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� Zmd{dfd�Zndgdh� Zod|djd�Zpdkd� Zqdldm� Zrd}dndo�Zsdpdq� ZtdS )~ax Lite version of scipy.linalg.
Notes
-----
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,
zgetrf, dpotrf, zpotrf, dgeqrf, zgeqrf, zungqr, dorgqr.
� )�division�absolute_import�print_function� matrix_power�solve� tensorsolve� tensorinv�inv�cholesky�eigvals�eigvalsh�pinv�slogdet�det�svd�eig�eigh�lstsq�norm�qr�cond� matrix_rank� LinAlgError� multi_dotN)*�array�asarray�zeros�empty�
empty_like� transpose�intc�single�double�csingle�cdouble�inexact�complexfloating�newaxis�ravel�all�Inf�dot�add�multiply�sqrt�maximum�fastCopyAndTranspose�sum�isfinite�size�finfo�errstate� geterrobj�
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_umath_linalg)r � N� V� A� S� Lc @ s e Zd ZdZdS )r a
Generic Python-exception-derived object raised by linalg functions.
General purpose exception class, derived from Python's exception.Exception
class, programmatically raised in linalg functions when a Linear
Algebra-related condition would prevent further correct execution of the
function.
Parameters
----------
None
Examples
--------
>>> from numpy import linalg as LA
>>> LA.inv(np.zeros((2,2)))
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "...linalg.py", line 350,
in inv return wrap(solve(a, identity(a.shape[0], dtype=a.dtype)))
File "...linalg.py", line 249,
in solve
raise LinAlgError('Singular matrix')
numpy.linalg.LinAlgError: Singular matrix
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isComplexTypeo s rk c C s t �| |�S )N)�_real_types_map�get)rj �defaultrR rR rS � _realType| s ro c C s t �| |�S )N)�_complex_types_maprm )rj rn rR rR rS � _complexType s rq c C s t S )z,Cast the type t to either double or cdouble.)r" )rj rR rR rS �_linalgRealType� s rr c G s� t }d}xf| D ]^}t|jjt�r\t|jj�r0d}t|jjd d�}|d kr`td|jjf ��nt }|t krt }qW |r�t
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r� c G s. x(| D ] }t |j�t|j�krtd��qW d S )NzArray must be square)�max�shape�minr )rw re rR rR rS �_assertSquareness� s
r� c G s> x8| D ]0}t |jdd � �t|jdd � �krtd��qW d S )N���z-Last 2 dimensions of the array must be square)r� r� r� r )rw re rR rR rS �_assertNdSquareness� s
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C s� t | �\} }t|�}| j}|dk r^ttd|��}x"|D ]}|�|� |�||� q6W | �|�} | j||j d� }d}x|D ] }||9 }q~W | � d|�} |�
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Solve the tensor equation ``a x = b`` for x.
It is assumed that all indices of `x` are summed over in the product,
together with the rightmost indices of `a`, as is done in, for example,
``tensordot(a, x, axes=b.ndim)``.
Parameters
----------
a : array_like
Coefficient tensor, of shape ``b.shape + Q``. `Q`, a tuple, equals
the shape of that sub-tensor of `a` consisting of the appropriate
number of its rightmost indices, and must be such that
``prod(Q) == prod(b.shape)`` (in which sense `a` is said to be
'square').
b : array_like
Right-hand tensor, which can be of any shape.
axes : tuple of ints, optional
Axes in `a` to reorder to the right, before inversion.
If None (default), no reordering is done.
Returns
-------
x : ndarray, shape Q
Raises
------
LinAlgError
If `a` is singular or not 'square' (in the above sense).
See Also
--------
numpy.tensordot, tensorinv, numpy.einsum
Examples
--------
>>> a = np.eye(2*3*4)
>>> a.shape = (2*3, 4, 2, 3, 4)
>>> b = np.random.randn(2*3, 4)
>>> x = np.linalg.tensorsolve(a, b)
>>> x.shape
(2, 3, 4)
>>> np.allclose(np.tensordot(a, x, axes=3), b)
True
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Solve a linear matrix equation, or system of linear scalar equations.
Computes the "exact" solution, `x`, of the well-determined, i.e., full
rank, linear matrix equation `ax = b`.
Parameters
----------
a : (..., M, M) array_like
Coefficient matrix.
b : {(..., M,), (..., M, K)}, array_like
Ordinate or "dependent variable" values.
Returns
-------
x : {(..., M,), (..., M, K)} ndarray
Solution to the system a x = b. Returned shape is identical to `b`.
Raises
------
LinAlgError
If `a` is singular or not square.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
The solutions are computed using LAPACK routine _gesv
`a` must be square and of full-rank, i.e., all rows (or, equivalently,
columns) must be linearly independent; if either is not true, use
`lstsq` for the least-squares best "solution" of the
system/equation.
References
----------
.. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
FL, Academic Press, Inc., 1980, pg. 22.
Examples
--------
Solve the system of equations ``3 * x0 + x1 = 9`` and ``x0 + 2 * x1 = 8``:
>>> a = np.array([[3,1], [1,2]])
>>> b = np.array([9,8])
>>> x = np.linalg.solve(a, b)
>>> x
array([ 2., 3.])
Check that the solution is correct:
>>> np.allclose(np.dot(a, x), b)
True
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Compute the 'inverse' of an N-dimensional array.
The result is an inverse for `a` relative to the tensordot operation
``tensordot(a, b, ind)``, i. e., up to floating-point accuracy,
``tensordot(tensorinv(a), a, ind)`` is the "identity" tensor for the
tensordot operation.
Parameters
----------
a : array_like
Tensor to 'invert'. Its shape must be 'square', i. e.,
``prod(a.shape[:ind]) == prod(a.shape[ind:])``.
ind : int, optional
Number of first indices that are involved in the inverse sum.
Must be a positive integer, default is 2.
Returns
-------
b : ndarray
`a`'s tensordot inverse, shape ``a.shape[ind:] + a.shape[:ind]``.
Raises
------
LinAlgError
If `a` is singular or not 'square' (in the above sense).
See Also
--------
numpy.tensordot, tensorsolve
Examples
--------
>>> a = np.eye(4*6)
>>> a.shape = (4, 6, 8, 3)
>>> ainv = np.linalg.tensorinv(a, ind=2)
>>> ainv.shape
(8, 3, 4, 6)
>>> b = np.random.randn(4, 6)
>>> np.allclose(np.tensordot(ainv, b), np.linalg.tensorsolve(a, b))
True
>>> a = np.eye(4*6)
>>> a.shape = (24, 8, 3)
>>> ainv = np.linalg.tensorinv(a, ind=1)
>>> ainv.shape
(8, 3, 24)
>>> b = np.random.randn(24)
>>> np.allclose(np.tensordot(ainv, b, 1), np.linalg.tensorsolve(a, b))
True
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Compute the (multiplicative) inverse of a matrix.
Given a square matrix `a`, return the matrix `ainv` satisfying
``dot(a, ainv) = dot(ainv, a) = eye(a.shape[0])``.
Parameters
----------
a : (..., M, M) array_like
Matrix to be inverted.
Returns
-------
ainv : (..., M, M) ndarray or matrix
(Multiplicative) inverse of the matrix `a`.
Raises
------
LinAlgError
If `a` is not square or inversion fails.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
Examples
--------
>>> from numpy.linalg import inv
>>> a = np.array([[1., 2.], [3., 4.]])
>>> ainv = inv(a)
>>> np.allclose(np.dot(a, ainv), np.eye(2))
True
>>> np.allclose(np.dot(ainv, a), np.eye(2))
True
If a is a matrix object, then the return value is a matrix as well:
>>> ainv = inv(np.matrix(a))
>>> ainv
matrix([[-2. , 1. ],
[ 1.5, -0.5]])
Inverses of several matrices can be computed at once:
>>> a = np.array([[[1., 2.], [3., 4.]], [[1, 3], [3, 5]]])
>>> inv(a)
array([[[-2. , 1. ],
[ 1.5, -0.5]],
[[-5. , 2. ],
[ 3. , -1. ]]])
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rh r� r� rx rk rc r[ rH r r� )re rg rj r� r� rb ZainvrR rR rS r � s 9 c C sf t t�}tj}t| �\} }t| � t| � t| �\}}t|�rBdnd}|| ||d�}||j |dd��S )a�
Cholesky decomposition.
Return the Cholesky decomposition, `L * L.H`, of the square matrix `a`,
where `L` is lower-triangular and .H is the conjugate transpose operator
(which is the ordinary transpose if `a` is real-valued). `a` must be
Hermitian (symmetric if real-valued) and positive-definite. Only `L` is
actually returned.
Parameters
----------
a : (..., M, M) array_like
Hermitian (symmetric if all elements are real), positive-definite
input matrix.
Returns
-------
L : (..., M, M) array_like
Upper or lower-triangular Cholesky factor of `a`. Returns a
matrix object if `a` is a matrix object.
Raises
------
LinAlgError
If the decomposition fails, for example, if `a` is not
positive-definite.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
The Cholesky decomposition is often used as a fast way of solving
.. math:: A \mathbf{x} = \mathbf{b}
(when `A` is both Hermitian/symmetric and positive-definite).
First, we solve for :math:`\mathbf{y}` in
.. math:: L \mathbf{y} = \mathbf{b},
and then for :math:`\mathbf{x}` in
.. math:: L.H \mathbf{x} = \mathbf{y}.
Examples
--------
>>> A = np.array([[1,-2j],[2j,5]])
>>> A
array([[ 1.+0.j, 0.-2.j],
[ 0.+2.j, 5.+0.j]])
>>> L = np.linalg.cholesky(A)
>>> L
array([[ 1.+0.j, 0.+0.j],
[ 0.+2.j, 1.+0.j]])
>>> np.dot(L, L.T.conj()) # verify that L * L.H = A
array([[ 1.+0.j, 0.-2.j],
[ 0.+2.j, 5.+0.j]])
>>> A = [[1,-2j],[2j,5]] # what happens if A is only array_like?
>>> np.linalg.cholesky(A) # an ndarray object is returned
array([[ 1.+0.j, 0.+0.j],
[ 0.+2.j, 1.+0.j]])
>>> # But a matrix object is returned if A is a matrix object
>>> LA.cholesky(np.matrix(A))
matrix([[ 1.+0.j, 0.+0.j],
[ 0.+2.j, 1.+0.j]])
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rc r\ rH Z cholesky_lorh r� r� rx rk r� )re rb r� rg rj r� r� r� rR rR rS r
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Compute the qr factorization of a matrix.
Factor the matrix `a` as *qr*, where `q` is orthonormal and `r` is
upper-triangular.
Parameters
----------
a : array_like, shape (M, N)
Matrix to be factored.
mode : {'reduced', 'complete', 'r', 'raw', 'full', 'economic'}, optional
If K = min(M, N), then
'reduced' : returns q, r with dimensions (M, K), (K, N) (default)
'complete' : returns q, r with dimensions (M, M), (M, N)
'r' : returns r only with dimensions (K, N)
'raw' : returns h, tau with dimensions (N, M), (K,)
'full' : alias of 'reduced', deprecated
'economic' : returns h from 'raw', deprecated.
The options 'reduced', 'complete, and 'raw' are new in numpy 1.8,
see the notes for more information. The default is 'reduced' and to
maintain backward compatibility with earlier versions of numpy both
it and the old default 'full' can be omitted. Note that array h
returned in 'raw' mode is transposed for calling Fortran. The
'economic' mode is deprecated. The modes 'full' and 'economic' may
be passed using only the first letter for backwards compatibility,
but all others must be spelled out. See the Notes for more
explanation.
Returns
-------
q : ndarray of float or complex, optional
A matrix with orthonormal columns. When mode = 'complete' the
result is an orthogonal/unitary matrix depending on whether or not
a is real/complex. The determinant may be either +/- 1 in that
case.
r : ndarray of float or complex, optional
The upper-triangular matrix.
(h, tau) : ndarrays of np.double or np.cdouble, optional
The array h contains the Householder reflectors that generate q
along with r. The tau array contains scaling factors for the
reflectors. In the deprecated 'economic' mode only h is returned.
Raises
------
LinAlgError
If factoring fails.
Notes
-----
This is an interface to the LAPACK routines dgeqrf, zgeqrf,
dorgqr, and zungqr.
For more information on the qr factorization, see for example:
http://en.wikipedia.org/wiki/QR_factorization
Subclasses of `ndarray` are preserved except for the 'raw' mode. So if
`a` is of type `matrix`, all the return values will be matrices too.
New 'reduced', 'complete', and 'raw' options for mode were added in
NumPy 1.8.0 and the old option 'full' was made an alias of 'reduced'. In
addition the options 'full' and 'economic' were deprecated. Because
'full' was the previous default and 'reduced' is the new default,
backward compatibility can be maintained by letting `mode` default.
The 'raw' option was added so that LAPACK routines that can multiply
arrays by q using the Householder reflectors can be used. Note that in
this case the returned arrays are of type np.double or np.cdouble and
the h array is transposed to be FORTRAN compatible. No routines using
the 'raw' return are currently exposed by numpy, but some are available
in lapack_lite and just await the necessary work.
Examples
--------
>>> a = np.random.randn(9, 6)
>>> q, r = np.linalg.qr(a)
>>> np.allclose(a, np.dot(q, r)) # a does equal qr
True
>>> r2 = np.linalg.qr(a, mode='r')
>>> r3 = np.linalg.qr(a, mode='economic')
>>> np.allclose(r, r2) # mode='r' returns the same r as mode='full'
True
>>> # But only triu parts are guaranteed equal when mode='economic'
>>> np.allclose(r, np.triu(r3[:6,:6], k=0))
True
Example illustrating a common use of `qr`: solving of least squares
problems
What are the least-squares-best `m` and `y0` in ``y = y0 + mx`` for
the following data: {(0,1), (1,0), (1,2), (2,1)}. (Graph the points
and you'll see that it should be y0 = 0, m = 1.) The answer is provided
by solving the over-determined matrix equation ``Ax = b``, where::
A = array([[0, 1], [1, 1], [1, 1], [2, 1]])
x = array([[y0], [m]])
b = array([[1], [0], [2], [1]])
If A = qr such that q is orthonormal (which is always possible via
Gram-Schmidt), then ``x = inv(r) * (q.T) * b``. (In numpy practice,
however, we simply use `lstsq`.)
>>> A = np.array([[0, 1], [1, 1], [1, 1], [2, 1]])
>>> A
array([[0, 1],
[1, 1],
[1, 1],
[2, 1]])
>>> b = np.array([1, 0, 2, 1])
>>> q, r = LA.qr(A)
>>> p = np.dot(q.T, b)
>>> np.dot(LA.inv(r), p)
array([ 1.1e-16, 1.0e+00])
)r� �completer� �raw)�fZfull� )z7The 'full' option is deprecated in favor of 'reduced'.
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Compute the eigenvalues of a general matrix.
Main difference between `eigvals` and `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.
Raises
------
LinAlgError
If the eigenvalue computation does not converge.
See Also
--------
eig : eigenvalues and right eigenvectors of general arrays
eigvalsh : eigenvalues of symmetric or Hermitian arrays.
eigh : eigenvalues and eigenvectors of symmetric/Hermitian arrays.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
This is implemented using the _geev LAPACK routines which compute
the eigenvalues and eigenvectors of general square arrays.
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``:
>>> from numpy import linalg as LA
>>> x = np.random.random()
>>> Q = np.array([[np.cos(x), -np.sin(x)], [np.sin(x), np.cos(x)]])
>>> LA.norm(Q[0, :]), LA.norm(Q[1, :]), np.dot(Q[0, :],Q[1, :])
(1.0, 1.0, 0.0)
Now multiply a diagonal matrix by Q on one side and by Q.T on the other:
>>> D = np.diag((-1,1))
>>> LA.eigvals(D)
array([-1., 1.])
>>> A = np.dot(Q, D)
>>> A = np.dot(A, Q.T)
>>> LA.eigvals(A)
array([ 1., -1.])
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Compute the eigenvalues of a Hermitian or real symmetric matrix.
Main difference from 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.
Raises
------
LinAlgError
If the eigenvalue computation does not converge.
See Also
--------
eigh : eigenvalues and eigenvectors of symmetric/Hermitian arrays.
eigvals : eigenvalues of general real or complex arrays.
eig : eigenvalues and right eigenvectors of general real or complex
arrays.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
The eigenvalues are computed using LAPACK routines _syevd, _heevd
Examples
--------
>>> from numpy import linalg as LA
>>> a = np.array([[1, -2j], [2j, 5]])
>>> LA.eigvalsh(a)
array([ 0.17157288, 5.82842712])
>>> # demonstrate the treatment of the imaginary part of the diagonal
>>> a = np.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.eigvals()
>>> # with:
>>> b = np.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 = LA.eigvalsh(a)
>>> wb = LA.eigvals(b)
>>> wa; wb
array([ 1., 6.])
array([ 6.+0.j, 1.+0.j])
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Compute the eigenvalues and right eigenvectors of a square array.
Parameters
----------
a : (..., M, M) array
Matrices for which the eigenvalues and right eigenvectors will
be computed
Returns
-------
w : (..., M) array
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. When `a`
is real the resulting eigenvalues will be real (0 imaginary
part) or occur in conjugate pairs
v : (..., M, M) array
The normalized (unit "length") eigenvectors, such that the
column ``v[:,i]`` is the eigenvector corresponding to the
eigenvalue ``w[i]``.
Raises
------
LinAlgError
If the eigenvalue computation does not converge.
See Also
--------
eigvals : eigenvalues of a non-symmetric array.
eigh : eigenvalues and eigenvectors of a symmetric or Hermitian
(conjugate symmetric) array.
eigvalsh : eigenvalues of a symmetric or Hermitian (conjugate symmetric)
array.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
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 ``dot(a,v) = w * v``. Thus, the arrays `a`, `w`, and
`v` satisfy the equations ``dot(a[:,:], v[:,i]) = w[i] * v[:,i]``
for :math:`i \in \{0,...,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
``dot(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.
References
----------
G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando, FL,
Academic Press, Inc., 1980, Various pp.
Examples
--------
>>> from numpy import linalg as LA
(Almost) trivial example with real e-values and e-vectors.
>>> w, v = LA.eig(np.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 = LA.eig(np.array([[1, -1], [1, 1]]))
>>> w; v
array([ 1. + 1.j, 1. - 1.j])
array([[ 0.70710678+0.j , 0.70710678+0.j ],
[ 0.00000000-0.70710678j, 0.00000000+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 = np.array([[1, 1j], [-1j, 1]])
>>> w, v = LA.eig(a)
>>> w; v
array([ 2.00000000e+00+0.j, 5.98651912e-36+0.j]) # i.e., {2, 0}
array([[ 0.00000000+0.70710678j, 0.70710678+0.j ],
[ 0.70710678+0.j , 0.00000000+0.70710678j]])
Be careful about round-off error!
>>> a = np.array([[1 + 1e-9, 0], [0, 1 - 1e-9]])
>>> # Theor. e-values are 1 +/- 1e-9
>>> w, v = LA.eig(a)
>>> w; v
array([ 1., 1.])
array([[ 1., 0.],
[ 0., 1.]])
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Return the eigenvalues and eigenvectors of a Hermitian or 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
Hermitian/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, (..., M, M) matrix}
The column ``v[:, i]`` is the normalized eigenvector corresponding
to the eigenvalue ``w[i]``. Will return a matrix object if `a` is
a matrix object.
Raises
------
LinAlgError
If the eigenvalue computation does not converge.
See Also
--------
eigvalsh : eigenvalues of symmetric or Hermitian arrays.
eig : eigenvalues and right eigenvectors for non-symmetric arrays.
eigvals : eigenvalues of non-symmetric arrays.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
The eigenvalues/eigenvectors are computed using LAPACK routines _syevd,
_heevd
The eigenvalues of real symmetric or complex Hermitian matrices are
always real. [1]_ The array `v` of (column) eigenvectors is unitary
and `a`, `w`, and `v` satisfy the equations
``dot(a, v[:, i]) = w[i] * v[:, i]``.
References
----------
.. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
FL, Academic Press, Inc., 1980, pg. 222.
Examples
--------
>>> from numpy import linalg as LA
>>> a = np.array([[1, -2j], [2j, 5]])
>>> a
array([[ 1.+0.j, 0.-2.j],
[ 0.+2.j, 5.+0.j]])
>>> w, v = LA.eigh(a)
>>> w; v
array([ 0.17157288, 5.82842712])
array([[-0.92387953+0.j , -0.38268343+0.j ],
[ 0.00000000+0.38268343j, 0.00000000-0.92387953j]])
>>> np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair
array([2.77555756e-17 + 0.j, 0. + 1.38777878e-16j])
>>> np.dot(a, v[:, 1]) - w[1] * v[:, 1] # verify 2nd e-val/vec pair
array([ 0.+0.j, 0.+0.j])
>>> A = np.matrix(a) # what happens if input is a matrix object
>>> A
matrix([[ 1.+0.j, 0.-2.j],
[ 0.+2.j, 5.+0.j]])
>>> w, v = LA.eigh(A)
>>> w; v
array([ 0.17157288, 5.82842712])
matrix([[-0.92387953+0.j , -0.38268343+0.j ],
[ 0.00000000+0.38268343j, 0.00000000-0.92387953j]])
>>> # demonstrate the treatment of the imaginary part of the diagonal
>>> a = np.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 = np.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 = LA.eigh(a)
>>> wb, vb = LA.eig(b)
>>> wa; wb
array([ 1., 6.])
array([ 6.+0.j, 1.+0.j])
>>> va; vb
array([[-0.44721360-0.j , -0.89442719+0.j ],
[ 0.00000000+0.89442719j, 0.00000000-0.4472136j ]])
array([[ 0.89442719+0.j , 0.00000000-0.4472136j],
[ 0.00000000-0.4472136j, 0.89442719+0.j ]])
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Singular Value Decomposition.
Factors the matrix `a` as ``u * np.diag(s) * v``, where `u` and `v`
are unitary and `s` is a 1-d array of `a`'s singular values.
Parameters
----------
a : (..., M, N) array_like
A real or complex matrix of shape (`M`, `N`) .
full_matrices : bool, optional
If True (default), `u` and `v` have the shapes (`M`, `M`) and
(`N`, `N`), respectively. Otherwise, the shapes are (`M`, `K`)
and (`K`, `N`), respectively, where `K` = min(`M`, `N`).
compute_uv : bool, optional
Whether or not to compute `u` and `v` in addition to `s`. True
by default.
Returns
-------
u : { (..., M, M), (..., M, K) } array
Unitary matrices. The actual shape depends on the value of
``full_matrices``. Only returned when ``compute_uv`` is True.
s : (..., K) array
The singular values for every matrix, sorted in descending order.
v : { (..., N, N), (..., K, N) } array
Unitary matrices. The actual shape depends on the value of
``full_matrices``. Only returned when ``compute_uv`` is True.
Raises
------
LinAlgError
If SVD computation does not converge.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
The decomposition is performed using LAPACK routine _gesdd
The SVD is commonly written as ``a = U S V.H``. The `v` returned
by this function is ``V.H`` and ``u = U``.
If ``U`` is a unitary matrix, it means that it
satisfies ``U.H = inv(U)``.
The rows of `v` are the eigenvectors of ``a.H a``. The columns
of `u` are the eigenvectors of ``a a.H``. For row ``i`` in
`v` and column ``i`` in `u`, the corresponding eigenvalue is
``s[i]**2``.
If `a` is a `matrix` object (as opposed to an `ndarray`), then so
are all the return values.
Examples
--------
>>> a = np.random.randn(9, 6) + 1j*np.random.randn(9, 6)
Reconstruction based on full SVD:
>>> U, s, V = np.linalg.svd(a, full_matrices=True)
>>> U.shape, V.shape, s.shape
((9, 9), (6, 6), (6,))
>>> S = np.zeros((9, 6), dtype=complex)
>>> S[:6, :6] = np.diag(s)
>>> np.allclose(a, np.dot(U, np.dot(S, V)))
True
Reconstruction based on reduced SVD:
>>> U, s, V = np.linalg.svd(a, full_matrices=False)
>>> U.shape, V.shape, s.shape
((9, 6), (6, 6), (6,))
>>> S = np.diag(s)
>>> np.allclose(a, np.dot(U, np.dot(S, V)))
True
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Compute the condition number of a matrix.
This function is capable of returning the condition number using
one of seven different norms, depending on the value of `p` (see
Parameters below).
Parameters
----------
x : (..., M, N) array_like
The matrix whose condition number is sought.
p : {None, 1, -1, 2, -2, inf, -inf, 'fro'}, optional
Order of the norm:
===== ============================
p norm for matrices
===== ============================
None 2-norm, computed directly using the ``SVD``
'fro' Frobenius norm
inf max(sum(abs(x), axis=1))
-inf min(sum(abs(x), axis=1))
1 max(sum(abs(x), axis=0))
-1 min(sum(abs(x), axis=0))
2 2-norm (largest sing. value)
-2 smallest singular value
===== ============================
inf means the numpy.inf object, and the Frobenius norm is
the root-of-sum-of-squares norm.
Returns
-------
c : {float, inf}
The condition number of the matrix. May be infinite.
See Also
--------
numpy.linalg.norm
Notes
-----
The condition number of `x` is defined as the norm of `x` times the
norm of the inverse of `x` [1]_; the norm can be the usual L2-norm
(root-of-sum-of-squares) or one of a number of other matrix norms.
References
----------
.. [1] G. Strang, *Linear Algebra and Its Applications*, Orlando, FL,
Academic Press, Inc., 1980, pg. 285.
Examples
--------
>>> from numpy import linalg as LA
>>> a = np.array([[1, 0, -1], [0, 1, 0], [1, 0, 1]])
>>> a
array([[ 1, 0, -1],
[ 0, 1, 0],
[ 1, 0, 1]])
>>> LA.cond(a)
1.4142135623730951
>>> LA.cond(a, 'fro')
3.1622776601683795
>>> LA.cond(a, np.inf)
2.0
>>> LA.cond(a, -np.inf)
1.0
>>> LA.cond(a, 1)
2.0
>>> LA.cond(a, -1)
1.0
>>> LA.cond(a, 2)
1.4142135623730951
>>> LA.cond(a, -2)
0.70710678118654746
>>> min(LA.svd(a, compute_uv=0))*min(LA.svd(LA.inv(a), compute_uv=0))
0.70710678118654746
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Return matrix rank of array using SVD method
Rank of the array is the number of SVD singular values of the array that are
greater than `tol`.
Parameters
----------
M : {(M,), (..., M, N)} array_like
input vector or stack of matrices
tol : {None, float}, optional
threshold below which SVD values are considered zero. If `tol` is
None, and ``S`` is an array with singular values for `M`, and
``eps`` is the epsilon value for datatype of ``S``, then `tol` is
set to ``S.max() * max(M.shape) * eps``.
Notes
-----
The default threshold to detect rank deficiency is a test on the magnitude
of the singular values of `M`. By default, we identify singular values less
than ``S.max() * max(M.shape) * eps`` as indicating rank deficiency (with
the symbols defined above). This is the algorithm MATLAB uses [1]. It also
appears in *Numerical recipes* in the discussion of SVD solutions for linear
least squares [2].
This default threshold is designed to detect rank deficiency accounting for
the numerical errors of the SVD computation. Imagine that there is a column
in `M` that is an exact (in floating point) linear combination of other
columns in `M`. Computing the SVD on `M` will not produce a singular value
exactly equal to 0 in general: any difference of the smallest SVD value from
0 will be caused by numerical imprecision in the calculation of the SVD.
Our threshold for small SVD values takes this numerical imprecision into
account, and the default threshold will detect such numerical rank
deficiency. The threshold may declare a matrix `M` rank deficient even if
the linear combination of some columns of `M` is not exactly equal to
another column of `M` but only numerically very close to another column of
`M`.
We chose our default threshold because it is in wide use. Other thresholds
are possible. For example, elsewhere in the 2007 edition of *Numerical
recipes* there is an alternative threshold of ``S.max() *
np.finfo(M.dtype).eps / 2. * np.sqrt(m + n + 1.)``. The authors describe
this threshold as being based on "expected roundoff error" (p 71).
The thresholds above deal with floating point roundoff error in the
calculation of the SVD. However, you may have more information about the
sources of error in `M` that would make you consider other tolerance values
to detect *effective* rank deficiency. The most useful measure of the
tolerance depends on the operations you intend to use on your matrix. For
example, if your data come from uncertain measurements with uncertainties
greater than floating point epsilon, choosing a tolerance near that
uncertainty may be preferable. The tolerance may be absolute if the
uncertainties are absolute rather than relative.
References
----------
.. [1] MATLAB reference documention, "Rank"
http://www.mathworks.com/help/techdoc/ref/rank.html
.. [2] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery,
"Numerical Recipes (3rd edition)", Cambridge University Press, 2007,
page 795.
Examples
--------
>>> from numpy.linalg import matrix_rank
>>> matrix_rank(np.eye(4)) # Full rank matrix
4
>>> I=np.eye(4); I[-1,-1] = 0. # rank deficient matrix
>>> matrix_rank(I)
3
>>> matrix_rank(np.ones((4,))) # 1 dimension - rank 1 unless all 0
1
>>> matrix_rank(np.zeros((4,)))
0
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Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate the generalized inverse of a matrix using its
singular-value decomposition (SVD) and including all
*large* singular values.
Parameters
----------
a : (M, N) array_like
Matrix to be pseudo-inverted.
rcond : float
Cutoff for small singular values.
Singular values smaller (in modulus) than
`rcond` * largest_singular_value (again, in modulus)
are set to zero.
Returns
-------
B : (N, M) ndarray
The pseudo-inverse of `a`. If `a` is a `matrix` instance, then so
is `B`.
Raises
------
LinAlgError
If the SVD computation does not converge.
Notes
-----
The pseudo-inverse of a matrix A, denoted :math:`A^+`, is
defined as: "the matrix that 'solves' [the least-squares problem]
:math:`Ax = b`," i.e., if :math:`\bar{x}` is said solution, then
:math:`A^+` is that matrix such that :math:`\bar{x} = A^+b`.
It can be shown that if :math:`Q_1 \Sigma Q_2^T = A` is the singular
value decomposition of A, then
:math:`A^+ = Q_2 \Sigma^+ Q_1^T`, where :math:`Q_{1,2}` are
orthogonal matrices, :math:`\Sigma` is a diagonal matrix consisting
of A's so-called singular values, (followed, typically, by
zeros), and then :math:`\Sigma^+` is simply the diagonal matrix
consisting of the reciprocals of A's singular values
(again, followed by zeros). [1]_
References
----------
.. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
FL, Academic Press, Inc., 1980, pp. 139-142.
Examples
--------
The following example checks that ``a * a+ * a == a`` and
``a+ * a * a+ == a+``:
>>> a = np.random.randn(9, 6)
>>> B = np.linalg.pinv(a)
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
True
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
True
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Compute the sign and (natural) logarithm of the determinant of an array.
If an array has a very small or very large determinant, then a call to
`det` may overflow or underflow. This routine is more robust against such
issues, because it computes the logarithm of the determinant rather than
the determinant itself.
Parameters
----------
a : (..., M, M) array_like
Input array, has to be a square 2-D array.
Returns
-------
sign : (...) array_like
A number representing the sign of the determinant. For a real matrix,
this is 1, 0, or -1. For a complex matrix, this is a complex number
with absolute value 1 (i.e., it is on the unit circle), or else 0.
logdet : (...) array_like
The natural log of the absolute value of the determinant.
If the determinant is zero, then `sign` will be 0 and `logdet` will be
-Inf. In all cases, the determinant is equal to ``sign * np.exp(logdet)``.
See Also
--------
det
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
.. versionadded:: 1.6.0
The determinant is computed via LU factorization using the LAPACK
routine z/dgetrf.
Examples
--------
The determinant of a 2-D array ``[[a, b], [c, d]]`` is ``ad - bc``:
>>> a = np.array([[1, 2], [3, 4]])
>>> (sign, logdet) = np.linalg.slogdet(a)
>>> (sign, logdet)
(-1, 0.69314718055994529)
>>> sign * np.exp(logdet)
-2.0
Computing log-determinants for a stack of matrices:
>>> a = np.array([ [[1, 2], [3, 4]], [[1, 2], [2, 1]], [[1, 3], [3, 1]] ])
>>> a.shape
(3, 2, 2)
>>> sign, logdet = np.linalg.slogdet(a)
>>> (sign, logdet)
(array([-1., -1., -1.]), array([ 0.69314718, 1.09861229, 2.07944154]))
>>> sign * np.exp(logdet)
array([-2., -3., -8.])
This routine succeeds where ordinary `det` does not:
>>> np.linalg.det(np.eye(500) * 0.1)
0.0
>>> np.linalg.slogdet(np.eye(500) * 0.1)
(1, -1151.2925464970228)
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Compute the determinant of an array.
Parameters
----------
a : (..., M, M) array_like
Input array to compute determinants for.
Returns
-------
det : (...) array_like
Determinant of `a`.
See Also
--------
slogdet : Another way to representing the determinant, more suitable
for large matrices where underflow/overflow may occur.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
The determinant is computed via LU factorization using the LAPACK
routine z/dgetrf.
Examples
--------
The determinant of a 2-D array [[a, b], [c, d]] is ad - bc:
>>> a = np.array([[1, 2], [3, 4]])
>>> np.linalg.det(a)
-2.0
Computing determinants for a stack of matrices:
>>> a = np.array([ [[1, 2], [3, 4]], [[1, 2], [2, 1]], [[1, 3], [3, 1]] ])
>>> a.shape
(3, 2, 2)
>>> np.linalg.det(a)
array([-2., -3., -8.])
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Return the least-squares solution to a linear matrix equation.
Solves the equation `a x = b` by computing a vector `x` that
minimizes the Euclidean 2-norm `|| b - a x ||^2`. The equation may
be under-, well-, or over- determined (i.e., the number of
linearly independent rows of `a` can be less than, equal to, or
greater than its number of linearly independent columns). If `a`
is square and of full rank, then `x` (but for round-off error) is
the "exact" solution of the equation.
Parameters
----------
a : (M, N) array_like
"Coefficient" matrix.
b : {(M,), (M, K)} array_like
Ordinate or "dependent variable" values. If `b` is two-dimensional,
the least-squares solution is calculated for each of the `K` columns
of `b`.
rcond : float, optional
Cut-off ratio for small singular values of `a`.
For the purposes of rank determination, singular values are treated
as zero if they are smaller than `rcond` times the largest singular
value of `a`.
Returns
-------
x : {(N,), (N, K)} ndarray
Least-squares solution. If `b` is two-dimensional,
the solutions are in the `K` columns of `x`.
residuals : {(), (1,), (K,)} ndarray
Sums of residuals; squared Euclidean 2-norm for each column in
``b - a*x``.
If the rank of `a` is < N or M <= N, this is an empty array.
If `b` is 1-dimensional, this is a (1,) shape array.
Otherwise the shape is (K,).
rank : int
Rank of matrix `a`.
s : (min(M, N),) ndarray
Singular values of `a`.
Raises
------
LinAlgError
If computation does not converge.
Notes
-----
If `b` is a matrix, then all array results are returned as matrices.
Examples
--------
Fit a line, ``y = mx + c``, through some noisy data-points:
>>> x = np.array([0, 1, 2, 3])
>>> y = np.array([-1, 0.2, 0.9, 2.1])
By examining the coefficients, we see that the line should have a
gradient of roughly 1 and cut the y-axis at, more or less, -1.
We can rewrite the line equation as ``y = Ap``, where ``A = [[x 1]]``
and ``p = [[m], [c]]``. Now use `lstsq` to solve for `p`:
>>> A = np.vstack([x, np.ones(len(x))]).T
>>> A
array([[ 0., 1.],
[ 1., 1.],
[ 2., 1.],
[ 3., 1.]])
>>> m, c = np.linalg.lstsq(A, y)[0]
>>> print(m, c)
1.0 -0.95
Plot the data along with the fitted line:
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y, 'o', label='Original data', markersize=10)
>>> plt.plot(x, m*x + c, 'r', label='Fitted line')
>>> plt.legend()
>>> plt.show()
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This is a private utility function used by numpy.linalg.norm().
Parameters
----------
x : ndarray
row_axis, col_axis : int
The axes of `x` that hold the 2-D matrices.
op : callable
This should be either numpy.amin or numpy.amax or numpy.sum.
Returns
-------
result : float or ndarray
If `x` is 2-D, the return values is a float.
Otherwise, it is an array with ``x.ndim - 2`` dimensions.
The return values are either the minimum or maximum or sum of the
singular values of the matrices, depending on whether `op`
is `numpy.amin` or `numpy.amax` or `numpy.sum`.
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Matrix or vector norm.
This function is able to return one of eight different matrix norms,
or one of an infinite number of vector norms (described below), depending
on the value of the ``ord`` parameter.
Parameters
----------
x : array_like
Input array. If `axis` is None, `x` must be 1-D or 2-D.
ord : {non-zero int, inf, -inf, 'fro', 'nuc'}, optional
Order of the norm (see table under ``Notes``). inf means numpy's
`inf` object.
axis : {int, 2-tuple of ints, None}, optional
If `axis` is an integer, it specifies the axis of `x` along which to
compute the vector norms. If `axis` is a 2-tuple, it specifies the
axes that hold 2-D matrices, and the matrix norms of these matrices
are computed. If `axis` is None then either a vector norm (when `x`
is 1-D) or a matrix norm (when `x` is 2-D) is returned.
keepdims : bool, optional
If this is set to True, the axes which are normed over are left in the
result as dimensions with size one. With this option the result will
broadcast correctly against the original `x`.
.. versionadded:: 1.10.0
Returns
-------
n : float or ndarray
Norm of the matrix or vector(s).
Notes
-----
For values of ``ord <= 0``, the result is, strictly speaking, not a
mathematical 'norm', but it may still be useful for various numerical
purposes.
The following norms can be calculated:
===== ============================ ==========================
ord norm for matrices norm for vectors
===== ============================ ==========================
None Frobenius norm 2-norm
'fro' Frobenius norm --
'nuc' nuclear norm --
inf max(sum(abs(x), axis=1)) max(abs(x))
-inf min(sum(abs(x), axis=1)) min(abs(x))
0 -- sum(x != 0)
1 max(sum(abs(x), axis=0)) as below
-1 min(sum(abs(x), axis=0)) as below
2 2-norm (largest sing. value) as below
-2 smallest singular value as below
other -- sum(abs(x)**ord)**(1./ord)
===== ============================ ==========================
The Frobenius norm is given by [1]_:
:math:`||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
The nuclear norm is the sum of the singular values.
References
----------
.. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
Examples
--------
>>> from numpy import linalg as LA
>>> a = np.arange(9) - 4
>>> a
array([-4, -3, -2, -1, 0, 1, 2, 3, 4])
>>> b = a.reshape((3, 3))
>>> b
array([[-4, -3, -2],
[-1, 0, 1],
[ 2, 3, 4]])
>>> LA.norm(a)
7.745966692414834
>>> LA.norm(b)
7.745966692414834
>>> LA.norm(b, 'fro')
7.745966692414834
>>> LA.norm(a, np.inf)
4.0
>>> LA.norm(b, np.inf)
9.0
>>> LA.norm(a, -np.inf)
0.0
>>> LA.norm(b, -np.inf)
2.0
>>> LA.norm(a, 1)
20.0
>>> LA.norm(b, 1)
7.0
>>> LA.norm(a, -1)
-4.6566128774142013e-010
>>> LA.norm(b, -1)
6.0
>>> LA.norm(a, 2)
7.745966692414834
>>> LA.norm(b, 2)
7.3484692283495345
>>> LA.norm(a, -2)
nan
>>> LA.norm(b, -2)
1.8570331885190563e-016
>>> LA.norm(a, 3)
5.8480354764257312
>>> LA.norm(a, -3)
nan
Using the `axis` argument to compute vector norms:
>>> c = np.array([[ 1, 2, 3],
... [-1, 1, 4]])
>>> LA.norm(c, axis=0)
array([ 1.41421356, 2.23606798, 5. ])
>>> LA.norm(c, axis=1)
array([ 3.74165739, 4.24264069])
>>> LA.norm(c, ord=1, axis=1)
array([ 6., 6.])
Using the `axis` argument to compute matrix norms:
>>> m = np.arange(8).reshape(2,2,2)
>>> LA.norm(m, axis=(1,2))
array([ 3.74165739, 11.22497216])
>>> LA.norm(m[0, :, :]), LA.norm(m[1, :, :])
(3.7416573867739413, 11.224972160321824)
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Compute the dot product of two or more arrays in a single function call,
while automatically selecting the fastest evaluation order.
`multi_dot` chains `numpy.dot` and uses optimal parenthesization
of the matrices [1]_ [2]_. Depending on the shapes of the matrices,
this can speed up the multiplication a lot.
If the first argument is 1-D it is treated as a row vector.
If the last argument is 1-D it is treated as a column vector.
The other arguments must be 2-D.
Think of `multi_dot` as::
def multi_dot(arrays): return functools.reduce(np.dot, arrays)
Parameters
----------
arrays : sequence of array_like
If the first argument is 1-D it is treated as row vector.
If the last argument is 1-D it is treated as column vector.
The other arguments must be 2-D.
Returns
-------
output : ndarray
Returns the dot product of the supplied arrays.
See Also
--------
dot : dot multiplication with two arguments.
References
----------
.. [1] Cormen, "Introduction to Algorithms", Chapter 15.2, p. 370-378
.. [2] http://en.wikipedia.org/wiki/Matrix_chain_multiplication
Examples
--------
`multi_dot` allows you to write::
>>> from numpy.linalg import multi_dot
>>> # Prepare some data
>>> A = np.random.random(10000, 100)
>>> B = np.random.random(100, 1000)
>>> C = np.random.random(1000, 5)
>>> D = np.random.random(5, 333)
>>> # the actual dot multiplication
>>> multi_dot([A, B, C, D])
instead of::
>>> np.dot(np.dot(np.dot(A, B), C), D)
>>> # or
>>> A.dot(B).dot(C).dot(D)
Notes
-----
The cost for a matrix multiplication can be calculated with the
following function::
def cost(A, B):
return A.shape[0] * A.shape[1] * B.shape[1]
Let's assume we have three matrices
:math:`A_{10x100}, B_{100x5}, C_{5x50}`.
The costs for the two different parenthesizations are as follows::
cost((AB)C) = 10*100*5 + 10*5*50 = 5000 + 2500 = 7500
cost(A(BC)) = 10*100*50 + 100*5*50 = 50000 + 25000 = 75000
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than `_multi_dot_matrix_chain_order`
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Return a np.array that encodes the optimal order of mutiplications.
The optimal order array is then used by `_multi_dot()` to do the
multiplication.
Also return the cost matrix if `return_costs` is `True`
The implementation CLOSELY follows Cormen, "Introduction to Algorithms",
Chapter 15.2, p. 370-378. Note that Cormen uses 1-based indices.
cost[i, j] = min([
cost[prefix] + cost[suffix] + cost_mult(prefix, suffix)
for k in range(i, j)])
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