mxnet.ndarray.numpy.linalg

Namespace for operators used in Gluon dispatched by F=ndarray.

Functions

cholesky(a[, upper])	Cholesky decomposition.
det(a)	Compute the determinant of an array.
eig(a)	Compute the eigenvalues and right eigenvectors of a square array.
eigh(a[, UPLO])	Return the eigenvalues and eigenvectors real symmetric matrix.
eigvals(a)	Compute the eigenvalues of a general matrix.
eigvalsh(a[, UPLO])	Compute the eigenvalues real symmetric matrix.
inv(a)	Compute the (multiplicative) inverse of a matrix.
lstsq(a, b[, rcond])	Return the least-squares solution to a linear matrix equation.
matrix_rank(M[, tol, hermitian])	Return matrix rank of array using SVD method
norm(x[, ord, axis, keepdims])	Matrix or vector norm.
pinv(a[, rcond, hermitian])	Compute the (Moore-Penrose) pseudo-inverse of a matrix.
qr(a[, mode])	Compute the qr factorization of a matrix a.
slogdet(a)	Compute the sign and (natural) logarithm of the determinant of an array.
solve(a, b)	Solve a linear matrix equation, or system of linear scalar equations.
svd(a)	Singular Value Decomposition.
tensorinv(a[, ind])	Compute the 'inverse' of an N-dimensional array.
tensorsolve(a, b[, axes])	Solve the tensor equation a x = b for x.

mxnet.ndarray.numpy.linalg.cholesky(a, upper=False)

Cholesky decomposition.

Notes

upper param is requested by API standardization in https://data-apis.org/array-api/latest/extensions/generated/signatures.linalg.cholesky.html instead of parameter in official NumPy operator.

Return the Cholesky decomposition, L * L.T, of the square matrix a, where L is lower-triangular and .T is the transpose operator. a must be symmetric and positive-definite. Only L is actually returned. Complex-valued input is currently not supported.

Parameters:

• a ((..., M, M) ndarray) – Symmetric, positive-definite input matrix.

• upper (bool) – If True, the result must be the upper-triangular Cholesky factor. If False, the result must be the lower-triangular Cholesky factor. Default: False.

Returns:

L – Lower-triangular Cholesky factor of a.

Return type:

(…, M, M) ndarray

Raises:

MXNetError – If the decomposition fails, for example, if a is not positive-definite.

Notes

Broadcasting rules apply.

The Cholesky decomposition is often used as a fast way of solving

\[A \mathbf{x} = \mathbf{b}\]

(when A is both symmetric and positive-definite).

First, we solve for \(\mathbf{y}\) in

\[L \mathbf{y} = \mathbf{b},\]

and then for \(\mathbf{x}\) in

\[L.T \mathbf{x} = \mathbf{y}.\]

Examples

>>> A = np.array([[16, 4], [4, 10]])
>>> A
array([[16.,  4.],
       [ 4., 10.]])
>>> L = np.linalg.cholesky(A)
>>> L
array([[4., 0.],
       [1., 3.]])
>>> np.dot(L, L.T)
array([[16.,  4.],
       [ 4., 10.]])

mxnet.ndarray.numpy.linalg.det(a)

Compute the determinant of an array.

Parameters:

a ((..., M, M) ndarray) – Input array to compute determinants for.

Returns:

det – Determinant of a.

Return type:

(…) ndarray

See also

slogdet

Another way to represent the determinant, more suitable

for

Notes

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.])

mxnet.ndarray.numpy.linalg.eig(a)

Compute the eigenvalues and right eigenvectors of a square array.

Parameters:

a ((..., M, M) ndarray) – 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.

• v ((…, M, M) ndarray) – The normalized (unit “length”) eigenvectors, such that the column v[:,i] is the eigenvector corresponding to the eigenvalue w[i].

Raises:

MXNetError – If the eigenvalue computation does not converge.

See also

eigvals

eigenvalues of a non-symmetric array.

eigh

eigenvalues and eigenvectors of a real symmetric array.

eigvalsh

eigenvalues of a real symmetric.

Notes

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 \(i \\in \\{0,...,M-1\\}\).

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.

This function differs from the original numpy.linalg.eig in the following way(s):

• Does not support complex input and output.

Examples

>>> from numpy import linalg as LA
>>> a = np.array([[-1.9147992 ,  6.054115  , 18.046988  ],
                  [ 0.77563655, -4.860152  ,  2.1012988 ],
                  [ 2.6083658 ,  2.3705218 ,  0.3192524 ]])
>>> w, v = LA.eig(a)
>>> w
array([ 6.9683027, -7.768063 , -5.655937 ])
>>> v
array([[ 0.90617794,  0.9543622 ,  0.2492316 ],
       [ 0.13086087, -0.04077047, -0.9325615 ],
       [ 0.4021404 , -0.29585576,  0.26117516]])

mxnet.ndarray.numpy.linalg.eigh(a, UPLO='L')

Return the eigenvalues and eigenvectors 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) ndarray) – 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, (…, 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:

MXNetError – If the eigenvalue computation does not converge.

See also

eig

eigenvalues and right eigenvectors of general arrays

eigvals

eigenvalues of a non-symmetric array.

eigvalsh

eigenvalues of a real symmetric.

Notes

The eigenvalues/eigenvectors are computed using LAPACK routines _syevd.

This function differs from the original numpy.linalg.eigh in the following way(s):

• Does not support complex input and output.

Examples

>>> from numpy import linalg as LA
>>> a = np.array([[ 6.8189726 , -3.926585  ,  4.3990498 ],
                  [-0.59656644, -1.9166266 ,  9.54532   ],
                  [ 2.1093285 ,  0.19688708, -1.1634291 ]])
>>> w, v = LA.eigh(a, UPLO='L')
>>> w
array([-2.175445 , -1.4581827,  7.3725457])
>>> v
array([[ 0.1805163 , -0.16569263,  0.9695154 ],
       [ 0.8242942 ,  0.56326365, -0.05721384],
       [-0.53661287,  0.80949366,  0.23825769]])

mxnet.ndarray.numpy.linalg.eigvals(a)

Compute the eigenvalues of a general matrix.

Main difference between eigvals and eig: the eigenvectors aren’t returned.

Parameters:

a ((..., M, M) ndarray) – A real-valued matrix whose eigenvalues will be computed.

Returns:

w – The eigenvalues, each repeated according to its multiplicity. They are not necessarily ordered.

Return type:

(…, M,) ndarray

Raises:

MXNetError – If the eigenvalue computation does not converge.

See also

eig

eigenvalues and right eigenvectors of general arrays

eigh

eigenvalues and eigenvectors of a real symmetric array.

eigvalsh

eigenvalues of a real symmetric.

Notes

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.

This function differs from the original numpy.linalg.eigvals in the following way(s):

• Does not support complex input and output.

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.]) # random

mxnet.ndarray.numpy.linalg.eigvalsh(a, UPLO='L')

Compute the eigenvalues real symmetric matrix.

Main difference from eigh: the eigenvectors are not computed.

Parameters:

• a ((..., M, M) ndarray) – A 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 – The eigenvalues in ascending order, each repeated according to its multiplicity.

Return type:

(…, M,) ndarray

Raises:

MXNetError – If the eigenvalue computation does not converge.

See also

eig

eigenvalues and right eigenvectors of general arrays

eigvals

eigenvalues of a non-symmetric array.

eigh

eigenvalues and eigenvectors of a real symmetric array.

Notes

Broadcasting rules apply, see the numpy.linalg documentation for details.

The eigenvalues are computed using LAPACK routines _syevd.

This function differs from the original numpy.linalg.eigvalsh in the following way(s):

• Does not support complex input and output.

Examples

>>> from numpy import linalg as LA
>>> a = np.array([[ 5.4119368 ,  8.996273  , -5.086096  ],
                  [ 0.8866155 ,  1.7490431 , -4.6107802 ],
                  [-0.08034172,  4.4172044 ,  1.4528792 ]])
>>> LA.eigvalsh(a, UPLO='L')
array([-2.87381886,  5.10144682,  6.38623114]) # in ascending order

mxnet.ndarray.numpy.linalg.inv(a)

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) ndarray) – Matrix to be inverted.

Returns:

ainv – (Multiplicative) inverse of the matrix a.

Return type:

(…, M, M) ndarray

Raises:

MXNetError – If a is not square or inversion fails.

Examples

>>> from mxnet import np
>>> a = np.array([[1., 2.], [3., 4.]])
array([[-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]]])
>>> np.linalg.inv(a)
array([[[-2.        ,  1.        ],
        [ 1.5       , -0.5       ]],

[[-1.2500001 , 0.75000006],

[ 0.75000006, -0.25000003]]])

mxnet.ndarray.numpy.linalg.lstsq(a, b, rcond='warn')

Return the least-squares solution to a linear matrix equation.

Solves the equation \(a x = b\) by computing a vector x that minimizes the squared Euclidean 2-norm \(\| b - a x \|^2_2\). 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) ndarray) – “Coefficient” matrix.

• b ({(M,), (M, K)} ndarray) – 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 The default of warn or -1 will use the machine precision as rcond parameter. The default of None will use the machine precision times max(M, N).

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,), (0,)} 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:

MXNetError – If computation does not converge.

Notes

If b is a matrix, then all array results are returned as matrices.

Examples

>>> x = np.array([0, 1, 2, 3])
>>> y = np.array([-1, 0.2, 0.9, 2.1])
>>> 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, rcond=None)[0]
>>> m, c
(1.0 -0.95) # may vary

mxnet.ndarray.numpy.linalg.matrix_rank(M, tol=None, hermitian=False)

Return matrix rank of array using SVD method

Rank of the array is the number of singular values of the array that are greater than tol.

Parameters M : {(M,), (…, M, N)} ndarray

Input vector or stack of matrices.

tol(…) ndarray, 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.

hermitianbool, optional

If True, M is assumed to be Hermitian (symmetric if real-valued), enabling a more efficient method for finding singular values. Defaults to False.

Returns:

rank – Rank of M.

Return type:

(…) ndarray

Examples

>>> from mxnet import np
>>> np.matrix_rank(np.eye(4)) # Full rank matrix
4
>>> I=np.eye(4); I[-1,-1] = 0. # rank deficient matrix
>>> np.matrix_rank(I)
3
>>> np.matrix_rank(np.ones((4,))) # 1 dimension - rank 1 unless all 0
1
>>> np.matrix_rank(np.zeros((4,)))
0

mxnet.ndarray.numpy.linalg.norm(x, ord=None, axis=None, keepdims=False)

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 (ndarray) – 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.

Returns:

n – Norm of the matrix or vector(s).

Return type:

ndarray

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’ – – 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 – as below -2 – as below other – sum(abs(x)**ord)**(1./ord) ===== ============================ ========================== The Frobenius norm is given by [1]_:

\(||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}\)

The nuclear norm is the sum of the singular values. When you want to operate norm for matrices,if you ord is (-1, 1, inf, -inf), you must give you axis, it is not support default axis.

References

[1]

G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

Examples

>>> from mxnet import np
>>> 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.]])
>>> np.linalg.norm(a)
array(7.745967)
>>> np.linalg.norm(b)
array(7.745967)
>>> np.linalg.norm(b, 'fro')
array(7.745967)
>>> np.linalg.norm(a, 'inf')
array(4.)
>>> np.linalg.norm(b, 'inf', axis=(0, 1))
array(9.)
>>> np.linalg.norm(a, '-inf')
array(0.)
>>> np.linalg.norm(b, '-inf', axis=(0, 1))
array(2.)
>>> np.linalg.norm(a, 1)
array(20.)
>>> np.linalg.norm(b, 1, axis=(0, 1))
array(7.)
>>> np.linalg.norm(a, -1)
array(0.)
>>> np.linalg.norm(b, -1, axis=(0, 1))
array(6.)
>>> np.linalg.norm(a, 2)
array(7.745967)
>>> np.linalg.norm(a, -2)
array(0.)
>>> np.linalg.norm(a, 3)
array(5.8480353)
>>> np.linalg.norm(a, -3)
array(0.)
Using the `axis` argument to compute vector norms:
>>> c = np.array([[ 1, 2, 3],
...               [-1, 1, 4]])
>>> np.linalg.norm(c, axis=0)
array([1.4142135, 2.236068 , 5.       ])
>>> np.linalg.norm(c, axis=1)
array([3.7416573, 4.2426405])
>>> np.linalg.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)
>>> np.linalg.norm(m, axis=(1,2))
array([ 3.7416573, 11.224973 ])
>>> np.linalg.norm(m[0, :, :]), np.linalg.norm(m[1, :, :])
(array(3.7416573), array(11.224973))

mxnet.ndarray.numpy.linalg.pinv(a, rcond=1e-15, hermitian=False)

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) ndarray) – Matrix or stack of matrices to be pseudo-inverted.

• rcond ((...) {float or ndarray of float}, optional) – Cutoff for small singular values. Singular values less than or equal to rcond * largest_singular_value are set to zero. Broadcasts against the stack of matrices.

• hermitian (bool, optional) – If True, a is assumed to be Hermitian (symmetric if real-valued), enabling a more efficient method for finding singular values. Defaults to False.

Returns:

B – The pseudo-inverse of a. If a is a matrix instance, then so is B.

Return type:

(…, N, M) ndarray

Raises:

MXNetError – If the SVD computation does not converge.

Notes

The pseudo-inverse of a matrix A, denoted \(A^+\), is defined as: “the matrix that ‘solves’ [the least-squares problem] \(Ax = b\),” i.e., if \(\\bar{x}\) is said solution, then \(A^+\) is that matrix such that \(\\bar{x} = A^+b\).

It can be shown that if \(Q_1 \\Sigma Q_2^T = A\) is the singular value decomposition of A, then \(A^+ = Q_2 \\Sigma^+ Q_1^T\), where \(Q_{1,2}\) are orthogonal matrices, \(\\Sigma\) is a diagonal matrix consisting of A’s so-called singular values, (followed, typically, by zeros), and then \(\\Sigma^+\) 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(2, 3) >>> pinv_a = np.linalg.pinv(a) >>> (a - np.dot(a, np.dot(pinv_a, a))).sum() array(0.) >>> (pinv_a - np.dot(pinv_a, np.dot(a, pinv_a))).sum() array(0.)

mxnet.ndarray.numpy.linalg.qr(a, mode='reduced')

Compute the qr factorization of a matrix a. Factor the matrix a as qr, where q is orthonormal and r is upper-triangular.

Parameters:

• a ((..., M, N) ndarray) – Matrix or stack of matrices to be qr factored.

• mode ({‘reduced’, ‘complete’, ‘r’, ‘raw’, ‘full’, ‘economic’}, optional) – Only default mode, ‘reduced’, is implemented. If K = min(M, N), then * ‘reduced’ : returns q, r with dimensions (M, K), (K, N) (default)

Returns:

• q ((…, M, K) ndarray) – A matrix or stack of matrices with K orthonormal columns, with K = min(M, N).

• r ((…, K, N) ndarray) – A matrix or stack of upper triangular matrices.

Raises:

MXNetError – If factoring fails.

Examples

>>> from mxnet import np
>>> a = np.random.uniform(-10, 10, (2, 2))
>>> q, r = np.linalg.qr(a)
>>> q
array([[-0.22121978, -0.97522414],
       [-0.97522414,  0.22121954]])
>>> r
array([[-4.4131265 , -7.1255064 ],
       [ 0.        , -0.28771925]])
>>> a = np.random.uniform(-10, 10, (2, 3))
>>> q, r = np.linalg.qr(a)
>>> q
array([[-0.28376842, -0.9588929 ],
       [-0.9588929 ,  0.28376836]])
>>> r
array([[-7.242763  , -0.5673361 , -2.624416  ],
       [ 0.        , -7.297918  , -0.15949416]])
>>> a = np.random.uniform(-10, 10, (3, 2))
>>> q, r = np.linalg.qr(a)
>>> q
array([[-0.34515655,  0.10919492],
       [ 0.14765628, -0.97452265],
       [-0.92685735, -0.19591334]])
>>> r
array([[-8.453794,  8.4175  ],
       [ 0.      ,  5.430561]])

mxnet.ndarray.numpy.linalg.slogdet(a)

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) ndarray) – Input array, has to be a square 2-D array.

Returns:

• sign ((…) ndarray) – A number representing the sign of the determinant. For a real matrix, this is 1, 0, or -1.

• 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

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]]) >>> (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)

mxnet.ndarray.numpy.linalg.solve(a, b)

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) ndarray) – Coefficient matrix.

• b ({(..., M,), (..., M, K)}, ndarray) – Ordinate or “dependent variable” values.

Returns:

x – Solution to the system a x = b. Returned shape is identical to b.

Return type:

{(…, M,), (…, M, K)} ndarray

Raises:

MXNetError – If a is singular or not square.

Notes

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.

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

mxnet.ndarray.numpy.linalg.svd(a)

Singular Value Decomposition.

When a is a 2D array, it is factorized as ut @ np.diag(s) @ v, where ut and v are 2D orthonormal arrays and s is a 1D array of a’s singular values. When a is higher-dimensional, SVD is applied in stacked mode as explained below.

Parameters:

a ((..., M, N) ndarray) – A real array with a.ndim >= 2 and M <= N.

Returns:

• ut ((…, M, M) ndarray) – Orthonormal array(s). The first a.ndim - 2 dimensions have the same size as those of the input a.

• s ((…, M) ndarray) – Vector(s) with the singular values, within each vector sorted in descending order. The first a.ndim - 2 dimensions have the same size as those of the input a.

• v ((…, M, N) ndarray) – Orthonormal array(s). The first a.ndim - 2 dimensions have the same size as those of the input a.

Notes

The decomposition is performed using LAPACK routine _gesvd.

SVD is usually described for the factorization of a 2D matrix \(A\). The higher-dimensional case will be discussed below. In the 2D case, SVD is written as \(A = U^T S V\), where \(A = a\), \(U^T = ut\), \(S= \mathtt{np.diag}(s)\) and \(V = v\). The 1D array s contains the singular values of a and ut and v are orthonormal. The rows of v are the eigenvectors of \(A^T A\) and the columns of ut are the eigenvectors of \(A A^T\). In both cases the corresponding (possibly non-zero) eigenvalues are given by s**2.

The sign of rows of u and v are determined as described in Auto-Differentiating Linear Algebra.

If a has more than two dimensions, then broadcasting rules apply. This means that SVD is working in “stacked” mode: it iterates over all indices of the first a.ndim - 2 dimensions and for each combination SVD is applied to the last two indices. The matrix a can be reconstructed from the decomposition with either (ut * s[..., None, :]) @ v or ut @ (s[..., None] * v). (The @ operator denotes batch matrix multiplication)

This function differs from the original numpy.linalg.svd in the following way(s):

• The sign of rows of u and v may differ.

• Does not support complex input.

Examples

>>> a = np.arange(54).reshape(6, 9)
>>> ut, s, v = np.linalg.svd(a)
>>> ut.shape, s.shape, v.shape
((6, 6), (6,), (6, 9))
>>> s = s.reshape(6, 1)
>>> ret = np.dot(ut, s * v)
>>> (ret - a > 1e-3).sum()
array(0.)
>>> (ret - a < -1e-3).sum()
array(0.)

mxnet.ndarray.numpy.linalg.tensorinv(a, ind=2)

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 – a’s tensordot inverse, shape a.shape[ind:] + a.shape[:ind].

Return type:

ndarray

Raises:

MXNetError – If a is singular or not ‘square’ (in the above sense).

See also

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

mxnet.ndarray.numpy.linalg.tensorsolve(a, b, axes=None)

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 (ndarray) – 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 (ndarray) – 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

Return type:

ndarray, shape Q

Raises:

MXNetError – 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