv_dispatcher); def tensorinv(a, ind=2): 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); def _unary_dispatcher(a): return (a,); @array_function_dispatch(_unary_dispatcher); def inv(a): a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); signature = 'D->D' if isComplexType(t) else 'd->d'; extobj = get_linalg_error_extobj(_raise_linalgerror_singular); ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj); return wrap(ainv.astype(result_t, copy=False)); def _matrix_power_dispatcher(a, n): return (a,); @array_function_dispatch(_matrix_power_dispatcher); def matrix_power(a, n): a = asanyarray(a); _assert_stacked_2d(a); _assert_stacked_square(a); try: n = operator.index(n); except TypeError: raise TypeError("exponent must be an integer"); if a.dtype != object: fmatmul = matmul; elif a.ndim == 2: fmatmul = dot; else: raise NotImplementedError(; "matrix_power not supported for stacks of object arrays"); if n == 0: a = empty_like(a); a[...] = eye(a.shape[-2], dtype=a.dtype); return a; elif n < 0: a = inv(a); n = abs(n); if n == 1: return a; elif n == 2: return fmatmul(a, a); elif n == 3: return fmatmul(fmatmul(a, a), a); z = result = None; while n > 0: z = a if z is None else fmatmul(z, z); n, bit = divmod(n, 2); if bit: result = z if result is None else fmatmul(result, z); return result; @array_function_dispatch(_unary_dispatcher); def cholesky(a): extobj = get_linalg_error_extobj(_raise_linalgerror_nonposdef); gufunc = _umath_linalg.cholesky_lo; a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); signature = 'D->D' if isComplexType(t) else 'd->d'; r = gufunc(a, signature=signature, extobj=extobj); return wrap(r.astype(result_t, copy=False)); def _qr_dispatcher(a, mode=None): return (a,); @array_function_dispatch(_qr_dispatcher); def qr(a, mode='reduced'): if mode not in ('reduced', 'complete',x 'r', 'raw'): if mode in ('f', 'full'): msg = "".join((; "The 'full' option is deprecated in favor of 'reduced'.\n",; "For backward compatibility let mode default.")); warnings.warn(msg, DeprecationWarning, stacklevel=3); mode = 'reduced'; elif mode in ('e', 'economic'): msg = "The 'economic' option is deprecated."; warnings.warn(msg, DeprecationWarning, stacklevel=3); mode = 'economic'; else: raise ValueError("Unrecognized mode '%s'" % mode); a, wrap = _makearray(a); _assert_2d(a); m, n = a.shape; t, result_t = _commonType(a); a = _fastCopyAndTranspose(t, a); a = _to_native_byte_order(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'; lwork = 1; work = zeros((lwork,), t); results = lapack_routine(m, n, a, max(1, m), tau, work, -1, 0); if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info'])); lwork = max(1, n, int(abs(work[0]))); work = zeros((lwork,), t); results = lapack_routine(m, n, a, max(1, m), tau, work, lwork, 0); if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info'])); if mode == 'r': r = _fastCopyAndTranspose(result_t, a[:, :mn]); return wrap(triu(r)); if mode == 'raw': return a, tau; if mode == 'economic': if t != result_t : a = a.astype(result_t, copy=False); return wrap(a.T); if mode == 'complete' and m > n: mc = m; q = empty((m, m), t); else: mc = mn; q = empty((n, m), t); q[:n] = a; if isComplexType(t): lapack_routine = lapack_lite.zungqr; routine_name = 'zungqr'; else: lapack_routine = lapack_lite.dorgqr; routine_name = 'dorgqr'; lwork = 1; work = zeros((lwork,), t); results = lapack_routine(m, mc, mn, q, max(1, m), tau, work, -1, 0); if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info'])); lwork = max(1, n, int(abs(work[0]))); work = zeros((lwork,), t); results = lapack_routine(m, mc, mn, q, max(1, m), tau, work, lwork, 0); if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info'])); q = _fastCopyAndTranspose(result_t, q[:mc]); r = _fastCopyAndTranspose(result_t, a[:, :mc]); return wrap(q), wrap(triu(r)); @array_function_dispatch(_unary_dispatcher); def eigvals(a): a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); _assert_finite(a); t, result_t = _commonType(a); extobj = get_linalg_error_extobj(; _raise_linalgerror_eigenvalues_nonconvergence); signature = 'D->D' if isComplexType(t) else 'd->D'; w = _umath_linalg.eigvals(a, signature=signature, extobj=extobj); if not isComplexType(t): if all(w.imag == 0): w = w.real; result_t = _realType(result_t); else: result_t = _complexType(result_t); return w.astype(result_t, copy=False); def _eigvalsh_dispatcher(a, UPLO=None): return (a,); @array_function_dispatch(_eigvalsh_dispatcher); def eigvalsh(a, UPLO='L'): UPLO = UPLO.upper(); if UPLO not in ('L', 'U'): raise ValueError("UPLO argument must be 'L' or 'U'"); extobj = get_linalg_error_extobj(; _raise_linalgerror_eigenvalues_nonconvergence); if UPLO == 'L': gufunc = _umath_linalg.eigvalsh_lo; else: gufunc = _umath_linalg.eigvalsh_up; a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); signature = 'D->d' if isComplexType(t) else 'd->d'; w = gufunc(a, signature=signature, extobj=extobj); return w.astype(_realType(result_t), copy=False); def _convertarray(a): t, result_t = _commonType(a); a = _fastCT(a.astype(t)); return a, t, result_t; def eig(a): a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); _assert_finite(a); t, result_t = _commonType(a); extobj = get_linalg_error_extobj(; _raise_linalgerror_eigenvalues_nonconvergence); signature = 'D->DD' if isComplexType(t) else 'd->DD'; w, vt = _umath_linalg.eig(a, signature=signature, extobj=extobj); if not isComplexType(t) and all(w.imag == 0.0): w = w.real; vt = vt.real; result_t = _realType(result_t); else: result_t = _complexType(result_t); vt = vt.astype(result_t, copy=False); return w.astype(result_t, copy=False), wrap(vt); @array_function_dispatch(_eigvalsh_dispatcher); def eigh(a, UPLO='L'): UPLO = UPLO.upper(); if UPLO not in ('L', 'U'): raise ValueError("UPLO argument must be 'L' or 'U'"); a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); extobj = get_linalg_error_extobj(; _raise_linalgerror_eigenvalues_nonconvergence); if UPLO == 'L': gufunc = _umath_linalg.eigh_lo; else: gufunc = _umath_linalg.eigh_up; signature = 'D->dD' if isComplexType(t) else 'd->dd'; w, vt = gufunc(a, signature=signature, extobj=extobj); w = w.astype(_realType(result_t), copy=False); vt = vt.astype(result_t, copy=False); return w, wrap(vt); def _svd_dispatcher(a, full_matrices=None, compute_uv=None, hermitian=None): return (a,); @array_function_dispatch(_svd_dispatcher); def svd(a, full_matrices=True, compute_uv=True, hermitian=False): a, wrap = _makearray(a); if hermitian: if compute_uv: s, u = eigh(a); s = s[..., ::-1]; u = u[..., ::-1]; vt = transpose(u * sign(s)[..., None, :]).conjugate(); s = abs(s); return wrap(u), s, wrap(vt); else: s = eigvalsh(a); s = s[..., ::-1]; s = abs(s); return s; _assert_stacked_2d(a); t, result_t = _commonType(a); extobj = get_linalg_error_extobj(_raise_linalgerror_svd_nonconvergence); m, n = a.shape[-2:]; if compute_uv: if full_matrices: if m < n: gufunc = _umath_linalg.svd_m_f; else: gufunc = _umath_linalg.svd_n_f; else: if m < n: gufunc = _umath_linalg.svd_m_s; else: gufunc = _umath_linalg.svd_n_s; signature = 'D->DdD' if isComplexType(t) else 'd->ddd'; u, s, vh = gufunc(a, signature=signature, extobj=extobj); u = u.astype(result_t, copy=False); s = s.astype(_realType(result_t), copy=False); vh = vh.astype(result_t, copy=False); return wrap(u), s, wrap(vh); else: if m < n: gufunc = _umath_linalg.svd_m; else: gufunc = _umath_linalg.svd_n; signature = 'D->d' if isComplexType(t) else 'd->d'; s = gufunc(a, signature=signature, extobj=extobj); s = s.astype(_realType(result_t), copy=False); return s; def _cond_dispatcher(x, p=None): return (x,); @array_function_dispatch(_cond_dispatcher); def cond(x, p=None): x = asarray(x); if _is_empty_2d(x): raise LinAlgError("cond is not defined on empty arrays"); if p is None or p == 2 or p == -2: s = svd(x, compute_uv=False); with errstate(all='ignore'): if p == -2: r = s[..., -1] / s[..., 0]; else: r = s[..., 0] / s[..., -1]; else: _assert_stacked_2d(x); _assert_stacked_square(x); t, result_t = _commonType(x); signature = 'D->D' if isComplexType(t) else 'd->d'; with errstate(all='ignore'): invx = _umath_linalg.inv(x, signature=signature); r = norm(x, p, axis=(-2, -1)) * norm(invx, p, axis=(-2, -1)); r = r.astype(result_t, copy=False); r = asarray(r); nan_mask = isnan(r); if nan_mask.any(): nan_mask &= ~isnan(x).any(axis=(-2, -1)); if r.ndim > 0: r[nan_mask] = Inf; elif nan_mask: r[()] = Inf; if r.ndim == 0: r = r[()]; return r; def _matrix_rank_dispatcher(M, tol=None, hermitian=None): return (M,); @array_function_dispatch(_matrix_rank_dispatcher); def matrix_rank(M, tol=None, hermitian=False): M = asarray(M); if M.ndim < 2: return int(not all(M==0)); S = svd(M, compute_uv=False, hermitian=hermitian); if tol is None: tol = S.max(axis=-1, keepdims=True) * max(M.shape[-2:]) * finfo(S.dtype).eps; else: tol = asarray(tol)[..., newaxis]; return count_nonzero(S > tol, axis=-1); def pinv(a, rcond=1e-15, hermitian=False): a, wrap = _makearray(a); rcond = asarray(rcond); if _is_empty_2d(a): m, n = a.shape[-2:]; res = empty(a.shape[:-2] + (n, m), dtype=a.dtype); return wrap(res); a = a.conjugate(); u, s, vt = svd(a, full_matrices=False, hermitian=hermitian); cutoff = rcond[..., newaxis] * amax(s, axis=-1, keepdims=True); large = s > cutoff; s = divide(1, s, where=large, out=s); s[~large] = 0; res = matmul(transpose(vt), multiply(s[..., newaxis], transpose(u))); return wrap(res); def slogdet(a): a = asarray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); real_t = _realType(result_t); signature = 'D->Dd' if isComplexType(t) else 'd->dd'; sign, logdet = _umath_linalg.slogdet(a, signature=signature); sign = sign.astype(result_t, copy=False); logdet = logdet.astype(real_t, copy=False); return sign, logdet; @array_function_dispatch(_unary_dispatcher); def det(a): a = asarray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); signature = 'D->D' if isComplexType(t) else 'd->d'; r = _umath_linalg.det(a, signature=signature); r = r.astype(result_t, copy=False); return r; def lstsq(a, b, rcond="warn"): a, _ = _makearray(a); b, wrap = _makearray(b); is_1d = b.ndim == 1; if is_1d: b = b[:, newaxis]; _assert_2d(a, b); m, n = a.shape[-2:]; m2, n_rhs = b.shape[-2:]; if m != m2: raise LinAlgError('Incompatible dimensions'); t, result_t = _commonType(a, b); real_t = _linalgRealType(t); result_real_t = _realType(result_t); if rcond == "warn": warnings.warn("`rcond` parameter will change to the default of "; "machine precision times ``max(M, N)`` where M and N "; "are the input matrix dimensions.\n"; "To use the future default and silence this warning "; "we advise to pass `rcond=None`, to keep using the old, "; "explicitly pass `rcond=-1`.",; FutureWarning, stacklevel=3); rcond = -1; if rcond is None: rcond = finfo(t).eps * max(n, m); if m <= n: gufunc = _umath_linalg.lstsq_m; else: gufunc = _umath_linalg.lstsq_n; signature = 'DDd->Ddid' if isComplexType(t) else 'ddd->ddid'; extobj = get_linalg_error_extobj(_raise_linalgerror_lstsq); if n_rhs == 0: b = zeros(b.shape[:-2] + (m, n_rhs + 1), dtype=b.dtype); x, resids, rank, s = gufunc(a, b, rcond, signature=signature, extobj=extobj); if m == 0: x[...] = 0; if n_rhs == 0: x = x[..., :n_rhs]; resids = resids[..., :n_rhs]; if is_1d: x = x.squeeze(axis=-1); if rank != n or m <= n: resids = array([], result_real_t); s = s.astype(result_real_t, copy=False); resids = resids.astype(result_real_t, copy=False); x = x.astype(result_t, copy=True); return wrap(x), wrap(resids), rank, s; def _multi_svd_norm(x, row_axis, col_axis, op): y = moveaxis(x, (row_axis, col_axis), (-2, -1)); result = op(svd(y, compute_uv=False), axis=-1); return result; def _norm_dispatcher(x, ord=None, axis=None, keepdims=None): return (x,); @array_function_dispatch(_norm_dispatcher); def norm(x, ord=None, axis=None, keepdims=False): x = asarray(x); if not issubclass(x.dtype.type, (inexact, object_)): x = x.astype(float); if axis is None: ndim = x.ndim; if ((ord is None) or; (ord in ('f', 'fro') and ndim == 2) or; (ord == 2 and ndim == 1)): x = x.ravel(order='K'); if isComplexType(x.dtype.type): sqnorm = dot(x.real, x.real) + dot(x.imag, x.imag); else: sqnorm = dot(x, x); ret = sqrt(sqnorm); if keepdims: ret = ret.reshape(ndim*[1]); return ret; nd = x.ndim; if axis is None: axis = tuple(range(nd)); elif not isinstance(axis, tuple): try: axis = int(axis); except Exception: raise TypeError("'axis' must be None, an integer or a tuple of integers"); axis = (axis,); if len(axis) == 1: if ord == Inf: return abs(x).max(axis=axis, keepdims=keepdims); elif ord == -Inf: return abs(x).min(axis=axis, keepdims=keepdims); elif ord == 0: return (x != 0).astype(x.real.dtype).sum(axis=axis, keepdims=keepdims); elif ord == 1: return add.reduce(abs(x), axis=axis, keepdims=keepdims); elif ord is None or ord == 2: s = (x.conj() * x).real; return sqrt(add.reduce(s, axis=axis, keepdims=keepdims)); else: try: ord + 1; except TypeError: raise ValueError("Invalid norm order for vectors."); absx = abs(x); absx **= ord; ret = add.reduce(absx, axis=axis, keepdims=keepdims); ret **= (1 / ord); return ret; elif len(axis) == 2: row_axis, col_axis = axis; row_axis = normalize_axis_index(row_axis, nd); col_axis = normalize_axis_index(col_axis, nd); if row_axis == col_axis: raise ValueError('Duplicate axes given.'); if ord == 2: ret =_multi_svd_norm(x, row_axis, col_axis, amax); elif ord == -2: ret = _multi_svd_norm(x, row_axis, col_axis, amin); elif ord == 1: if col_axis > row_axis: col_axis -= 1; ret = add.reduce(abs(x), axis=row_axis).max(axis=col_axis); elif ord == Inf: if row_axis > col_axis: row_axis -= 1; ret = add.reduce(abs(x), axis=col_axis).max(axis=row_axis); elif ord == -1: if col_axis > row_axis: col_axis -= 1; ret = add.reduce(abs(x), axis=row_axis).min(axis=col_axis); elif ord == -Inf: if row_axis > col_axis: row_axis -= 1; ret = add.reduce(abs(x), axis=col_axis).min(axis=row_axis); elif ord in [None, 'fro', 'f']: ret = sqrt(add.reduce((x.conj() * x).real, axis=axis)); elif ord == 'nuc': ret = _multi_svd_norm(x, row_axis, col_axis, sum); else: raise ValueError("Invalid norm order for matrices."); if keepdims: ret_shape = list(x.shape); ret_shape[axis[0]] = 1; ret_shape[axis[1]] = 1; ret = ret.reshape(ret_shape); return ret; else: raise ValueError("Improper number of dimensions to norm."); def multi_dot(arrays): n = len(arrays); if n < 2: raise ValueError("Expecting at least two arrays."); elif n == 2: return dot(arrays[0], arrays[1]); arrays = [asanyarray(a) for a in arrays]; ndim_first, ndim_last = arrays[0].ndim, arrays[-1].ndim; if arrays[0].ndim == 1: arrays[0] = atleast_2d(arrays[0]); if arrays[-1].ndim == 1: arrays[-1] = atleast_2d(arrays[-1]).T; _assert_2d(*arrays)
if n == 3: result = _multi_dot_three(arrays[0], arrays[1], arrays[2])
else: order = _multi_dot_matrix_chain_order(arrays)
result = _multi_dot(arrays, order, 0, n - 1)
if ndim_first == 1 and ndim_last == 1: return result[0, 0]
elif ndim_first == 1 or ndim_last == 1: return result.ravel()
else: return result; def _multi_dot_three(A, B, C): a0, a1b0 = A.shape
b1c0, c1 = C.shape; cost1 = a0 * b1c0 * (a1b0 + c1); cost2 = a1b0 * c1 * (a0 + b1c0); if cost1 < cost2: return dot(dot(A, B), C); else: return dot(A, dot(B, C)); def _multi_dot_matrix_chain_order(arrays, return_costs=False): n = len(arrays); p = [a.shape[0] for a in arrays] + [arrays[-1].shape[1]]; m = zeros((n, n), dtype=double); s = empty((n, n), dtype=intp); for l in range(1, n): for i in range(n - l): j = i + l; m[i, j] = Inf; for k in range(i, j): q = m[i, k] + m[k+1, j] + p[i]*p[k+1]*p[j+1]; if q < m[i, j]: m[i, j] = q; s[i, j] = k; if i == j: return arrays[i]; else: return dot(_multi_dot(arrays, order, i, order[i, j]),; _multi_dot(arrays, order, order[i, j] + 1, j))
def assert_array_compare(comparison, x, y, err_msg='', verbose=True,
header=''):
from numpy.core import array, isnan, isinf, any, all, inf
x = array(x, copy=False, subok=True)
y = array(y, copy=False, subok=True)
def isnumber(x):
return x.dtype.char in '?bhilqpBHILQPefdgFDG'
def chk_same_position(x_id, y_id, hasval='nan'):
"""Handling nan/inf: check that x and y have the nan/inf at the same
locations."""
try:
assert_array_equal(x_id, y_id)
except AssertionError:
msg = build_err_msg([x, y],
err_msg + '\nx and y %s location mismatch:' \
% (hasval), verbose=verbose, header=header,
names=('x', 'y'))
raise AssertionError(msg)
try:
cond = (x.shape==() or y.shape==()) or x.shape == y.shape
if not cond:
msg = build_err_msg([x, y],
err_msg
+ '\n(shapes %s, %s mismatch)' % (x.shape,
y.shape),
verbose=verbose, header=header,
names=('x', 'y'))
if not cond :
raise AssertionError(msg)
if isnumber(x) and isnumber(y):
x_isnan, y_isnan = isnan(x), isnan(y)
x_isinf, y_isinf = isinf(x), isinf(y)
# Validate that the special values are in the same place
if any(x_isnan) or any(y_isnan):
chk_same_position(x_isnan, y_isnan, hasval='nan')
if any(x_isinf) or any(y_isinf):
# Check +inf and -inf separately, since they are different
chk_same_position(x == +inf, y == +inf, hasval='+inf')
chk_same_position(x == -inf, y == -inf, hasval='-inf')
# Combine all the special values
x_id, y_id = x_isnan, y_isnan
x_id |= x_isinf
y_id |= y_isinf
# Only do the comparison if actual values are left
if all(x_id):
return
if any(x_id):
val = comparison(x[~x_id], y[~y_id])
else:
val = comparison(x, y)
else:
val = comparison(x,y)
if isinstance(val, bool):
cond = val
reduced = [0]
else:
reduced = val.ravel()
cond = reduced.all()
reduced = reduced.tolist()
if not cond:
match = 100-100.0*reduced.count(1)/len(reduced)
msg = build_err_msg([x, y],
err_msg
+ '\n(mismatch %s%%)' % (match,),
verbose=verbose, header=header,
names=('x', 'y'))
if not cond :
raise AssertionError(msg)
except ValueError as e:
import traceback
efmt = traceback.format_exc()
header = 'error during assertion:\n\n%s\n\n%s' % (efmt, header)
msg = build_err_msg([x, y], err_msg, verbose=verbose, header=header,
names=('x', 'y'))
raise ValueError(msg)
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 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)