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import nlcpy
import numpy
from nlcpy import veo
from nlcpy.request import request
from . import util
def _geev(a, jobvr):
a = nlcpy.asarray(a)
util._assertRankAtLeast2(a)
util._assertNdSquareness(a)
# used to match the contiguous of result to numpy.
c_order = a.flags.c_contiguous or sum([i > 1 for i in a.shape[:-2]]) < 2
a_complex = a.dtype.char in 'FD'
if a.dtype.char == 'F':
dtype = 'D'
f_dtype = 'f'
c_dtype = 'F'
elif a.dtype.char == 'D':
dtype = 'D'
f_dtype = 'd'
c_dtype = 'D'
else:
dtype = 'd'
if a.dtype.char == 'f':
f_dtype = 'f'
c_dtype = 'F'
else:
f_dtype = 'd'
c_dtype = 'D'
if a.size == 0:
dtype = c_dtype if a_complex else f_dtype
w = nlcpy.empty(shape=a.shape[:-1], dtype=dtype)
if jobvr:
vr = nlcpy.empty(shape=a.shape, dtype=dtype)
return w, vr
else:
return w
a = nlcpy.array(nlcpy.moveaxis(a, (-1, -2), (1, 0)), dtype=dtype, order='F')
wr = nlcpy.empty(a.shape[1:], dtype=dtype, order='F')
wi = nlcpy.empty(a.shape[1:], dtype=dtype, order='F')
vr = nlcpy.empty(a.shape if jobvr else 1, dtype=dtype, order='F')
vc = nlcpy.empty(a.shape if jobvr else 1, dtype='D', order='F')
n = a.shape[0]
work = nlcpy.empty(
65 * n if a_complex else 66 * n, dtype=dtype, order='F')
rwork = nlcpy.empty(2 * n if a_complex else 1, dtype='d', order='F')
info = numpy.empty(1, dtype='l')
fpe = request._get_fpe_flag()
args = (
a,
wr,
wi,
vr,
vc,
work,
rwork,
ord('V') if jobvr else ord('N'),
veo.OnStack(info, inout=veo.INTENT_OUT),
veo.OnStack(fpe, inout=veo.INTENT_OUT),
)
request._push_and_flush_request(
'nlcpy_eig',
args,
)
if a_complex:
w_complex = True
w = wr
vc = vr
else:
w_complex = nlcpy.any(wi)
w = wr + wi * 1.0j
if w_complex:
if c_order:
w = nlcpy.asarray(nlcpy.moveaxis(w, 0, -1), dtype=c_dtype, order='C')
else:
w = nlcpy.moveaxis(nlcpy.asarray(w, dtype=c_dtype), 0, -1)
else:
wr = w.real
w = nlcpy.moveaxis(nlcpy.asarray(wr, dtype=f_dtype), 0, -1)
if jobvr:
if w_complex:
if c_order:
vr = nlcpy.asarray(
nlcpy.moveaxis(vc, (1, 0), (-1, -2)), dtype=c_dtype, order='C')
else:
vr = nlcpy.moveaxis(
nlcpy.asarray(vc, dtype=c_dtype), (1, 0), (-1, -2))
else:
if c_dtype == "F":
vr = nlcpy.asarray(vc.real, dtype=f_dtype, order='C')
else:
vc = nlcpy.moveaxis(
nlcpy.asarray(vc, dtype=c_dtype), (1, 0), (-1, -2))
vr = vc.real
if jobvr:
return w, vr
else:
return w
def _syevd(a, jobz, UPLO):
a = nlcpy.asarray(a)
util._assertRankAtLeast2(a)
util._assertNdSquareness(a)
UPLO = UPLO.upper()
if UPLO not in 'UL':
raise ValueError("UPLO argument must be 'L' or 'U'")
# used to match the contiguous of result to numpy.
c_order = a.flags.c_contiguous or sum([i > 1 for i in a.shape[:-2]]) < 2
a_complex = a.dtype.char in 'FD'
if a.dtype.char == 'F':
dtype = 'F'
f_dtype = 'f'
elif a.dtype.char == 'D':
dtype = 'D'
f_dtype = 'd'
else:
if a.dtype.char == 'f':
dtype = 'f'
f_dtype = 'f'
else:
dtype = 'd'
f_dtype = 'd'
if a.size == 0:
w = nlcpy.empty(shape=a.shape[:-1], dtype=f_dtype)
if jobz:
vr = nlcpy.empty(shape=a.shape, dtype=dtype)
return w, vr
else:
return w
a = nlcpy.array(nlcpy.moveaxis(a, (-1, -2), (1, 0)), dtype=dtype, order='F')
w = nlcpy.empty(a.shape[1:], dtype=f_dtype, order='F')
n = a.shape[0]
if a.size > 1:
if a_complex:
lwork = max(2 * n + n * n, n + 48)
lrwork = 1 + 5 * n + 2 * n * n if jobz else n
else:
lwork = max(2 * n + 32, 1 + 6 * n + 2 * n * n) if jobz else 2 * n + 32
lrwork = 1
liwork = 3 + 5 * n if jobz else 1
else:
lwork = 1
lrwork = 1
liwork = 1
work = nlcpy.empty(lwork, dtype=dtype)
rwork = nlcpy.empty(lrwork, dtype=f_dtype)
iwork = nlcpy.empty(liwork, dtype='l')
info = numpy.empty(1, dtype='l')
fpe = request._get_fpe_flag()
args = (
a,
w,
work,
rwork,
iwork,
ord('V') if jobz else ord('N'),
ord(UPLO),
veo.OnStack(info, inout=veo.INTENT_OUT),
veo.OnStack(fpe, inout=veo.INTENT_OUT),
)
request._push_and_flush_request(
'nlcpy_eigh',
args,
)
if c_order:
w = nlcpy.asarray(nlcpy.moveaxis(w, 0, -1), order='C')
else:
w = nlcpy.moveaxis(w, 0, -1)
if jobz:
if c_order:
a = nlcpy.asarray(nlcpy.moveaxis(a, (1, 0), (-1, -2)), order='C')
else:
a = nlcpy.moveaxis(a, (1, 0), (-1, -2))
return w, a
else:
return w
[ドキュメント]def eig(a):
"""Computes the eigenvalues and right eigenvectors of a square array.
Parameters
----------
a : (..., M, M) array_like
Matrices for which the eigenvalues and right eigenvectors will be computed.
Returns
-------
w : (..., M) ndarray
The eigenvalues, each repeated according to its multiplicity. The eigenvalues are
not necessarily ordered. The resulting array will be of complex type, unless the
imaginary part is zero in which case it will be cast to a real type. Please note
that there are cases where rounding errors affect the dtype of the array. When
*a* is real the resulting eigenvalues will be real (0 imaginary part) or occur in
conjugate pairs.
v : (..., M, M) ndarray
The normalized (unit "length") eigenvectors, such that the column ``v[:,i]`` is
the eigenvector corresponding to the eigenvalue ``w[i]``.
Note
----
This is implemented using the ``_geev`` LAPACK routines which compute the eigenvalues
and eigenvectors of general square arrays.
The number `w` is an eigenvalue of `a` if there exists a vector `v` such that
``a @ v = w * v``.
Thus, the arrays `a`, `w`, and `v` satisfy the equations
``a @ v[:,i] = w[i] * v[:,i]`` for `i` in {0, 1, ..., M-1}.
The array `v` of eigenvectors may not be of maximum rank, that is, some of the
columns may be linearly dependent, although round-off error may obscure that fact. If
the eigenvalues are all different, then theoretically the eigenvectors are linearly
independent. Likewise, the (complex-valued) matrix of eigenvectors `v` is unitary if
the matrix `a` is normal, i.e., if ``dot(a, a.H) = dot(a.H, a)``, where `a.H` denotes
the conjugate transpose of `a.`
Finally, it is emphasized that `v` consists of the `right` (as in right-hand side)
eigenvectors of `a.` A vector `y` satisfying ``y.T @ a = z * y.T`` for some number
`z` is called a `left` eigenvector of `a,` and, in general, the left and right
eigenvectors of a matrix are not necessarily the (perhaps conjugate) transposes of
each other.
See Also
--------
eigvals : Computes the eigenvalues of a general matrix.
eigh : Computes the eigenvalues and eigenvectors of a complex Hermitian or a real
symmetric matrix.
eigvalsh : Computes the eigenvalues of a complex Hermitian or real symmetric matrix.
Examples
--------
(Almost) trivial example with real e-values and e-vectors.
>>> import nlcpy as vp
>>> w, v = vp.linalg.eig(vp.diag((1, 2, 3)))
>>> w; v
array([1., 2., 3.])
array([[1., 0., 0.],
[0., 1., 0.],
[0., 0., 1.]])
Real matrix possessing complex e-values and e-vectors; note that the e-values are
complex conjugates of each other.
>>> w, v = vp.linalg.eig(vp.array([[1, -1], [1, 1]]))
>>> w; v # doctest: +SKIP
array([1.+1.j, 1.-1.j])
array([[0.70710678+0.j , 0.70710678-0.j ],
[0. -0.70710678j, 0. +0.70710678j]])
Complex-valued matrix with real e-values (but complex-valued e-vectors); note that
``a.conj().T == a``, i.e., *a* is Hermitian.
>>> a = vp.array([[1, 1j], [-1j, 1]])
>>> w, v = vp.linalg.eig(a)
>>> w; v # doctest: +SKIP
array([2.+0.j, 0.+0.j])
array([[ 0. +0.70710678j, 0.70710678+0.j ], # may vary
[ 0.70710678+0.j , -0. +0.70710678j]])
Be careful about round-off error!
>>> a = vp.array([[1 + 1e-9, 0], [0, 1 - 1e-9]])
>>> # Theor. e-values are 1 +/- 1e-9
>>> w, v = vp.linalg.eig(a)
>>> w; v # doctest: +SKIP
array([1., 1.])
array([[1., 0.],
[0., 1.]])
"""
return _geev(a, True)
[ドキュメント]def eigvals(a):
"""Computes the eigenvalues of a general matrix.
Main difference from :func:`eig` : the eigenvectors aren't returned.
Parameters
----------
a : (..., M, M) array_like
A complex- or real-valued matrix whose eigenvalues will be computed.
Returns
-------
w : (..., M) ndarray
The eigenvalues, each repeated according to its multiplicity. They are not
necessarily ordered, nor are they necessarily real for real matrices.
Note
----
This is implemented using the ``_geev`` LAPACK routines which compute the eigenvalues
and eigenvectors of general square arrays.
See Also
--------
eig : Computes the eigenvalues and right eigenvectors of a square array.
eigh : Computes the eigenvalues and eigenvectors of a complex Hermitian or a real
symmetric matrix.
eigvalsh : Computes the eigenvalues of a complex Hermitian or real symmetric matrix.
Examples
--------
Illustration, using the fact that the eigenvalues of a diagonal matrix are its
diagonal elements, that multiplying a matrix on the left by an orthogonal matrix,
*Q*, and on the right by *Q.T* (the transpose of *Q*), preserves the eigenvalues of
the "middle" matrix. In other words, if *Q* is orthogonal, then ``Q * A * Q.T`` has
the same eigenvalues as ``A``:
>>> import nlcpy as vp
>>> x = vp.random.random()
>>> Q = vp.array([[vp.cos(x), -vp.sin(x)], [vp.sin(x), vp.cos(x)]])
>>> vp.linalg.norm(Q[0, :]), vp.linalg.norm(Q[1, :]), vp.dot(Q[0, :],Q[1, :])
(array(1.), array(1.), array(0.))
Now multiply a diagonal matrix by ``Q`` on one side and by ``Q.T`` on the other:
>>> D = vp.diag((-1,1))
>>> vp.linalg.eigvals(D)
array([-1., 1.])
>>> A = vp.dot(Q, D)
>>> A = vp.dot(A, Q.T)
>>> vp.linalg.eigvals(A) # doctest: +SKIP
array([ 1., -1.]) # random
"""
return _geev(a, False)
[ドキュメント]def eigh(a, UPLO='L'):
"""Computes the eigenvalues and eigenvectors of a complex Hermitian or a real
symmetric matrix.
Returns two objects, a 1-D array containing the eigenvalues of *a*, and a 2-D square
array or matrix (depending on the input type) of the corresponding eigenvectors (in
columns).
Parameters
----------
a : (..., M, M) array_like
Hermitian or real symmetric matrices whose eigenvalues and eigenvectors are to be
computed.
UPLO : {'L', 'U'}, optional
Specifies whether the calculation is done with the lower triangular part of *a*
('L', default) or the upper triangular part ('U'). Irrespective of this value
only the real parts of the diagonal will be considered in the computation to
preserve the notion of a Hermitian matrix. It therefore follows that the
imaginary part of the diagonal will always be treated as zero.
Returns
-------
w : (..., M) ndarray
The eigenvalues in ascending order, each repeated according to its multiplicity.
v : (..., M, M) ndarray
The column ``v[:, i]`` is the normalized eigenvector corresponding to the
eigenvalue ``w[i]``.
Note
----
The eigenvalues/eigenvectors are computed using LAPACK routines ``_syevd``,
``_heevd``.
The eigenvalues of real symmetric or complex Hermitian matrices are always real. The
array `v` of (column) eigenvectors is unitary and `a,` `w,` and `v` satisfy the
equations ``dot(a, v[:, i]) = w[i] * v[:, i]``.
See Also
--------
eigvalsh : Computes the eigenvalues of a complex Hermitian or real symmetric matrix.
eig : Computes the eigenvalues and right eigenvectors of a square array.
eigvals : Computes the eigenvalues of a general matrix.
Examples
--------
>>> import nlcpy as vp
>>> a = vp.array([[1, -2j], [2j, 5]])
>>> a
array([[ 1.+0.j, -0.-2.j],
[ 0.+2.j, 5.+0.j]])
>>> w, v = vp.linalg.eigh(a)
>>> w; v # doctest: +SKIP
array([0.17157288, 5.82842712])
array([[-0.92387953+0.j , -0.38268343+0.j ], # may vary
[ 0. +0.38268343j, 0. -0.92387953j]])
>>> # demonstrate the treatment of the imaginary part of the diagonal
>>> a = vp.array([[5+2j, 9-2j], [0+2j, 2-1j]])
>>> a
array([[5.+2.j, 9.-2.j],
[0.+2.j, 2.-1.j]])
>>> # with UPLO='L' this is numerically equivalent to using LA.eig() with:
>>> b = vp.array([[5.+0.j, 0.-2.j], [0.+2.j, 2.-0.j]])
>>> b
array([[5.+0.j, 0.-2.j],
[0.+2.j, 2.+0.j]])
>>> wa, va = vp.linalg.eigh(a)
>>> wb, vb = vp.linalg.eig(b)
>>> wa; wb # doctest: +SKIP
array([1., 6.])
array([6.+0.j, 1.+0.j])
>>> va; vb # doctest: +SKIP
array([[-0.4472136 -0.j , -0.89442719+0.j ],
[ 0. +0.89442719j, 0. -0.4472136j ]])
array([[ 0.89442719+0.j , 0. +0.4472136j],
[-0. +0.4472136j, 0.89442719+0.j ]])
"""
return _syevd(a, True, UPLO)
[ドキュメント]def eigvalsh(a, UPLO='L'):
"""Computes the eigenvalues of a complex Hermitian or real symmetric matrix.
Main difference from :func:`eigh` : the eigenvectors are not computed.
Parameters
----------
a : (..., M, M) array_like
A complex- or real-valued matrix whose eigenvalues are to be computed.
UPLO : {'L', 'U'}, optional
Specifies whether the calculation is done with the lower triangular part of a
('L', default) or the upper triangular part ('U'). Irrespective of this value
only the real parts of the diagonal will be considered in the computation to
preserve the notion of a Hermitian matrix. It therefore follows that the
imaginary part of the diagonal will always be treated as zero.
Returns
-------
w : (..., M) ndarray
The eigenvalues in ascending order, each repeated according to its multiplicity.
Note
----
The eigenvalues are computed using LAPACK routines ``_syevd``, ``_heevd``.
See Also
--------
eig : Computes the eigenvalues and right eigenvectors of a square array.
eigvals : Computes the eigenvalues of a general matrix.
eigh : Computes the eigenvalues and eigenvectors of a complex Hermitian or a real
symmetric matrix.
Examples
--------
>>> import nlcpy as vp
>>> a = vp.array([[1, -2j], [2j, 5]])
>>> vp.linalg.eigvalsh(a) # doctest: +SKIP
array([0.17157288, 5.82842712]) # may vary
>>> # demonstrate the treatment of the imaginary part of the diagonal
>>> a = vp.array([[5+2j, 9-2j], [0+2j, 2-1j]])
>>> a # doctest: +SKIP
array([[5.+2.j, 9.-2.j],
[0.+2.j, 2.-1.j]])
>>> # with UPLO='L' this is numerically equivalent to using vp.linalg.eigvals()
>>> # with:
>>> b = vp.array([[5.+0.j, 0.-2.j], [0.+2.j, 2.-0.j]])
>>> b # doctest: +SKIP
array([[5.+0.j, 0.-2.j],
[0.+2.j, 2.+0.j]])
>>> wa = vp.linalg.eigvalsh(a)
>>> wb = vp.linalg.eigvals(b)
>>> wa; wb # doctest: +SKIP
array([1., 6.])
array([6.+0.j, 1.+0.j])
"""
return _syevd(a, False, UPLO)