package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

absolute(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True, signature, extobj)

Calculate the absolute value element-wise.

``np.abs`` is a shorthand for this function.

Parameters ---------- x : array_like Input array. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional This condition is broadcast over the input. At locations where the condition is True, the `out` array will be set to the ufunc result. Elsewhere, the `out` array will retain its original value. Note that if an uninitialized `out` array is created via the default ``out=None``, locations within it where the condition is False will remain uninitialized. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs <ufuncs.kwargs>`.

Returns ------- absolute : ndarray An ndarray containing the absolute value of each element in `x`. For complex input, ``a + ib``, the absolute value is :math:`\sqrt a^2 + b^2 `. This is a scalar if `x` is a scalar.

Examples -------- >>> x = np.array(-1.2, 1.2) >>> np.absolute(x) array( 1.2, 1.2) >>> np.absolute(1.2 + 1j) 1.5620499351813308

Plot the function over ``-10, 10``:

>>> import matplotlib.pyplot as plt

>>> x = np.linspace(start=-10, stop=10, num=101) >>> plt.plot(x, np.absolute(x)) >>> plt.show()

Plot the function over the complex plane:

>>> xx = x + 1j * x:, np.newaxis >>> plt.imshow(np.abs(xx), extent=-10, 10, -10, 10, cmap='gray') >>> plt.show()

Return the maximum of an array or maximum along an axis.

Parameters ---------- a : array_like Input data. axis : None or int or tuple of ints, optional Axis or axes along which to operate. By default, flattened input is used.

.. versionadded:: 1.7.0

If this is a tuple of ints, the maximum is selected over multiple axes, instead of a single axis or all the axes as before. out : ndarray, optional Alternative output array in which to place the result. Must be of the same shape and buffer length as the expected output. See `ufuncs-output-type` for more details.

keepdims : bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

If the default value is passed, then `keepdims` will not be passed through to the `amax` method of sub-classes of `ndarray`, however any non-default value will be. If the sub-class' method does not implement `keepdims` any exceptions will be raised.

initial : scalar, optional The minimum value of an output element. Must be present to allow computation on empty slice. See `~numpy.ufunc.reduce` for details.

.. versionadded:: 1.15.0

where : array_like of bool, optional Elements to compare for the maximum. See `~numpy.ufunc.reduce` for details.

.. versionadded:: 1.17.0

Returns ------- amax : ndarray or scalar Maximum of `a`. If `axis` is None, the result is a scalar value. If `axis` is given, the result is an array of dimension ``a.ndim - 1``.

See Also -------- amin : The minimum value of an array along a given axis, propagating any NaNs. nanmax : The maximum value of an array along a given axis, ignoring any NaNs. maximum : Element-wise maximum of two arrays, propagating any NaNs. fmax : Element-wise maximum of two arrays, ignoring any NaNs. argmax : Return the indices of the maximum values.

nanmin, minimum, fmin

Notes ----- NaN values are propagated, that is if at least one item is NaN, the corresponding max value will be NaN as well. To ignore NaN values (MATLAB behavior), please use nanmax.

Don't use `amax` for element-wise comparison of 2 arrays; when ``a.shape0`` is 2, ``maximum(a0, a1)`` is faster than ``amax(a, axis=0)``.

Examples -------- >>> a = np.arange(4).reshape((2,2)) >>> a array([0, 1], [2, 3]) >>> np.amax(a) # Maximum of the flattened array 3 >>> np.amax(a, axis=0) # Maxima along the first axis array(2, 3) >>> np.amax(a, axis=1) # Maxima along the second axis array(1, 3) >>> np.amax(a, where=False, True, initial=-1, axis=0) array(-1, 3) >>> b = np.arange(5, dtype=float) >>> b2 = np.NaN >>> np.amax(b) nan >>> np.amax(b, where=~np.isnan(b), initial=-1) 4.0 >>> np.nanmax(b) 4.0

You can use an initial value to compute the maximum of an empty slice, or to initialize it to a different value:

>>> np.max([-50], [10], axis=-1, initial=0) array( 0, 10)

Notice that the initial value is used as one of the elements for which the maximum is determined, unlike for the default argument Python's max function, which is only used for empty iterables.

>>> np.max(5, initial=6) 6 >>> max(5, default=6) 5

conjugate(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True, signature, extobj)

Return the complex conjugate, element-wise.

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part.

Parameters ---------- x : array_like Input value. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional This condition is broadcast over the input. At locations where the condition is True, the `out` array will be set to the ufunc result. Elsewhere, the `out` array will retain its original value. Note that if an uninitialized `out` array is created via the default ``out=None``, locations within it where the condition is False will remain uninitialized. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs <ufuncs.kwargs>`.

Returns ------- y : ndarray The complex conjugate of `x`, with same dtype as `y`. This is a scalar if `x` is a scalar.

Notes ----- `conj` is an alias for `conjugate`:

>>> np.conj is np.conjugate True

Examples -------- >>> np.conjugate(1+2j) (1-2j)

>>> x = np.eye(2) + 1j * np.eye(2) >>> np.conjugate(x) array([ 1.-1.j, 0.-0.j], [ 0.-0.j, 1.-1.j])

Compute the hyperbolic matrix cosine.

This routine uses expm to compute the matrix exponentials.

Parameters ---------- A : (N, N) array_like Input array.

Returns ------- coshm : (N, N) ndarray Hyperbolic matrix cosine of `A`

Examples -------- >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([1.0, 3.0], [1.0, 4.0]) >>> c = coshm(a) >>> c array([ 11.24592233, 38.76236492], [ 12.92078831, 50.00828725])

Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))

>>> t = tanhm(a) >>> s = sinhm(a) >>> t - s.dot(np.linalg.inv(c)) array([ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16])

Compute the matrix cosine.

This routine uses expm to compute the matrix exponentials.

Parameters ---------- A : (N, N) array_like Input array

Returns ------- cosm : (N, N) ndarray Matrix cosine of A

Examples -------- >>> from scipy.linalg import expm, sinm, cosm

Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) applied to a matrix:

>>> a = np.array([1.0, 2.0], [-1.0, 3.0]) >>> expm(1j*a) array([ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]) >>> cosm(a) + 1j*sinm(a) array([ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j])

Extract a diagonal or construct a diagonal array.

See the more detailed documentation for ``numpy.diagonal`` if you use this function to extract a diagonal and wish to write to the resulting array; whether it returns a copy or a view depends on what version of numpy you are using.

Parameters ---------- v : array_like If `v` is a 2-D array, return a copy of its `k`-th diagonal. If `v` is a 1-D array, return a 2-D array with `v` on the `k`-th diagonal. k : int, optional Diagonal in question. The default is 0. Use `k>0` for diagonals above the main diagonal, and `k<0` for diagonals below the main diagonal.

Returns ------- out : ndarray The extracted diagonal or constructed diagonal array.

See Also -------- diagonal : Return specified diagonals. diagflat : Create a 2-D array with the flattened input as a diagonal. trace : Sum along diagonals. triu : Upper triangle of an array. tril : Lower triangle of an array.

Examples -------- >>> x = np.arange(9).reshape((3,3)) >>> x array([0, 1, 2], [3, 4, 5], [6, 7, 8])

>>> np.diag(x) array(0, 4, 8) >>> np.diag(x, k=1) array(1, 5) >>> np.diag(x, k=-1) array(3, 7)

>>> np.diag(np.diag(x)) array([0, 0, 0], [0, 4, 0], [0, 0, 8])

dot(a, b, out=None)

Dot product of two arrays. Specifically,

• If both `a` and `b` are 1-D arrays, it is inner product of vectors (without complex conjugation).

• If both `a` and `b` are 2-D arrays, it is matrix multiplication, but using :func:`matmul` or ``a @ b`` is preferred.

• If either `a` or `b` is 0-D (scalar), it is equivalent to :func:`multiply` and using ``numpy.multiply(a, b)`` or ``a * b`` is preferred.

• If `a` is an N-D array and `b` is a 1-D array, it is a sum product over the last axis of `a` and `b`.

• If `a` is an N-D array and `b` is an M-D array (where ``M>=2``), it is a sum product over the last axis of `a` and the second-to-last axis of `b`::

dot(a, b)i,j,k,m = sum(ai,j,: * bk,:,m)

Parameters ---------- a : array_like First argument. b : array_like Second argument. out : ndarray, optional Output argument. This must have the exact kind that would be returned if it was not used. In particular, it must have the right type, must be C-contiguous, and its dtype must be the dtype that would be returned for `dot(a,b)`. This is a performance feature. Therefore, if these conditions are not met, an exception is raised, instead of attempting to be flexible.

Returns ------- output : ndarray Returns the dot product of `a` and `b`. If `a` and `b` are both scalars or both 1-D arrays then a scalar is returned; otherwise an array is returned. If `out` is given, then it is returned.

Raises ------ ValueError If the last dimension of `a` is not the same size as the second-to-last dimension of `b`.

See Also -------- vdot : Complex-conjugating dot product. tensordot : Sum products over arbitrary axes. einsum : Einstein summation convention. matmul : '@' operator as method with out parameter.

Examples -------- >>> np.dot(3, 4) 12

Neither argument is complex-conjugated:

>>> np.dot(2j, 3j, 2j, 3j) (-13+0j)

For 2-D arrays it is the matrix product:

>>> a = [1, 0], [0, 1] >>> b = [4, 1], [2, 2] >>> np.dot(a, b) array([4, 1], [2, 2])

>>> a = np.arange(3*4*5*6).reshape((3,4,5,6)) >>> b = np.arange(3*4*5*6)::-1.reshape((5,4,6,3)) >>> np.dot(a, b)2,3,2,1,2,2 499128 >>> sum(a2,3,2,: * b1,2,:,2) 499128

Compute the matrix exponential using Pade approximation.

Parameters ---------- A : (N, N) array_like or sparse matrix Matrix to be exponentiated.

Returns ------- expm : (N, N) ndarray Matrix exponential of `A`.

References ---------- .. 1 Awad H. Al-Mohy and Nicholas J. Higham (2009) 'A New Scaling and Squaring Algorithm for the Matrix Exponential.' SIAM Journal on Matrix Analysis and Applications. 31 (3). pp. 970-989. ISSN 1095-7162

Examples -------- >>> from scipy.linalg import expm, sinm, cosm

Matrix version of the formula exp(0) = 1:

>>> expm(np.zeros((2,2))) array([ 1., 0.], [ 0., 1.])

Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) applied to a matrix:

>>> a = np.array([1.0, 2.0], [-1.0, 3.0]) >>> expm(1j*a) array([ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]) >>> cosm(a) + 1j*sinm(a) array([ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j])

Relative condition number of the matrix exponential in the Frobenius norm.

Parameters ---------- A : 2-D array_like Square input matrix with shape (N, N). check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

Returns ------- kappa : float The relative condition number of the matrix exponential in the Frobenius norm

Notes ----- A faster estimate for the condition number in the 1-norm has been published but is not yet implemented in SciPy.

.. versionadded:: 0.14.0

See also -------- expm : Compute the exponential of a matrix. expm_frechet : Compute the Frechet derivative of the matrix exponential.

Examples -------- >>> from scipy.linalg import expm_cond >>> A = np.array([-0.3, 0.2, 0.6], [0.6, 0.3, -0.1], [-0.7, 1.2, 0.9]) >>> k = expm_cond(A) >>> k 1.7787805864469866

Frechet derivative of the matrix exponential of A in the direction E.

Parameters ---------- A : (N, N) array_like Matrix of which to take the matrix exponential. E : (N, N) array_like Matrix direction in which to take the Frechet derivative. method : str, optional Choice of algorithm. Should be one of

• `SPS` (default)
• `blockEnlarge`

compute_expm : bool, optional Whether to compute also `expm_A` in addition to `expm_frechet_AE`. Default is True. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

Returns ------- expm_A : ndarray Matrix exponential of A. expm_frechet_AE : ndarray Frechet derivative of the matrix exponential of A in the direction E.

For ``compute_expm = False``, only `expm_frechet_AE` is returned.

See also -------- expm : Compute the exponential of a matrix.

Notes ----- This section describes the available implementations that can be selected by the `method` parameter. The default method is *SPS*.

Method *blockEnlarge* is a naive algorithm.

Method *SPS* is Scaling-Pade-Squaring 1_. It is a sophisticated implementation which should take only about 3/8 as much time as the naive implementation. The asymptotics are the same.

.. versionadded:: 0.13.0

References ---------- .. 1 Awad H. Al-Mohy and Nicholas J. Higham (2009) Computing the Frechet Derivative of the Matrix Exponential, with an application to Condition Number Estimation. SIAM Journal On Matrix Analysis and Applications., 30 (4). pp. 1639-1657. ISSN 1095-7162

Examples -------- >>> import scipy.linalg >>> A = np.random.randn(3, 3) >>> E = np.random.randn(3, 3) >>> expm_A, expm_frechet_AE = scipy.linalg.expm_frechet(A, E) >>> expm_A.shape, expm_frechet_AE.shape ((3, 3), (3, 3))

>>> import scipy.linalg >>> A = np.random.randn(3, 3) >>> E = np.random.randn(3, 3) >>> expm_A, expm_frechet_AE = scipy.linalg.expm_frechet(A, E) >>> M = np.zeros((6, 6)) >>> M:3, :3 = A; M:3, 3: = E; M3:, 3: = A >>> expm_M = scipy.linalg.expm(M) >>> np.allclose(expm_A, expm_M:3, :3) True >>> np.allclose(expm_frechet_AE, expm_M:3, 3:) True

Compute the fractional power of a matrix.

Proceeds according to the discussion in section (6) of 1_.

Parameters ---------- A : (N, N) array_like Matrix whose fractional power to evaluate. t : float Fractional power.

Returns ------- X : (N, N) array_like The fractional power of the matrix.

References ---------- .. 1 Nicholas J. Higham and Lijing lin (2011) 'A Schur-Pade Algorithm for Fractional Powers of a Matrix.' SIAM Journal on Matrix Analysis and Applications, 32 (3). pp. 1056-1078. ISSN 0895-4798

Examples -------- >>> from scipy.linalg import fractional_matrix_power >>> a = np.array([1.0, 3.0], [1.0, 4.0]) >>> b = fractional_matrix_power(a, 0.5) >>> b array([ 0.75592895, 1.13389342], [ 0.37796447, 1.88982237]) >>> np.dot(b, b) # Verify square root array([ 1., 3.], [ 1., 4.])

Evaluate a matrix function specified by a callable.

Returns the value of matrix-valued function ``f`` at `A`. The function ``f`` is an extension of the scalar-valued function `func` to matrices.

Parameters ---------- A : (N, N) array_like Matrix at which to evaluate the function func : callable Callable object that evaluates a scalar function f. Must be vectorized (eg. using vectorize). disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True)

Returns ------- funm : (N, N) ndarray Value of the matrix function specified by func evaluated at `A` errest : float (if disp == False)

1-norm of the estimated error, ||err||_1 / ||A||_1

Examples -------- >>> from scipy.linalg import funm >>> a = np.array([1.0, 3.0], [1.0, 4.0]) >>> funm(a, lambda x: x*x) array([ 4., 15.], [ 5., 19.]) >>> a.dot(a) array([ 4., 15.], [ 5., 19.])

Notes ----- This function implements the general algorithm based on Schur decomposition (Algorithm 9.1.1. in 1_).

If the input matrix is known to be diagonalizable, then relying on the eigendecomposition is likely to be faster. For example, if your matrix is Hermitian, you can do

>>> from scipy.linalg import eigh >>> def funm_herm(a, func, check_finite=False): ... w, v = eigh(a, check_finite=check_finite) ... ## if you further know that your matrix is positive semidefinite, ... ## you can optionally guard against precision errors by doing ... # w = np.maximum(w, 0) ... w = func(w) ... return (v * w).dot(v.conj().T)

References ---------- .. 1 Gene H. Golub, Charles F. van Loan, Matrix Computations 4th ed.

Compute the inverse of a matrix.

Parameters ---------- a : array_like Square matrix to be inverted. overwrite_a : bool, optional Discard data in `a` (may improve performance). Default is False. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

Returns ------- ainv : ndarray Inverse of the matrix `a`.

Raises ------ LinAlgError If `a` is singular. ValueError If `a` is not square, or not 2D.

Examples -------- >>> from scipy import linalg >>> a = np.array([1., 2.], [3., 4.]) >>> linalg.inv(a) array([-2. , 1. ], [ 1.5, -0.5]) >>> np.dot(a, linalg.inv(a)) array([ 1., 0.], [ 0., 1.])

isfinite(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True, signature, extobj)

Test element-wise for finiteness (not infinity or not Not a Number).

The result is returned as a boolean array.

Parameters ---------- x : array_like Input values. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional This condition is broadcast over the input. At locations where the condition is True, the `out` array will be set to the ufunc result. Elsewhere, the `out` array will retain its original value. Note that if an uninitialized `out` array is created via the default ``out=None``, locations within it where the condition is False will remain uninitialized. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs <ufuncs.kwargs>`.

Returns ------- y : ndarray, bool True where ``x`` is not positive infinity, negative infinity, or NaN; false otherwise. This is a scalar if `x` is a scalar.

See Also -------- isinf, isneginf, isposinf, isnan

Notes ----- Not a Number, positive infinity and negative infinity are considered to be non-finite.

NumPy uses the IEEE Standard for Binary Floating-Point for Arithmetic (IEEE 754). This means that Not a Number is not equivalent to infinity. Also that positive infinity is not equivalent to negative infinity. But infinity is equivalent to positive infinity. Errors result if the second argument is also supplied when `x` is a scalar input, or if first and second arguments have different shapes.

Examples -------- >>> np.isfinite(1) True >>> np.isfinite(0) True >>> np.isfinite(np.nan) False >>> np.isfinite(np.inf) False >>> np.isfinite(np.NINF) False >>> np.isfinite(np.log(-1.),1.,np.log(0)) array(False, True, False)

>>> x = np.array(-np.inf, 0., np.inf) >>> y = np.array(2, 2, 2) >>> np.isfinite(x, y) array(0, 1, 0) >>> y array(0, 1, 0)

Khatri-rao product

A column-wise Kronecker product of two matrices

Parameters ---------- a: (n, k) array_like Input array b: (m, k) array_like Input array

Returns ------- c: (n*m, k) ndarray Khatri-rao product of `a` and `b`.

Notes ----- The mathematical definition of the Khatri-Rao product is:

.. math::

(A_j \bigotimes B_j)_j

which is the Kronecker product of every column of A and B, e.g.::

c = np.vstack(np.kron(a[:, k], b[:, k]) for k in range(b.shape[1])).T

See Also -------- kron : Kronecker product

Examples -------- >>> from scipy import linalg >>> a = np.array([1, 2, 3], [4, 5, 6]) >>> b = np.array([3, 4, 5], [6, 7, 8], [2, 3, 9]) >>> linalg.khatri_rao(a, b) array([ 3, 8, 15], [ 6, 14, 24], [ 2, 6, 27], [12, 20, 30], [24, 35, 48], [ 8, 15, 54])

logical_not(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True, signature, extobj)

Compute the truth value of NOT x element-wise.

Parameters ---------- x : array_like Logical NOT is applied to the elements of `x`. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional This condition is broadcast over the input. At locations where the condition is True, the `out` array will be set to the ufunc result. Elsewhere, the `out` array will retain its original value. Note that if an uninitialized `out` array is created via the default ``out=None``, locations within it where the condition is False will remain uninitialized. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs <ufuncs.kwargs>`.

Returns ------- y : bool or ndarray of bool Boolean result with the same shape as `x` of the NOT operation on elements of `x`. This is a scalar if `x` is a scalar.

See Also -------- logical_and, logical_or, logical_xor

Examples -------- >>> np.logical_not(3) False >>> np.logical_not(True, False, 0, 1) array(False, True, True, False)

>>> x = np.arange(5) >>> np.logical_not(x<3) array(False, False, False, True, True)

Compute matrix logarithm.

The matrix logarithm is the inverse of expm: expm(logm(`A`)) == `A`

Parameters ---------- A : (N, N) array_like Matrix whose logarithm to evaluate disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True)

Returns ------- logm : (N, N) ndarray Matrix logarithm of `A` errest : float (if disp == False)

1-norm of the estimated error, ||err||_1 / ||A||_1

References ---------- .. 1 Awad H. Al-Mohy and Nicholas J. Higham (2012) 'Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm.' SIAM Journal on Scientific Computing, 34 (4). C152-C169. ISSN 1095-7197

.. 2 Nicholas J. Higham (2008) 'Functions of Matrices: Theory and Computation' ISBN 978-0-898716-46-7

.. 3 Nicholas J. Higham and Lijing lin (2011) 'A Schur-Pade Algorithm for Fractional Powers of a Matrix.' SIAM Journal on Matrix Analysis and Applications, 32 (3). pp. 1056-1078. ISSN 0895-4798

Examples -------- >>> from scipy.linalg import logm, expm >>> a = np.array([1.0, 3.0], [1.0, 4.0]) >>> b = logm(a) >>> b array([-1.02571087, 2.05142174], [ 0.68380725, 1.02571087]) >>> expm(b) # Verify expm(logm(a)) returns a array([ 1., 3.], [ 1., 4.])

Matrix or vector norm.

This function is able to return one of seven different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the ``ord`` parameter.

Parameters ---------- a : (M,) or (M, N) array_like Input array. If `axis` is None, `a` must be 1D or 2D. ord : non-zero int, inf, -inf, 'fro', optional Order of the norm (see table under ``Notes``). inf means NumPy's `inf` object axis : nt, 2-tuple of ints, None, optional If `axis` is an integer, it specifies the axis of `a` 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 `a` is 1-D) or a matrix norm (when `a` 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 `a`. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

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 -- 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^

/2

`

The ``axis`` and ``keepdims`` arguments are passed directly to ``numpy.linalg.norm`` and are only usable if they are supported by the version of numpy in use.

References ---------- .. 1 G. H. Golub and C. F. Van Loan, *Matrix Computations*, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

Examples -------- >>> from scipy.linalg import norm >>> a = np.arange(9) - 4.0 >>> 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.])

>>> norm(a) 7.745966692414834 >>> norm(b) 7.745966692414834 >>> norm(b, 'fro') 7.745966692414834 >>> norm(a, np.inf) 4 >>> norm(b, np.inf) 9 >>> norm(a, -np.inf) 0 >>> norm(b, -np.inf) 2

>>> norm(a, 1) 20 >>> norm(b, 1) 7 >>> norm(a, -1) -4.6566128774142013e-010 >>> norm(b, -1) 6 >>> norm(a, 2) 7.745966692414834 >>> norm(b, 2) 7.3484692283495345

>>> norm(a, -2) 0 >>> norm(b, -2) 1.8570331885190563e-016 >>> norm(a, 3) 5.8480354764257312 >>> norm(a, -3) 0

Return the product of array elements over a given axis.

Parameters ---------- a : array_like Input data. axis : None or int or tuple of ints, optional Axis or axes along which a product is performed. The default, axis=None, will calculate the product of all the elements in the input array. If axis is negative it counts from the last to the first axis.

.. versionadded:: 1.7.0

If axis is a tuple of ints, a product is performed on all of the axes specified in the tuple instead of a single axis or all the axes as before. dtype : dtype, optional The type of the returned array, as well as of the accumulator in which the elements are multiplied. The dtype of `a` is used by default unless `a` has an integer dtype of less precision than the default platform integer. In that case, if `a` is signed then the platform integer is used while if `a` is unsigned then an unsigned integer of the same precision as the platform integer is used. out : ndarray, optional Alternative output array in which to place the result. It must have the same shape as the expected output, but the type of the output values will be cast if necessary. keepdims : bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

If the default value is passed, then `keepdims` will not be passed through to the `prod` method of sub-classes of `ndarray`, however any non-default value will be. If the sub-class' method does not implement `keepdims` any exceptions will be raised. initial : scalar, optional The starting value for this product. See `~numpy.ufunc.reduce` for details.

.. versionadded:: 1.15.0

where : array_like of bool, optional Elements to include in the product. See `~numpy.ufunc.reduce` for details.

.. versionadded:: 1.17.0

Returns ------- product_along_axis : ndarray, see `dtype` parameter above. An array shaped as `a` but with the specified axis removed. Returns a reference to `out` if specified.

See Also -------- ndarray.prod : equivalent method ufuncs-output-type

Notes ----- Arithmetic is modular when using integer types, and no error is raised on overflow. That means that, on a 32-bit platform:

>>> x = np.array(536870910, 536870910, 536870910, 536870910) >>> np.prod(x) 16 # may vary

The product of an empty array is the neutral element 1:

>>> np.prod() 1.0

Examples -------- By default, calculate the product of all elements:

>>> np.prod(1.,2.) 2.0

Even when the input array is two-dimensional:

>>> np.prod([1.,2.],[3.,4.]) 24.0

But we can also specify the axis over which to multiply:

>>> np.prod([1.,2.],[3.,4.], axis=1) array( 2., 12.)

Or select specific elements to include:

>>> np.prod(1., np.nan, 3., where=True, False, True) 3.0

If the type of `x` is unsigned, then the output type is the unsigned platform integer:

>>> x = np.array(1, 2, 3, dtype=np.uint8) >>> np.prod(x).dtype == np.uint True

If `x` is of a signed integer type, then the output type is the default platform integer:

>>> x = np.array(1, 2, 3, dtype=np.int8) >>> np.prod(x).dtype == int True

You can also start the product with a value other than one:

>>> np.prod(1, 2, initial=5) 10

Return a contiguous flattened array.

A 1-D array, containing the elements of the input, is returned. A copy is made only if needed.

As of NumPy 1.10, the returned array will have the same type as the input array. (for example, a masked array will be returned for a masked array input)

Parameters ---------- a : array_like Input array. The elements in `a` are read in the order specified by `order`, and packed as a 1-D array. order : 'C','F', 'A', 'K', optional

The elements of `a` are read using this index order. 'C' means to index the elements in row-major, C-style order, with the last axis index changing fastest, back to the first axis index changing slowest. 'F' means to index the elements in column-major, Fortran-style order, with the first index changing fastest, and the last index changing slowest. Note that the 'C' and 'F' options take no account of the memory layout of the underlying array, and only refer to the order of axis indexing. 'A' means to read the elements in Fortran-like index order if `a` is Fortran *contiguous* in memory, C-like order otherwise. 'K' means to read the elements in the order they occur in memory, except for reversing the data when strides are negative. By default, 'C' index order is used.

Returns ------- y : array_like y is an array of the same subtype as `a`, with shape ``(a.size,)``. Note that matrices are special cased for backward compatibility, if `a` is a matrix, then y is a 1-D ndarray.

See Also -------- ndarray.flat : 1-D iterator over an array. ndarray.flatten : 1-D array copy of the elements of an array in row-major order. ndarray.reshape : Change the shape of an array without changing its data.

Notes ----- In row-major, C-style order, in two dimensions, the row index varies the slowest, and the column index the quickest. This can be generalized to multiple dimensions, where row-major order implies that the index along the first axis varies slowest, and the index along the last quickest. The opposite holds for column-major, Fortran-style index ordering.

When a view is desired in as many cases as possible, ``arr.reshape(-1)`` may be preferable.

Examples -------- It is equivalent to ``reshape(-1, order=order)``.

>>> x = np.array([1, 2, 3], [4, 5, 6]) >>> np.ravel(x) array(1, 2, 3, 4, 5, 6)

>>> x.reshape(-1) array(1, 2, 3, 4, 5, 6)

>>> np.ravel(x, order='F') array(1, 4, 2, 5, 3, 6)

When ``order`` is 'A', it will preserve the array's 'C' or 'F' ordering:

>>> np.ravel(x.T) array(1, 4, 2, 5, 3, 6) >>> np.ravel(x.T, order='A') array(1, 2, 3, 4, 5, 6)

When ``order`` is 'K', it will preserve orderings that are neither 'C' nor 'F', but won't reverse axes:

>>> a = np.arange(3)::-1; a array(2, 1, 0) >>> a.ravel(order='C') array(2, 1, 0) >>> a.ravel(order='K') array(2, 1, 0)

>>> a = np.arange(12).reshape(2,3,2).swapaxes(1,2); a array([[ 0, 2, 4], [ 1, 3, 5]], [[ 6, 8, 10], [ 7, 9, 11]]) >>> a.ravel(order='C') array( 0, 2, 4, 1, 3, 5, 6, 8, 10, 7, 9, 11) >>> a.ravel(order='K') array( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)

Convert real Schur form to complex Schur form.

Convert a quasi-diagonal real-valued Schur form to the upper-triangular complex-valued Schur form.

Parameters ---------- T : (M, M) array_like Real Schur form of the original array Z : (M, M) array_like Schur transformation matrix check_finite : bool, optional Whether to check that the input arrays contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

Returns ------- T : (M, M) ndarray Complex Schur form of the original array Z : (M, M) ndarray Schur transformation matrix corresponding to the complex form

See Also -------- schur : Schur decomposition of an array

Examples -------- >>> from scipy.linalg import schur, rsf2csf >>> A = np.array([0, 2, 2], [0, 1, 2], [1, 0, 1]) >>> T, Z = schur(A) >>> T array([ 2.65896708, 1.42440458, -1.92933439], [ 0. , -0.32948354, -0.49063704], [ 0. , 1.31178921, -0.32948354]) >>> Z array([0.72711591, -0.60156188, 0.33079564], [0.52839428, 0.79801892, 0.28976765], [0.43829436, 0.03590414, -0.89811411]) >>> T2 , Z2 = rsf2csf(T, Z) >>> T2 array([2.65896708+0.j, -1.64592781+0.743164187j, -1.21516887+1.00660462j], [0.+0.j , -0.32948354+8.02254558e-01j, -0.82115218-2.77555756e-17j], [0.+0.j , 0.+0.j, -0.32948354-0.802254558j]) >>> Z2 array([0.72711591+0.j, 0.28220393-0.31385693j, 0.51319638-0.17258824j], [0.52839428+0.j, 0.24720268+0.41635578j, -0.68079517-0.15118243j], [0.43829436+0.j, -0.76618703+0.01873251j, -0.03063006+0.46857912j])

Compute Schur decomposition of a matrix.

The Schur decomposition is::

A = Z T Z^H

where Z is unitary and T is either upper-triangular, or for real Schur decomposition (output='real'), quasi-upper triangular. In the quasi-triangular form, 2x2 blocks describing complex-valued eigenvalue pairs may extrude from the diagonal.

Parameters ---------- a : (M, M) array_like Matrix to decompose output : 'real', 'complex', optional Construct the real or complex Schur decomposition (for real matrices). lwork : int, optional Work array size. If None or -1, it is automatically computed. overwrite_a : bool, optional Whether to overwrite data in a (may improve performance). sort : None, callable, 'lhp', 'rhp', 'iuc', 'ouc', optional Specifies whether the upper eigenvalues should be sorted. A callable may be passed that, given a eigenvalue, returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True). Alternatively, string parameters may be used::

'lhp' Left-hand plane (x.real < 0.0) 'rhp' Right-hand plane (x.real > 0.0) 'iuc' Inside the unit circle (x*x.conjugate() <= 1.0) 'ouc' Outside the unit circle (x*x.conjugate() > 1.0)

Defaults to None (no sorting). check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

Returns ------- T : (M, M) ndarray Schur form of A. It is real-valued for the real Schur decomposition. Z : (M, M) ndarray An unitary Schur transformation matrix for A. It is real-valued for the real Schur decomposition. sdim : int If and only if sorting was requested, a third return value will contain the number of eigenvalues satisfying the sort condition.

Raises ------ LinAlgError Error raised under three conditions:

1. The algorithm failed due to a failure of the QR algorithm to compute all eigenvalues. 2. If eigenvalue sorting was requested, the eigenvalues could not be reordered due to a failure to separate eigenvalues, usually because of poor conditioning. 3. If eigenvalue sorting was requested, roundoff errors caused the leading eigenvalues to no longer satisfy the sorting condition.

See also -------- rsf2csf : Convert real Schur form to complex Schur form

Examples -------- >>> from scipy.linalg import schur, eigvals >>> A = np.array([0, 2, 2], [0, 1, 2], [1, 0, 1]) >>> T, Z = schur(A) >>> T array([ 2.65896708, 1.42440458, -1.92933439], [ 0. , -0.32948354, -0.49063704], [ 0. , 1.31178921, -0.32948354]) >>> Z array([0.72711591, -0.60156188, 0.33079564], [0.52839428, 0.79801892, 0.28976765], [0.43829436, 0.03590414, -0.89811411])

>>> T2, Z2 = schur(A, output='complex') >>> T2 array([ 2.65896708, -1.22839825+1.32378589j, 0.42590089+1.51937378j], [ 0. , -0.32948354+0.80225456j, -0.59877807+0.56192146j], [ 0. , 0. , -0.32948354-0.80225456j]) >>> eigvals(T2) array(2.65896708, -0.32948354+0.80225456j, -0.32948354-0.80225456j)

An arbitrary custom eig-sorting condition, having positive imaginary part, which is satisfied by only one eigenvalue

>>> T3, Z3, sdim = schur(A, output='complex', sort=lambda x: x.imag > 0) >>> sdim 1

sign(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True, signature, extobj)

Returns an element-wise indication of the sign of a number.

The `sign` function returns ``-1 if x < 0, 0 if x==0, 1 if x > 0``. nan is returned for nan inputs.

For complex inputs, the `sign` function returns ``sign(x.real) + 0j if x.real != 0 else sign(x.imag) + 0j``.

complex(nan, 0) is returned for complex nan inputs.

Parameters ---------- x : array_like Input values. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional This condition is broadcast over the input. At locations where the condition is True, the `out` array will be set to the ufunc result. Elsewhere, the `out` array will retain its original value. Note that if an uninitialized `out` array is created via the default ``out=None``, locations within it where the condition is False will remain uninitialized. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs <ufuncs.kwargs>`.

Returns ------- y : ndarray The sign of `x`. This is a scalar if `x` is a scalar.

Notes ----- There is more than one definition of sign in common use for complex numbers. The definition used here is equivalent to :math:`x/\sqrtx*x` which is different from a common alternative, :math:`x/|x|`.

Examples -------- >>> np.sign(-5., 4.5) array(-1., 1.) >>> np.sign(0) 0 >>> np.sign(5-2j) (1+0j)

Matrix sign function.

Extension of the scalar sign(x) to matrices.

Parameters ---------- A : (N, N) array_like Matrix at which to evaluate the sign function disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True)

Returns ------- signm : (N, N) ndarray Value of the sign function at `A` errest : float (if disp == False)

1-norm of the estimated error, ||err||_1 / ||A||_1

Examples -------- >>> from scipy.linalg import signm, eigvals >>> a = [1,2,3], [1,2,1], [1,1,1] >>> eigvals(a) array( 4.12488542+0.j, -0.76155718+0.j, 0.63667176+0.j) >>> eigvals(signm(a)) array(-1.+0.j, 1.+0.j, 1.+0.j)

Compute the hyperbolic matrix sine.

This routine uses expm to compute the matrix exponentials.

Parameters ---------- A : (N, N) array_like Input array.

Returns ------- sinhm : (N, N) ndarray Hyperbolic matrix sine of `A`

Examples -------- >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([1.0, 3.0], [1.0, 4.0]) >>> s = sinhm(a) >>> s array([ 10.57300653, 39.28826594], [ 13.09608865, 49.86127247])

Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))

>>> t = tanhm(a) >>> c = coshm(a) >>> t - s.dot(np.linalg.inv(c)) array([ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16])

Compute the matrix sine.

This routine uses expm to compute the matrix exponentials.

Parameters ---------- A : (N, N) array_like Input array.

Returns ------- sinm : (N, N) ndarray Matrix sine of `A`

Examples -------- >>> from scipy.linalg import expm, sinm, cosm

Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) applied to a matrix:

>>> a = np.array([1.0, 2.0], [-1.0, 3.0]) >>> expm(1j*a) array([ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]) >>> cosm(a) + 1j*sinm(a) array([ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j])

Solves the linear equation set ``a * x = b`` for the unknown ``x`` for square ``a`` matrix.

If the data matrix is known to be a particular type then supplying the corresponding string to ``assume_a`` key chooses the dedicated solver. The available options are

=================== ======== generic matrix 'gen' symmetric 'sym' hermitian 'her' positive definite 'pos' =================== ========

If omitted, ``'gen'`` is the default structure.

The datatype of the arrays define which solver is called regardless of the values. In other words, even when the complex array entries have precisely zero imaginary parts, the complex solver will be called based on the data type of the array.

Parameters ---------- a : (N, N) array_like Square input data b : (N, NRHS) array_like Input data for the right hand side. sym_pos : bool, optional Assume `a` is symmetric and positive definite. This key is deprecated and assume_a = 'pos' keyword is recommended instead. The functionality is the same. It will be removed in the future. lower : bool, optional If True, only the data contained in the lower triangle of `a`. Default is to use upper triangle. (ignored for ``'gen'``) overwrite_a : bool, optional Allow overwriting data in `a` (may enhance performance). Default is False. overwrite_b : bool, optional Allow overwriting data in `b` (may enhance performance). Default is False. check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. assume_a : str, optional Valid entries are explained above. transposed: bool, optional If True, ``a^T x = b`` for real matrices, raises `NotImplementedError` for complex matrices (only for True).

Returns ------- x : (N, NRHS) ndarray The solution array.

Raises ------ ValueError If size mismatches detected or input a is not square. LinAlgError If the matrix is singular. LinAlgWarning If an ill-conditioned input a is detected. NotImplementedError If transposed is True and input a is a complex matrix.

Examples -------- Given `a` and `b`, solve for `x`:

>>> a = np.array([3, 2, 0], [1, -1, 0], [0, 5, 1]) >>> b = np.array(2, 4, -1) >>> from scipy import linalg >>> x = linalg.solve(a, b) >>> x array( 2., -2., 9.) >>> np.dot(a, x) == b array( True, True, True, dtype=bool)

Notes ----- If the input b matrix is a 1-D array with N elements, when supplied together with an NxN input a, it is assumed as a valid column vector despite the apparent size mismatch. This is compatible with the numpy.dot() behavior and the returned result is still 1-D array.

The generic, symmetric, hermitian and positive definite solutions are obtained via calling ?GESV, ?SYSV, ?HESV, and ?POSV routines of LAPACK respectively.

Matrix square root.

Parameters ---------- A : (N, N) array_like Matrix whose square root to evaluate disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True) blocksize : integer, optional If the blocksize is not degenerate with respect to the size of the input array, then use a blocked algorithm. (Default: 64)

Returns ------- sqrtm : (N, N) ndarray Value of the sqrt function at `A`

errest : float (if disp == False)

Frobenius norm of the estimated error, ||err||_F / ||A||_F

References ---------- .. 1 Edvin Deadman, Nicholas J. Higham, Rui Ralha (2013) 'Blocked Schur Algorithms for Computing the Matrix Square Root, Lecture Notes in Computer Science, 7782. pp. 171-182.

Examples -------- >>> from scipy.linalg import sqrtm >>> a = np.array([1.0, 3.0], [1.0, 4.0]) >>> r = sqrtm(a) >>> r array([ 0.75592895, 1.13389342], [ 0.37796447, 1.88982237]) >>> r.dot(r) array([ 1., 3.], [ 1., 4.])

Singular Value Decomposition.

Factorizes the matrix `a` into two unitary matrices ``U`` and ``Vh``, and a 1-D array ``s`` of singular values (real, non-negative) such that ``a == U @ S @ Vh``, where ``S`` is a suitably shaped matrix of zeros with main diagonal ``s``.

Parameters ---------- a : (M, N) array_like Matrix to decompose. full_matrices : bool, optional If True (default), `U` and `Vh` are of shape ``(M, M)``, ``(N, N)``. If False, the shapes are ``(M, K)`` and ``(K, N)``, where ``K = min(M, N)``. compute_uv : bool, optional Whether to compute also ``U`` and ``Vh`` in addition to ``s``. Default is True. overwrite_a : bool, optional Whether to overwrite `a`; may improve performance. Default is False. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. lapack_driver : 'gesdd', 'gesvd', optional Whether to use the more efficient divide-and-conquer approach (``'gesdd'``) or general rectangular approach (``'gesvd'``) to compute the SVD. MATLAB and Octave use the ``'gesvd'`` approach. Default is ``'gesdd'``.

.. versionadded:: 0.18

Returns ------- U : ndarray Unitary matrix having left singular vectors as columns. Of shape ``(M, M)`` or ``(M, K)``, depending on `full_matrices`. s : ndarray The singular values, sorted in non-increasing order. Of shape (K,), with ``K = min(M, N)``. Vh : ndarray Unitary matrix having right singular vectors as rows. Of shape ``(N, N)`` or ``(K, N)`` depending on `full_matrices`.

For ``compute_uv=False``, only ``s`` is returned.

Raises ------ LinAlgError If SVD computation does not converge.

See also -------- svdvals : Compute singular values of a matrix. diagsvd : Construct the Sigma matrix, given the vector s.

Examples -------- >>> from scipy import linalg >>> m, n = 9, 6 >>> a = np.random.randn(m, n) + 1.j*np.random.randn(m, n) >>> U, s, Vh = linalg.svd(a) >>> U.shape, s.shape, Vh.shape ((9, 9), (6,), (6, 6))

Reconstruct the original matrix from the decomposition:

>>> sigma = np.zeros((m, n)) >>> for i in range(min(m, n)): ... sigmai, i = si >>> a1 = np.dot(U, np.dot(sigma, Vh)) >>> np.allclose(a, a1) True

Alternatively, use ``full_matrices=False`` (notice that the shape of ``U`` is then ``(m, n)`` instead of ``(m, m)``):

>>> U, s, Vh = linalg.svd(a, full_matrices=False) >>> U.shape, s.shape, Vh.shape ((9, 6), (6,), (6, 6)) >>> S = np.diag(s) >>> np.allclose(a, np.dot(U, np.dot(S, Vh))) True

>>> s2 = linalg.svd(a, compute_uv=False) >>> np.allclose(s, s2) True

Compute the hyperbolic matrix tangent.

This routine uses expm to compute the matrix exponentials.

Parameters ---------- A : (N, N) array_like Input array

Returns ------- tanhm : (N, N) ndarray Hyperbolic matrix tangent of `A`

Examples -------- >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([1.0, 3.0], [1.0, 4.0]) >>> t = tanhm(a) >>> t array([ 0.3428582 , 0.51987926], [ 0.17329309, 0.86273746])

Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))

>>> s = sinhm(a) >>> c = coshm(a) >>> t - s.dot(np.linalg.inv(c)) array([ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16])

Compute the matrix tangent.

This routine uses expm to compute the matrix exponentials.

Parameters ---------- A : (N, N) array_like Input array.

Returns ------- tanm : (N, N) ndarray Matrix tangent of `A`

Examples -------- >>> from scipy.linalg import tanm, sinm, cosm >>> a = np.array([1.0, 3.0], [1.0, 4.0]) >>> t = tanm(a) >>> t array([ -2.00876993, -8.41880636], [ -2.80626879, -10.42757629])

Verify tanm(a) = sinm(a).dot(inv(cosm(a)))

>>> s = sinm(a) >>> c = cosm(a) >>> s.dot(np.linalg.inv(c)) array([ -2.00876993, -8.41880636], [ -2.80626879, -10.42757629])

Reverse or permute the axes of an array; returns the modified array.

For an array a with two axes, transpose(a) gives the matrix transpose.

Parameters ---------- a : array_like Input array. axes : tuple or list of ints, optional If specified, it must be a tuple or list which contains a permutation of 0,1,..,N-1 where N is the number of axes of a. The i'th axis of the returned array will correspond to the axis numbered ``axesi`` of the input. If not specified, defaults to ``range(a.ndim)::-1``, which reverses the order of the axes.

Returns ------- p : ndarray `a` with its axes permuted. A view is returned whenever possible.

See Also -------- moveaxis argsort

Notes ----- Use `transpose(a, argsort(axes))` to invert the transposition of tensors when using the `axes` keyword argument.

Transposing a 1-D array returns an unchanged view of the original array.

Examples -------- >>> x = np.arange(4).reshape((2,2)) >>> x array([0, 1], [2, 3])

>>> np.transpose(x) array([0, 2], [1, 3])

>>> x = np.ones((1, 2, 3)) >>> np.transpose(x, (1, 0, 2)).shape (2, 1, 3)

Make a copy of a matrix with elements below the kth diagonal zeroed.

Parameters ---------- m : array_like Matrix whose elements to return k : int, optional Diagonal below which to zero elements. `k` == 0 is the main diagonal, `k` < 0 subdiagonal and `k` > 0 superdiagonal.

Returns ------- triu : ndarray Return matrix with zeroed elements below the kth diagonal and has same shape and type as `m`.

Examples -------- >>> from scipy.linalg import triu >>> triu([1,2,3],[4,5,6],[7,8,9],[10,11,12], -1) array([ 1, 2, 3], [ 4, 5, 6], [ 0, 8, 9], [ 0, 0, 12])