230 lines
7 KiB
Python
230 lines
7 KiB
Python
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"""
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====================================
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Linear algebra (:mod:`scipy.linalg`)
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====================================
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.. currentmodule:: scipy.linalg
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Linear algebra functions.
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.. seealso::
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`numpy.linalg` for more linear algebra functions. Note that
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although `scipy.linalg` imports most of them, identically named
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functions from `scipy.linalg` may offer more or slightly differing
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functionality.
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Basics
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======
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.. autosummary::
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:toctree: generated/
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inv - Find the inverse of a square matrix
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solve - Solve a linear system of equations
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solve_banded - Solve a banded linear system
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solveh_banded - Solve a Hermitian or symmetric banded system
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solve_circulant - Solve a circulant system
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solve_triangular - Solve a triangular matrix
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solve_toeplitz - Solve a toeplitz matrix
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det - Find the determinant of a square matrix
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norm - Matrix and vector norm
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lstsq - Solve a linear least-squares problem
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pinv - Pseudo-inverse (Moore-Penrose) using lstsq
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pinv2 - Pseudo-inverse using svd
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pinvh - Pseudo-inverse of hermitian matrix
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kron - Kronecker product of two arrays
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tril - Construct a lower-triangular matrix from a given matrix
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triu - Construct an upper-triangular matrix from a given matrix
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orthogonal_procrustes - Solve an orthogonal Procrustes problem
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matrix_balance - Balance matrix entries with a similarity transformation
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subspace_angles - Compute the subspace angles between two matrices
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LinAlgError
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LinAlgWarning
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Eigenvalue Problems
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===================
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.. autosummary::
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:toctree: generated/
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eig - Find the eigenvalues and eigenvectors of a square matrix
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eigvals - Find just the eigenvalues of a square matrix
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eigh - Find the e-vals and e-vectors of a Hermitian or symmetric matrix
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eigvalsh - Find just the eigenvalues of a Hermitian or symmetric matrix
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eig_banded - Find the eigenvalues and eigenvectors of a banded matrix
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eigvals_banded - Find just the eigenvalues of a banded matrix
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eigh_tridiagonal - Find the eigenvalues and eigenvectors of a tridiagonal matrix
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eigvalsh_tridiagonal - Find just the eigenvalues of a tridiagonal matrix
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Decompositions
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==============
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.. autosummary::
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:toctree: generated/
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lu - LU decomposition of a matrix
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lu_factor - LU decomposition returning unordered matrix and pivots
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lu_solve - Solve Ax=b using back substitution with output of lu_factor
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svd - Singular value decomposition of a matrix
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svdvals - Singular values of a matrix
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diagsvd - Construct matrix of singular values from output of svd
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orth - Construct orthonormal basis for the range of A using svd
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null_space - Construct orthonormal basis for the null space of A using svd
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ldl - LDL.T decomposition of a Hermitian or a symmetric matrix.
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cholesky - Cholesky decomposition of a matrix
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cholesky_banded - Cholesky decomp. of a sym. or Hermitian banded matrix
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cho_factor - Cholesky decomposition for use in solving a linear system
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cho_solve - Solve previously factored linear system
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cho_solve_banded - Solve previously factored banded linear system
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polar - Compute the polar decomposition.
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qr - QR decomposition of a matrix
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qr_multiply - QR decomposition and multiplication by Q
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qr_update - Rank k QR update
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qr_delete - QR downdate on row or column deletion
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qr_insert - QR update on row or column insertion
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rq - RQ decomposition of a matrix
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qz - QZ decomposition of a pair of matrices
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ordqz - QZ decomposition of a pair of matrices with reordering
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schur - Schur decomposition of a matrix
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rsf2csf - Real to complex Schur form
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hessenberg - Hessenberg form of a matrix
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cdf2rdf - Complex diagonal form to real diagonal block form
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.. seealso::
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`scipy.linalg.interpolative` -- Interpolative matrix decompositions
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Matrix Functions
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================
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.. autosummary::
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:toctree: generated/
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expm - Matrix exponential
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logm - Matrix logarithm
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cosm - Matrix cosine
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sinm - Matrix sine
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tanm - Matrix tangent
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coshm - Matrix hyperbolic cosine
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sinhm - Matrix hyperbolic sine
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tanhm - Matrix hyperbolic tangent
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signm - Matrix sign
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sqrtm - Matrix square root
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funm - Evaluating an arbitrary matrix function
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expm_frechet - Frechet derivative of the matrix exponential
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expm_cond - Relative condition number of expm in the Frobenius norm
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fractional_matrix_power - Fractional matrix power
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Matrix Equation Solvers
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=======================
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.. autosummary::
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:toctree: generated/
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solve_sylvester - Solve the Sylvester matrix equation
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solve_continuous_are - Solve the continuous-time algebraic Riccati equation
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solve_discrete_are - Solve the discrete-time algebraic Riccati equation
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solve_continuous_lyapunov - Solve the continous-time Lyapunov equation
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solve_discrete_lyapunov - Solve the discrete-time Lyapunov equation
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Sketches and Random Projections
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===============================
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.. autosummary::
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:toctree: generated/
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clarkson_woodruff_transform - Applies the Clarkson Woodruff Sketch (a.k.a CountMin Sketch)
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Special Matrices
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================
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.. autosummary::
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:toctree: generated/
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block_diag - Construct a block diagonal matrix from submatrices
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circulant - Circulant matrix
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companion - Companion matrix
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dft - Discrete Fourier transform matrix
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hadamard - Hadamard matrix of order 2**n
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hankel - Hankel matrix
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helmert - Helmert matrix
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hilbert - Hilbert matrix
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invhilbert - Inverse Hilbert matrix
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leslie - Leslie matrix
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pascal - Pascal matrix
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invpascal - Inverse Pascal matrix
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toeplitz - Toeplitz matrix
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tri - Construct a matrix filled with ones at and below a given diagonal
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Low-level routines
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==================
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.. autosummary::
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:toctree: generated/
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get_blas_funcs
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get_lapack_funcs
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find_best_blas_type
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.. seealso::
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`scipy.linalg.blas` -- Low-level BLAS functions
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`scipy.linalg.lapack` -- Low-level LAPACK functions
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`scipy.linalg.cython_blas` -- Low-level BLAS functions for Cython
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`scipy.linalg.cython_lapack` -- Low-level LAPACK functions for Cython
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""" # noqa: E501
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from __future__ import division, print_function, absolute_import
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from .linalg_version import linalg_version as __version__
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from .misc import *
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from .basic import *
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from .decomp import *
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from .decomp_lu import *
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from ._decomp_ldl import *
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from .decomp_cholesky import *
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from .decomp_qr import *
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from ._decomp_qz import *
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from .decomp_svd import *
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from .decomp_schur import *
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from ._decomp_polar import *
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from .matfuncs import *
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from .blas import *
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from .lapack import *
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from .special_matrices import *
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from ._solvers import *
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from ._procrustes import *
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from ._decomp_update import *
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from ._sketches import *
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__all__ = [s for s in dir() if not s.startswith('_')]
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from numpy.dual import register_func
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for k in ['norm', 'inv', 'svd', 'solve', 'det', 'eig', 'eigh', 'eigvals',
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'eigvalsh', 'lstsq', 'cholesky']:
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try:
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register_func(k, eval(k))
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except ValueError:
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pass
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try:
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register_func('pinv', pinv2)
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except ValueError:
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pass
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del k, register_func
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from scipy._lib._testutils import PytestTester
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test = PytestTester(__name__)
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del PytestTester
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