package scipy

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

array(object, dtype=None, copy=True, order='K', subok=False, ndmin=0)

Create an array.

Parameters ---------- object : array_like An array, any object exposing the array interface, an object whose __array__ method returns an array, or any (nested) sequence. dtype : data-type, optional The desired data-type for the array. If not given, then the type will be determined as the minimum type required to hold the objects in the sequence. copy : bool, optional If true (default), then the object is copied. Otherwise, a copy will only be made if __array__ returns a copy, if obj is a nested sequence, or if a copy is needed to satisfy any of the other requirements (`dtype`, `order`, etc.). order : 'K', 'A', 'C', 'F', optional Specify the memory layout of the array. If object is not an array, the newly created array will be in C order (row major) unless 'F' is specified, in which case it will be in Fortran order (column major). If object is an array the following holds.

===== ========= =================================================== order no copy copy=True ===== ========= =================================================== 'K' unchanged F & C order preserved, otherwise most similar order 'A' unchanged F order if input is F and not C, otherwise C order 'C' C order C order 'F' F order F order ===== ========= ===================================================

When ``copy=False`` and a copy is made for other reasons, the result is the same as if ``copy=True``, with some exceptions for `A`, see the Notes section. The default order is 'K'. subok : bool, optional If True, then sub-classes will be passed-through, otherwise the returned array will be forced to be a base-class array (default). ndmin : int, optional Specifies the minimum number of dimensions that the resulting array should have. Ones will be pre-pended to the shape as needed to meet this requirement.

Returns ------- out : ndarray An array object satisfying the specified requirements.

See Also -------- empty_like : Return an empty array with shape and type of input. ones_like : Return an array of ones with shape and type of input. zeros_like : Return an array of zeros with shape and type of input. full_like : Return a new array with shape of input filled with value. empty : Return a new uninitialized array. ones : Return a new array setting values to one. zeros : Return a new array setting values to zero. full : Return a new array of given shape filled with value.

Notes ----- When order is 'A' and `object` is an array in neither 'C' nor 'F' order, and a copy is forced by a change in dtype, then the order of the result is not necessarily 'C' as expected. This is likely a bug.

Examples -------- >>> np.array(1, 2, 3) array(1, 2, 3)

Upcasting:

>>> np.array(1, 2, 3.0) array( 1., 2., 3.)

More than one dimension:

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

Minimum dimensions 2:

>>> np.array(1, 2, 3, ndmin=2) array([1, 2, 3])

Type provided:

>>> np.array(1, 2, 3, dtype=complex) array( 1.+0.j, 2.+0.j, 3.+0.j)

Data-type consisting of more than one element:

>>> x = np.array((1,2),(3,4),dtype=('a','<i4'),('b','<i4')) >>> x'a' array(1, 3)

Creating an array from sub-classes:

>>> np.array(np.mat('1 2; 3 4')) array([1, 2], [3, 4])

>>> np.array(np.mat('1 2; 3 4'), subok=True) matrix([1, 2], [3, 4])

Convert the input to an array.

Parameters ---------- a : array_like Input data, in any form that can be converted to an array. This includes lists, lists of tuples, tuples, tuples of tuples, tuples of lists and ndarrays. dtype : data-type, optional By default, the data-type is inferred from the input data. order : 'C', 'F', optional Whether to use row-major (C-style) or column-major (Fortran-style) memory representation. Defaults to 'C'.

Returns ------- out : ndarray Array interpretation of `a`. No copy is performed if the input is already an ndarray with matching dtype and order. If `a` is a subclass of ndarray, a base class ndarray is returned.

See Also -------- asanyarray : Similar function which passes through subclasses. ascontiguousarray : Convert input to a contiguous array. asfarray : Convert input to a floating point ndarray. asfortranarray : Convert input to an ndarray with column-major memory order. asarray_chkfinite : Similar function which checks input for NaNs and Infs. fromiter : Create an array from an iterator. fromfunction : Construct an array by executing a function on grid positions.

Examples -------- Convert a list into an array:

>>> a = 1, 2 >>> np.asarray(a) array(1, 2)

Existing arrays are not copied:

>>> a = np.array(1, 2) >>> np.asarray(a) is a True

If `dtype` is set, array is copied only if dtype does not match:

>>> a = np.array(1, 2, dtype=np.float32) >>> np.asarray(a, dtype=np.float32) is a True >>> np.asarray(a, dtype=np.float64) is a False

Contrary to `asanyarray`, ndarray subclasses are not passed through:

>>> issubclass(np.recarray, np.ndarray) True >>> a = np.array((1.0, 2), (3.0, 4), dtype='f4,i4').view(np.recarray) >>> np.asarray(a) is a False >>> np.asanyarray(a) is a True

Convert the input to an array, checking for NaNs or Infs.

Parameters ---------- a : array_like Input data, in any form that can be converted to an array. This includes lists, lists of tuples, tuples, tuples of tuples, tuples of lists and ndarrays. Success requires no NaNs or Infs. dtype : data-type, optional By default, the data-type is inferred from the input data. order : 'C', 'F', optional Whether to use row-major (C-style) or column-major (Fortran-style) memory representation. Defaults to 'C'.

Returns ------- out : ndarray Array interpretation of `a`. No copy is performed if the input is already an ndarray. If `a` is a subclass of ndarray, a base class ndarray is returned.

Raises ------ ValueError Raises ValueError if `a` contains NaN (Not a Number) or Inf (Infinity).

See Also -------- asarray : Create and array. asanyarray : Similar function which passes through subclasses. ascontiguousarray : Convert input to a contiguous array. asfarray : Convert input to a floating point ndarray. asfortranarray : Convert input to an ndarray with column-major memory order. fromiter : Create an array from an iterator. fromfunction : Construct an array by executing a function on grid positions.

Examples -------- Convert a list into an array. If all elements are finite ``asarray_chkfinite`` is identical to ``asarray``.

>>> a = 1, 2 >>> np.asarray_chkfinite(a, dtype=float) array(1., 2.)

Raises ValueError if array_like contains Nans or Infs.

>>> a = 1, 2, np.inf >>> try: ... np.asarray_chkfinite(a) ... except ValueError: ... print('ValueError') ... ValueError

None

Compute eigenvalues from an ordinary or generalized eigenvalue problem.

Find eigenvalues of a general matrix::

a vr:,i = wi b vr:,i

Parameters ---------- a : (M, M) array_like A complex or real matrix whose eigenvalues and eigenvectors will be computed. b : (M, M) array_like, optional Right-hand side matrix in a generalized eigenvalue problem. If omitted, identity matrix is assumed. overwrite_a : bool, optional Whether to overwrite data in a (may improve performance) 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. homogeneous_eigvals : bool, optional If True, return the eigenvalues in homogeneous coordinates. In this case ``w`` is a (2, M) array so that::

w1,i a vr:,i = w0,i b vr:,i

Default is False.

Returns ------- w : (M,) or (2, M) double or complex ndarray The eigenvalues, each repeated according to its multiplicity but not in any specific order. The shape is (M,) unless ``homogeneous_eigvals=True``.

Raises ------ LinAlgError If eigenvalue computation does not converge

See Also -------- eig : eigenvalues and right eigenvectors of general arrays. eigvalsh : eigenvalues of symmetric or Hermitian arrays eigvals_banded : eigenvalues for symmetric/Hermitian band matrices eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal matrices

Examples -------- >>> from scipy import linalg >>> a = np.array([0., -1.], [1., 0.]) >>> linalg.eigvals(a) array(0.+1.j, 0.-1.j)

>>> b = np.array([0., 1.], [1., 1.]) >>> linalg.eigvals(a, b) array( 1.+0.j, -1.+0.j)

>>> a = np.array([3., 0., 0.], [0., 8., 0.], [0., 0., 7.]) >>> linalg.eigvals(a, homogeneous_eigvals=True) array([3.+0.j, 8.+0.j, 7.+0.j], [1.+0.j, 1.+0.j, 1.+0.j])

Return available LAPACK function objects from names.

Arrays are used to determine the optimal prefix of LAPACK routines.

Parameters ---------- names : str or sequence of str Name(s) of LAPACK functions without type prefix.

arrays : sequence of ndarrays, optional Arrays can be given to determine optimal prefix of LAPACK routines. If not given, double-precision routines will be used, otherwise the most generic type in arrays will be used.

dtype : str or dtype, optional Data-type specifier. Not used if `arrays` is non-empty.

Returns ------- funcs : list List containing the found function(s).

Notes ----- This routine automatically chooses between Fortran/C interfaces. Fortran code is used whenever possible for arrays with column major order. In all other cases, C code is preferred.

In LAPACK, the naming convention is that all functions start with a type prefix, which depends on the type of the principal matrix. These can be one of 's', 'd', 'c', 'z' for the numpy types float32, float64, complex64, complex128 respectively, and are stored in attribute ``typecode`` of the returned functions.

Examples -------- Suppose we would like to use '?lange' routine which computes the selected norm of an array. We pass our array in order to get the correct 'lange' flavor.

>>> import scipy.linalg as LA >>> a = np.random.rand(3,2) >>> x_lange = LA.get_lapack_funcs('lange', (a,)) >>> x_lange.typecode 'd' >>> x_lange = LA.get_lapack_funcs('lange',(a*1j,)) >>> x_lange.typecode 'z'

Several LAPACK routines work best when its internal WORK array has the optimal size (big enough for fast computation and small enough to avoid waste of memory). This size is determined also by a dedicated query to the function which is often wrapped as a standalone function and commonly denoted as ``###_lwork``. Below is an example for ``?sysv``

>>> import scipy.linalg as LA >>> a = np.random.rand(1000,1000) >>> b = np.random.rand(1000,1)*1j >>> # We pick up zsysv and zsysv_lwork due to b array ... xsysv, xlwork = LA.get_lapack_funcs(('sysv', 'sysv_lwork'), (a, b)) >>> opt_lwork, _ = xlwork(a.shape0) # returns a complex for 'z' prefix >>> udut, ipiv, x, info = xsysv(a, b, lwork=int(opt_lwork.real))

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, unless `ord` is None. If both `axis` and `ord` are None, the 2-norm of ``x.ravel`` will be returned. ord : non-zero int, inf, -inf, 'fro', 'nuc', optional Order of the norm (see table under ``Notes``). inf means numpy's `inf` object. The default is None. axis : None, int, 2-tuple of ints, 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. The default is None.

.. versionadded:: 1.8.0

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^

/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, ..., 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) 0.0 >>> LA.norm(b, -2) 1.8570331885190563e-016 # may vary >>> LA.norm(a, 3) 5.8480354764257312 # may vary >>> LA.norm(a, -3) 0.0

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(m0, :, :), LA.norm(m1, :, :) (3.7416573867739413, 11.224972160321824)

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