package scipy

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

Find a root of a function, using (extended) Anderson mixing.

The Jacobian is formed by for a 'best' solution in the space spanned by last `M` vectors. As a result, only a MxM matrix inversions and MxN multiplications are required. Ey_

Parameters ---------- F : function(x) -> f Function whose root to find; should take and return an array-like object. xin : array_like Initial guess for the solution alpha : float, optional Initial guess for the Jacobian is (-1/alpha). M : float, optional Number of previous vectors to retain. Defaults to 5. w0 : float, optional Regularization parameter for numerical stability. Compared to unity, good values of the order of 0.01. iter : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose : bool, optional Print status to stdout on every iteration. maxiter : int, optional Maximum number of iterations to make. If more are needed to meet convergence, `NoConvergence` is raised. f_tol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol : float, optional Relative tolerance for the residual. If omitted, not used. x_tol : float, optional Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used. x_rtol : float, optional Relative minimum step size. If omitted, not used. tol_norm : function(vector) -> scalar, optional Norm to use in convergence check. Default is the maximum norm. line_search : None, 'armijo' (default), 'wolfe', optional Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'. callback : function, optional Optional callback function. It is called on every iteration as ``callback(x, f)`` where `x` is the current solution and `f` the corresponding residual.

Returns ------- sol : ndarray An array (of similar array type as `x0`) containing the final solution.

Raises ------ NoConvergence When a solution was not found.

See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method=='anderson'`` in particular.

References ---------- .. Ey V. Eyert, J. Comp. Phys., 124, 271 (1996).

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 given object to one suitable for use as a Jacobian.

Find a root of a function, using Broyden's first Jacobian approximation.

This method is also known as \'Broyden's good method\'.

Parameters ---------- F : function(x) -> f Function whose root to find; should take and return an array-like object. xin : array_like Initial guess for the solution alpha : float, optional Initial guess for the Jacobian is ``(-1/alpha)``. reduction_method : str or tuple, optional Method used in ensuring that the rank of the Broyden matrix stays low. Can either be a string giving the name of the method, or a tuple of the form ``(method, param1, param2, ...)`` that gives the name of the method and values for additional parameters.

Methods available:

• ``restart``: drop all matrix columns. Has no extra parameters.
• ``simple``: drop oldest matrix column. Has no extra parameters.
• ``svd``: keep only the most significant SVD components. Takes an extra parameter, ``to_retain``, which determines the number of SVD components to retain when rank reduction is done. Default is ``max_rank - 2``.

max_rank : int, optional Maximum rank for the Broyden matrix. Default is infinity (ie., no rank reduction). iter : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose : bool, optional Print status to stdout on every iteration. maxiter : int, optional Maximum number of iterations to make. If more are needed to meet convergence, `NoConvergence` is raised. f_tol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol : float, optional Relative tolerance for the residual. If omitted, not used. x_tol : float, optional Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used. x_rtol : float, optional Relative minimum step size. If omitted, not used. tol_norm : function(vector) -> scalar, optional Norm to use in convergence check. Default is the maximum norm. line_search : None, 'armijo' (default), 'wolfe', optional Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'. callback : function, optional Optional callback function. It is called on every iteration as ``callback(x, f)`` where `x` is the current solution and `f` the corresponding residual.

Returns ------- sol : ndarray An array (of similar array type as `x0`) containing the final solution.

Raises ------ NoConvergence When a solution was not found.

See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method=='broyden1'`` in particular.

Notes ----- This algorithm implements the inverse Jacobian Quasi-Newton update

.. math:: H_+ = H + (dx - H df) dx^\dagger H / ( dx^\dagger H df)

which corresponds to Broyden's first Jacobian update

.. math:: J_+ = J + (df - J dx) dx^\dagger / dx^\dagger dx

References ---------- .. 1 B.A. van der Rotten, PhD thesis, \'A limited memory Broyden method to solve high-dimensional systems of nonlinear equations\'. Mathematisch Instituut, Universiteit Leiden, The Netherlands (2003).

https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf

Find a root of a function, using Broyden's second Jacobian approximation.

This method is also known as 'Broyden's bad method'.

Parameters ---------- F : function(x) -> f Function whose root to find; should take and return an array-like object. xin : array_like Initial guess for the solution alpha : float, optional Initial guess for the Jacobian is ``(-1/alpha)``. reduction_method : str or tuple, optional Method used in ensuring that the rank of the Broyden matrix stays low. Can either be a string giving the name of the method, or a tuple of the form ``(method, param1, param2, ...)`` that gives the name of the method and values for additional parameters.

Methods available:

• ``restart``: drop all matrix columns. Has no extra parameters.
• ``simple``: drop oldest matrix column. Has no extra parameters.
• ``svd``: keep only the most significant SVD components. Takes an extra parameter, ``to_retain``, which determines the number of SVD components to retain when rank reduction is done. Default is ``max_rank - 2``.

max_rank : int, optional Maximum rank for the Broyden matrix. Default is infinity (ie., no rank reduction). iter : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose : bool, optional Print status to stdout on every iteration. maxiter : int, optional Maximum number of iterations to make. If more are needed to meet convergence, `NoConvergence` is raised. f_tol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol : float, optional Relative tolerance for the residual. If omitted, not used. x_tol : float, optional Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used. x_rtol : float, optional Relative minimum step size. If omitted, not used. tol_norm : function(vector) -> scalar, optional Norm to use in convergence check. Default is the maximum norm. line_search : None, 'armijo' (default), 'wolfe', optional Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'. callback : function, optional Optional callback function. It is called on every iteration as ``callback(x, f)`` where `x` is the current solution and `f` the corresponding residual.

Returns ------- sol : ndarray An array (of similar array type as `x0`) containing the final solution.

Raises ------ NoConvergence When a solution was not found.

See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method=='broyden2'`` in particular.

Notes ----- This algorithm implements the inverse Jacobian Quasi-Newton update

.. math:: H_+ = H + (dx - H df) df^\dagger / ( df^\dagger df)

corresponding to Broyden's second method.

References ---------- .. 1 B.A. van der Rotten, PhD thesis, 'A limited memory Broyden method to solve high-dimensional systems of nonlinear equations'. Mathematisch Instituut, Universiteit Leiden, The Netherlands (2003).

https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf

None

Find a root of a function, using diagonal Broyden Jacobian approximation.

The Jacobian approximation is derived from previous iterations, by retaining only the diagonal of Broyden matrices.

.. warning::

This algorithm may be useful for specific problems, but whether it will work may depend strongly on the problem.

Parameters ---------- F : function(x) -> f Function whose root to find; should take and return an array-like object. xin : array_like Initial guess for the solution alpha : float, optional Initial guess for the Jacobian is (-1/alpha). iter : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose : bool, optional Print status to stdout on every iteration. maxiter : int, optional Maximum number of iterations to make. If more are needed to meet convergence, `NoConvergence` is raised. f_tol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol : float, optional Relative tolerance for the residual. If omitted, not used. x_tol : float, optional Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used. x_rtol : float, optional Relative minimum step size. If omitted, not used. tol_norm : function(vector) -> scalar, optional Norm to use in convergence check. Default is the maximum norm. line_search : None, 'armijo' (default), 'wolfe', optional Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'. callback : function, optional Optional callback function. It is called on every iteration as ``callback(x, f)`` where `x` is the current solution and `f` the corresponding residual.

Returns ------- sol : ndarray An array (of similar array type as `x0`) containing the final solution.

Raises ------ NoConvergence When a solution was not found.

See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method=='diagbroyden'`` in particular.

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

Find a root of a function, using a tuned diagonal Jacobian approximation.

The Jacobian matrix is diagonal and is tuned on each iteration.

.. warning::

This algorithm may be useful for specific problems, but whether it will work may depend strongly on the problem.

See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method=='excitingmixing'`` in particular.

Parameters ---------- F : function(x) -> f Function whose root to find; should take and return an array-like object. xin : array_like Initial guess for the solution alpha : float, optional Initial Jacobian approximation is (-1/alpha). alphamax : float, optional The entries of the diagonal Jacobian are kept in the range ``alpha, alphamax``. iter : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose : bool, optional Print status to stdout on every iteration. maxiter : int, optional Maximum number of iterations to make. If more are needed to meet convergence, `NoConvergence` is raised. f_tol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol : float, optional Relative tolerance for the residual. If omitted, not used. x_tol : float, optional Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used. x_rtol : float, optional Relative minimum step size. If omitted, not used. tol_norm : function(vector) -> scalar, optional Norm to use in convergence check. Default is the maximum norm. line_search : None, 'armijo' (default), 'wolfe', optional Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'. callback : function, optional Optional callback function. It is called on every iteration as ``callback(x, f)`` where `x` is the current solution and `f` the corresponding residual.

Returns ------- sol : ndarray An array (of similar array type as `x0`) containing the final solution.

Raises ------ NoConvergence When a solution was not found.

Return available BLAS function objects from names.

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

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

arrays : sequence of ndarrays, optional Arrays can be given to determine optimal prefix of BLAS 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 BLAS, 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. The code and the dtype are stored in attributes `typecode` and `dtype` of the returned functions.

Examples -------- >>> import scipy.linalg as LA >>> a = np.random.rand(3,2) >>> x_gemv = LA.get_blas_funcs('gemv', (a,)) >>> x_gemv.typecode 'd' >>> x_gemv = LA.get_blas_funcs('gemv',(a*1j,)) >>> x_gemv.typecode 'z'

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 2-dimensional.

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.])

Find a root of a function, using a scalar Jacobian approximation.

.. warning::

This algorithm may be useful for specific problems, but whether it will work may depend strongly on the problem.

Parameters ---------- F : function(x) -> f Function whose root to find; should take and return an array-like object. xin : array_like Initial guess for the solution alpha : float, optional The Jacobian approximation is (-1/alpha). iter : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose : bool, optional Print status to stdout on every iteration. maxiter : int, optional Maximum number of iterations to make. If more are needed to meet convergence, `NoConvergence` is raised. f_tol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol : float, optional Relative tolerance for the residual. If omitted, not used. x_tol : float, optional Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used. x_rtol : float, optional Relative minimum step size. If omitted, not used. tol_norm : function(vector) -> scalar, optional Norm to use in convergence check. Default is the maximum norm. line_search : None, 'armijo' (default), 'wolfe', optional Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'. callback : function, optional Optional callback function. It is called on every iteration as ``callback(x, f)`` where `x` is the current solution and `f` the corresponding residual.

Returns ------- sol : ndarray An array (of similar array type as `x0`) containing the final solution.

Raises ------ NoConvergence When a solution was not found.

See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method=='linearmixing'`` in particular.

None

Find a root of a function, using Krylov approximation for inverse Jacobian.

This method is suitable for solving large-scale problems.

Parameters ---------- F : function(x) -> f Function whose root to find; should take and return an array-like object. xin : array_like Initial guess for the solution rdiff : float, optional Relative step size to use in numerical differentiation. method : 'lgmres', 'gmres', 'bicgstab', 'cgs', 'minres' or function Krylov method to use to approximate the Jacobian. Can be a string, or a function implementing the same interface as the iterative solvers in `scipy.sparse.linalg`.

The default is `scipy.sparse.linalg.lgmres`. inner_M : LinearOperator or InverseJacobian Preconditioner for the inner Krylov iteration. Note that you can use also inverse Jacobians as (adaptive) preconditioners. For example,

>>> from scipy.optimize.nonlin import BroydenFirst, KrylovJacobian >>> from scipy.optimize.nonlin import InverseJacobian >>> jac = BroydenFirst() >>> kjac = KrylovJacobian(inner_M=InverseJacobian(jac))

If the preconditioner has a method named 'update', it will be called as ``update(x, f)`` after each nonlinear step, with ``x`` giving the current point, and ``f`` the current function value. inner_tol, inner_maxiter, ... Parameters to pass on to the \'inner\' Krylov solver. See `scipy.sparse.linalg.gmres` for details. outer_k : int, optional Size of the subspace kept across LGMRES nonlinear iterations. See `scipy.sparse.linalg.lgmres` for details. iter : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose : bool, optional Print status to stdout on every iteration. maxiter : int, optional Maximum number of iterations to make. If more are needed to meet convergence, `NoConvergence` is raised. f_tol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol : float, optional Relative tolerance for the residual. If omitted, not used. x_tol : float, optional Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used. x_rtol : float, optional Relative minimum step size. If omitted, not used. tol_norm : function(vector) -> scalar, optional Norm to use in convergence check. Default is the maximum norm. line_search : None, 'armijo' (default), 'wolfe', optional Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'. callback : function, optional Optional callback function. It is called on every iteration as ``callback(x, f)`` where `x` is the current solution and `f` the corresponding residual.

Returns ------- sol : ndarray An array (of similar array type as `x0`) containing the final solution.

Raises ------ NoConvergence When a solution was not found.

See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method=='krylov'`` in particular. scipy.sparse.linalg.gmres scipy.sparse.linalg.lgmres

Notes ----- This function implements a Newton-Krylov solver. The basic idea is to compute the inverse of the Jacobian with an iterative Krylov method. These methods require only evaluating the Jacobian-vector products, which are conveniently approximated by a finite difference:

.. math:: J v \approx (f(x + \omega*v/|v|) - f(x)) / \omega

Due to the use of iterative matrix inverses, these methods can deal with large nonlinear problems.

SciPy's `scipy.sparse.linalg` module offers a selection of Krylov solvers to choose from. The default here is `lgmres`, which is a variant of restarted GMRES iteration that reuses some of the information obtained in the previous Newton steps to invert Jacobians in subsequent steps.

For a review on Newton-Krylov methods, see for example 1_, and for the LGMRES sparse inverse method, see 2_.

References ---------- .. 1 D.A. Knoll and D.E. Keyes, J. Comp. Phys. 193, 357 (2004). :doi:`10.1016/j.jcp.2003.08.010` .. 2 A.H. Baker and E.R. Jessup and T. Manteuffel, SIAM J. Matrix Anal. Appl. 26, 962 (2005). :doi:`10.1137/S0895479803422014`

Find a root of a function, in a way suitable for large-scale problems.

Parameters ---------- F : function(x) -> f Function whose root to find; should take and return an array-like object. xin : array_like Initial guess for the solution jacobian : Jacobian A Jacobian approximation: `Jacobian` object or something that `asjacobian` can transform to one. Alternatively, a string specifying which of the builtin Jacobian approximations to use:

krylov, broyden1, broyden2, anderson diagbroyden, linearmixing, excitingmixing

iter : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose : bool, optional Print status to stdout on every iteration. maxiter : int, optional Maximum number of iterations to make. If more are needed to meet convergence, `NoConvergence` is raised. f_tol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol : float, optional Relative tolerance for the residual. If omitted, not used. x_tol : float, optional Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used. x_rtol : float, optional Relative minimum step size. If omitted, not used. tol_norm : function(vector) -> scalar, optional Norm to use in convergence check. Default is the maximum norm. line_search : None, 'armijo' (default), 'wolfe', optional Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'. callback : function, optional Optional callback function. It is called on every iteration as ``callback(x, f)`` where `x` is the current solution and `f` the corresponding residual.

Returns ------- sol : ndarray An array (of similar array type as `x0`) containing the final solution.

Raises ------ NoConvergence When a solution was not found. full_output : bool If true, returns a dictionary `info` containing convergence information. raise_exception : bool If True, a `NoConvergence` exception is raise if no solution is found.

See Also -------- asjacobian, Jacobian

Notes ----- This algorithm implements the inexact Newton method, with backtracking or full line searches. Several Jacobian approximations are available, including Krylov and Quasi-Newton methods.

References ---------- .. KIM C. T. Kelley, 'Iterative Methods for Linear and Nonlinear Equations'. Society for Industrial and Applied Mathematics. (1995) https://archive.siam.org/books/kelley/fr16/

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 1-D or 2-D. 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

Compute QR decomposition of a matrix.

Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal and R upper triangular.

Parameters ---------- a : (M, N) array_like Matrix to be decomposed overwrite_a : bool, optional Whether data in a is overwritten (may improve performance) lwork : int, optional Work array size, lwork >= a.shape1. If None or -1, an optimal size is computed. mode : 'full', 'r', 'economic', 'raw', optional Determines what information is to be returned: either both Q and R ('full', default), only R ('r') or both Q and R but computed in economy-size ('economic', see Notes). The final option 'raw' (added in SciPy 0.11) makes the function return two matrices (Q, TAU) in the internal format used by LAPACK. pivoting : bool, optional Whether or not factorization should include pivoting for rank-revealing qr decomposition. If pivoting, compute the decomposition ``A P = Q R`` as above, but where P is chosen such that the diagonal of R is non-increasing. 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 ------- Q : float or complex ndarray Of shape (M, M), or (M, K) for ``mode='economic'``. Not returned if ``mode='r'``. R : float or complex ndarray Of shape (M, N), or (K, N) for ``mode='economic'``. ``K = min(M, N)``. P : int ndarray Of shape (N,) for ``pivoting=True``. Not returned if ``pivoting=False``.

Raises ------ LinAlgError Raised if decomposition fails

Notes ----- This is an interface to the LAPACK routines dgeqrf, zgeqrf, dorgqr, zungqr, dgeqp3, and zgeqp3.

If ``mode=economic``, the shapes of Q and R are (M, K) and (K, N) instead of (M,M) and (M,N), with ``K=min(M,N)``.

Examples -------- >>> from scipy import linalg >>> a = np.random.randn(9, 6)

>>> q, r = linalg.qr(a) >>> np.allclose(a, np.dot(q, r)) True >>> q.shape, r.shape ((9, 9), (9, 6))

>>> r2 = linalg.qr(a, mode='r') >>> np.allclose(r, r2) True

>>> q3, r3 = linalg.qr(a, mode='economic') >>> q3.shape, r3.shape ((9, 6), (6, 6))

>>> q4, r4, p4 = linalg.qr(a, pivoting=True) >>> d = np.abs(np.diag(r4)) >>> np.all(d1: <= d:-1) True >>> np.allclose(a:, p4, np.dot(q4, r4)) True >>> q4.shape, r4.shape, p4.shape ((9, 9), (9, 6), (6,))

>>> q5, r5, p5 = linalg.qr(a, mode='economic', pivoting=True) >>> q5.shape, r5.shape, p5.shape ((9, 6), (6, 6), (6,))

Minimize over alpha, the function ``phi(alpha)``.

Uses the interpolation algorithm (Armijo backtracking) as suggested by Wright and Nocedal in 'Numerical Optimization', 1999, pg. 56-57

alpha > 0 is assumed to be a descent direction.

Returns ------- alpha phi1

Scalar function search for alpha that satisfies strong Wolfe conditions

alpha > 0 is assumed to be a descent direction.

Parameters ---------- phi : callable phi(alpha) Function at point `alpha` derphi : callable phi'(alpha) Objective function derivative. Returns a scalar. phi0 : float, optional Value of phi at 0 old_phi0 : float, optional Value of phi at previous point derphi0 : float, optional Value derphi at 0 c1 : float, optional Parameter for Armijo condition rule. c2 : float, optional Parameter for curvature condition rule. amax, amin : float, optional Maximum and minimum step size xtol : float, optional Relative tolerance for an acceptable step.

Returns ------- alpha : float Step size, or None if no suitable step was found phi : float Value of `phi` at the new point `alpha` phi0 : float Value of `phi` at `alpha=0`

Notes ----- Uses routine DCSRCH from MINPACK.

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 1D 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 1D array.

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

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

vdot(a, b)

Return the dot product of two vectors.

The vdot(`a`, `b`) function handles complex numbers differently than dot(`a`, `b`). If the first argument is complex the complex conjugate of the first argument is used for the calculation of the dot product.

Note that `vdot` handles multidimensional arrays differently than `dot`: it does *not* perform a matrix product, but flattens input arguments to 1-D vectors first. Consequently, it should only be used for vectors.

Parameters ---------- a : array_like If `a` is complex the complex conjugate is taken before calculation of the dot product. b : array_like Second argument to the dot product.

Returns ------- output : ndarray Dot product of `a` and `b`. Can be an int, float, or complex depending on the types of `a` and `b`.

See Also -------- dot : Return the dot product without using the complex conjugate of the first argument.

Examples -------- >>> a = np.array(1+2j,3+4j) >>> b = np.array(5+6j,7+8j) >>> np.vdot(a, b) (70-8j) >>> np.vdot(b, a) (70+8j)

Note that higher-dimensional arrays are flattened!

>>> a = np.array([1, 4], [5, 6]) >>> b = np.array([4, 1], [2, 2]) >>> np.vdot(a, b) 30 >>> np.vdot(b, a) 30 >>> 1*4 + 4*1 + 5*2 + 6*2 30