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1"""
2====================================
3Linear algebra (:mod:`scipy.linalg`)
4====================================
6.. currentmodule:: scipy.linalg
8Linear algebra functions.
10.. eventually, we should replace the numpy.linalg HTML link with just `numpy.linalg`
12.. seealso::
14 `numpy.linalg <https://www.numpy.org/devdocs/reference/routines.linalg.html>`__
15 for more linear algebra functions. Note that
16 although `scipy.linalg` imports most of them, identically named
17 functions from `scipy.linalg` may offer more or slightly differing
18 functionality.
21Basics
22======
24.. autosummary::
25 :toctree: generated/
27 inv - Find the inverse of a square matrix
28 solve - Solve a linear system of equations
29 solve_banded - Solve a banded linear system
30 solveh_banded - Solve a Hermitian or symmetric banded system
31 solve_circulant - Solve a circulant system
32 solve_triangular - Solve a triangular matrix
33 solve_toeplitz - Solve a toeplitz matrix
34 det - Find the determinant of a square matrix
35 norm - Matrix and vector norm
36 lstsq - Solve a linear least-squares problem
37 pinv - Pseudo-inverse (Moore-Penrose) using lstsq
38 pinv2 - Pseudo-inverse using svd
39 pinvh - Pseudo-inverse of hermitian matrix
40 kron - Kronecker product of two arrays
41 khatri_rao - Khatri-Rao product of two arrays
42 tril - Construct a lower-triangular matrix from a given matrix
43 triu - Construct an upper-triangular matrix from a given matrix
44 orthogonal_procrustes - Solve an orthogonal Procrustes problem
45 matrix_balance - Balance matrix entries with a similarity transformation
46 subspace_angles - Compute the subspace angles between two matrices
47 LinAlgError
48 LinAlgWarning
50Eigenvalue Problems
51===================
53.. autosummary::
54 :toctree: generated/
56 eig - Find the eigenvalues and eigenvectors of a square matrix
57 eigvals - Find just the eigenvalues of a square matrix
58 eigh - Find the e-vals and e-vectors of a Hermitian or symmetric matrix
59 eigvalsh - Find just the eigenvalues of a Hermitian or symmetric matrix
60 eig_banded - Find the eigenvalues and eigenvectors of a banded matrix
61 eigvals_banded - Find just the eigenvalues of a banded matrix
62 eigh_tridiagonal - Find the eigenvalues and eigenvectors of a tridiagonal matrix
63 eigvalsh_tridiagonal - Find just the eigenvalues of a tridiagonal matrix
65Decompositions
66==============
68.. autosummary::
69 :toctree: generated/
71 lu - LU decomposition of a matrix
72 lu_factor - LU decomposition returning unordered matrix and pivots
73 lu_solve - Solve Ax=b using back substitution with output of lu_factor
74 svd - Singular value decomposition of a matrix
75 svdvals - Singular values of a matrix
76 diagsvd - Construct matrix of singular values from output of svd
77 orth - Construct orthonormal basis for the range of A using svd
78 null_space - Construct orthonormal basis for the null space of A using svd
79 ldl - LDL.T decomposition of a Hermitian or a symmetric matrix.
80 cholesky - Cholesky decomposition of a matrix
81 cholesky_banded - Cholesky decomp. of a sym. or Hermitian banded matrix
82 cho_factor - Cholesky decomposition for use in solving a linear system
83 cho_solve - Solve previously factored linear system
84 cho_solve_banded - Solve previously factored banded linear system
85 polar - Compute the polar decomposition.
86 qr - QR decomposition of a matrix
87 qr_multiply - QR decomposition and multiplication by Q
88 qr_update - Rank k QR update
89 qr_delete - QR downdate on row or column deletion
90 qr_insert - QR update on row or column insertion
91 rq - RQ decomposition of a matrix
92 qz - QZ decomposition of a pair of matrices
93 ordqz - QZ decomposition of a pair of matrices with reordering
94 schur - Schur decomposition of a matrix
95 rsf2csf - Real to complex Schur form
96 hessenberg - Hessenberg form of a matrix
97 cdf2rdf - Complex diagonal form to real diagonal block form
98 cossin - Cosine sine decomposition of a unitary or orthogonal matrix
100.. seealso::
102 `scipy.linalg.interpolative` -- Interpolative matrix decompositions
105Matrix Functions
106================
108.. autosummary::
109 :toctree: generated/
111 expm - Matrix exponential
112 logm - Matrix logarithm
113 cosm - Matrix cosine
114 sinm - Matrix sine
115 tanm - Matrix tangent
116 coshm - Matrix hyperbolic cosine
117 sinhm - Matrix hyperbolic sine
118 tanhm - Matrix hyperbolic tangent
119 signm - Matrix sign
120 sqrtm - Matrix square root
121 funm - Evaluating an arbitrary matrix function
122 expm_frechet - Frechet derivative of the matrix exponential
123 expm_cond - Relative condition number of expm in the Frobenius norm
124 fractional_matrix_power - Fractional matrix power
127Matrix Equation Solvers
128=======================
130.. autosummary::
131 :toctree: generated/
133 solve_sylvester - Solve the Sylvester matrix equation
134 solve_continuous_are - Solve the continuous-time algebraic Riccati equation
135 solve_discrete_are - Solve the discrete-time algebraic Riccati equation
136 solve_continuous_lyapunov - Solve the continuous-time Lyapunov equation
137 solve_discrete_lyapunov - Solve the discrete-time Lyapunov equation
140Sketches and Random Projections
141===============================
143.. autosummary::
144 :toctree: generated/
146 clarkson_woodruff_transform - Applies the Clarkson Woodruff Sketch (a.k.a CountMin Sketch)
148Special Matrices
149================
151.. autosummary::
152 :toctree: generated/
154 block_diag - Construct a block diagonal matrix from submatrices
155 circulant - Circulant matrix
156 companion - Companion matrix
157 convolution_matrix - Convolution matrix
158 dft - Discrete Fourier transform matrix
159 fiedler - Fiedler matrix
160 fiedler_companion - Fiedler companion matrix
161 hadamard - Hadamard matrix of order 2**n
162 hankel - Hankel matrix
163 helmert - Helmert matrix
164 hilbert - Hilbert matrix
165 invhilbert - Inverse Hilbert matrix
166 leslie - Leslie matrix
167 pascal - Pascal matrix
168 invpascal - Inverse Pascal matrix
169 toeplitz - Toeplitz matrix
170 tri - Construct a matrix filled with ones at and below a given diagonal
172Low-level routines
173==================
175.. autosummary::
176 :toctree: generated/
178 get_blas_funcs
179 get_lapack_funcs
180 find_best_blas_type
182.. seealso::
184 `scipy.linalg.blas` -- Low-level BLAS functions
186 `scipy.linalg.lapack` -- Low-level LAPACK functions
188 `scipy.linalg.cython_blas` -- Low-level BLAS functions for Cython
190 `scipy.linalg.cython_lapack` -- Low-level LAPACK functions for Cython
192""" # noqa: E501
194from .misc import *
195from .basic import *
196from .decomp import *
197from .decomp_lu import *
198from ._decomp_ldl import *
199from .decomp_cholesky import *
200from .decomp_qr import *
201from ._decomp_qz import *
202from .decomp_svd import *
203from .decomp_schur import *
204from ._decomp_polar import *
205from .matfuncs import *
206from .blas import *
207from .lapack import *
208from .special_matrices import *
209from ._solvers import *
210from ._procrustes import *
211from ._decomp_update import *
212from ._sketches import *
213from ._decomp_cossin import *
215__all__ = [s for s in dir() if not s.startswith('_')]
218from scipy._lib._testutils import PytestTester
219test = PytestTester(__name__)
220del PytestTester