scipy.sparse.linalg

Sparse linear algebra (scipy.sparse.linalg)

Abstract linear operators

LinearOperator(dtype, shape) Common interface for performing matrix vector products Many iterative methods (e.g.
aslinearoperator(A) Return A as a LinearOperator.

Matrix Operations

inv(A) Compute the inverse of a sparse matrix :Parameters: A : (M,M) ndarray or sparse matrix square matrix to be inverted :Returns: Ainv : (M,M) ndarray or sparse matrix inverse of A ..
expm(A) Compute the matrix exponential using Pade approximation.
expm_multiply(A, B[, start, stop, num, endpoint]) Compute the action of the matrix exponential of A on B.

Matrix norms

norm(x[, ord, axis]) Norm of a sparse matrix This function is able to return one of seven different matrix norms, depending on the value of the ord parameter.
onenormest(A[, t, itmax, compute_v, compute_w]) Compute a lower bound of the 1-norm of a sparse matrix.

Solving linear problems

Direct methods for linear equation systems:

spsolve(A, b[, permc_spec, use_umfpack]) Solve the sparse linear system Ax=b, where b may be a vector or a matrix.
factorized(A) Return a fuction for solving a sparse linear system, with A pre-factorized.
MatrixRankWarning
use_solver(**kwargs) Select default sparse direct solver to be used.

Iterative methods for linear equation systems:

bicg(A, b[, x0, tol, maxiter, xtype, M, ...]) Use BIConjugate Gradient iteration to solve A x = b :Parameters: A : {sparse matrix, dense matrix, LinearOperator} The real or complex N-by-N matrix of the linear system It is required that the linear operator can produce Ax and A^T x.
bicgstab(A, b[, x0, tol, maxiter, xtype, M, ...]) Use BIConjugate Gradient STABilized iteration to solve A x = b :Parameters: A : {sparse matrix, dense matrix, LinearOperator} The real or complex N-by-N matrix of the linear system A must represent a hermitian, positive definite matrix b : {array, matrix} Right hand side of the linear system.
cg(A, b[, x0, tol, maxiter, xtype, M, callback]) Use Conjugate Gradient iteration to solve A x = b :Parameters: A : {sparse matrix, dense matrix, LinearOperator} The real or complex N-by-N matrix of the linear system A must represent a hermitian, positive definite matrix b : {array, matrix} Right hand side of the linear system.
cgs(A, b[, x0, tol, maxiter, xtype, M, callback]) Use Conjugate Gradient Squared iteration to solve A x = b :Parameters: A : {sparse matrix, dense matrix, LinearOperator} The real-valued N-by-N matrix of the linear system b : {array, matrix} Right hand side of the linear system.
gmres(A, b[, x0, tol, restart, maxiter, ...]) Use Generalized Minimal RESidual iteration to solve A x = b.
lgmres(A, b[, x0, tol, maxiter, M, ...]) Solve a matrix equation using the LGMRES algorithm.
minres(A, b[, x0, shift, tol, maxiter, ...]) Use MINimum RESidual iteration to solve Ax=b MINRES minimizes norm(A*x - b) for a real symmetric matrix A.
qmr(A, b[, x0, tol, maxiter, xtype, M1, M2, ...]) Use Quasi-Minimal Residual iteration to solve A x = b :Parameters: A : {sparse matrix, dense matrix, LinearOperator} The real-valued N-by-N matrix of the linear system.

Iterative methods for least-squares problems:

lsqr(A, b[, damp, atol, btol, conlim, ...]) Find the least-squares solution to a large, sparse, linear system of equations.
lsmr(A, b[, damp, atol, btol, conlim, ...]) Iterative solver for least-squares problems.

Matrix factorizations

Eigenvalue problems:

eigs(A[, k, M, sigma, which, v0, ncv, ...]) Find k eigenvalues and eigenvectors of the square matrix A.
eigsh(A[, k, M, sigma, which, v0, ncv, ...]) Find k eigenvalues and eigenvectors of the real symmetric square matrix or complex hermitian matrix A.
lobpcg(A, X[, B, M, Y, tol, maxiter, ...]) Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG) LOBPCG is a preconditioned eigensolver for large symmetric positive definite (SPD) generalized eigenproblems.

Singular values problems:

svds(A[, k, ncv, tol, which, v0, maxiter, ...]) Compute the largest k singular values/vectors for a sparse matrix.

Complete or incomplete LU factorizations

splu(A[, permc_spec, diag_pivot_thresh, ...]) Compute the LU decomposition of a sparse, square matrix.
spilu(A[, drop_tol, fill_factor, drop_rule, ...]) Compute an incomplete LU decomposition for a sparse, square matrix.
SuperLU LU factorization of a sparse matrix.

Exceptions

ArpackNoConvergence(msg, eigenvalues, ...) ARPACK iteration did not converge ..
ArpackError(info[, infodict]) ARPACK error

Functions

aslinearoperator(A) Return A as a LinearOperator.
bicg(A, b[, x0, tol, maxiter, xtype, M, ...]) Use BIConjugate Gradient iteration to solve A x = b :Parameters: A : {sparse matrix, dense matrix, LinearOperator} The real or complex N-by-N matrix of the linear system It is required that the linear operator can produce Ax and A^T x.
bicgstab(A, b[, x0, tol, maxiter, xtype, M, ...]) Use BIConjugate Gradient STABilized iteration to solve A x = b :Parameters: A : {sparse matrix, dense matrix, LinearOperator} The real or complex N-by-N matrix of the linear system A must represent a hermitian, positive definite matrix b : {array, matrix} Right hand side of the linear system.
cg(A, b[, x0, tol, maxiter, xtype, M, callback]) Use Conjugate Gradient iteration to solve A x = b :Parameters: A : {sparse matrix, dense matrix, LinearOperator} The real or complex N-by-N matrix of the linear system A must represent a hermitian, positive definite matrix b : {array, matrix} Right hand side of the linear system.
cgs(A, b[, x0, tol, maxiter, xtype, M, callback]) Use Conjugate Gradient Squared iteration to solve A x = b :Parameters: A : {sparse matrix, dense matrix, LinearOperator} The real-valued N-by-N matrix of the linear system b : {array, matrix} Right hand side of the linear system.
eigs(A[, k, M, sigma, which, v0, ncv, ...]) Find k eigenvalues and eigenvectors of the square matrix A.
eigsh(A[, k, M, sigma, which, v0, ncv, ...]) Find k eigenvalues and eigenvectors of the real symmetric square matrix or complex hermitian matrix A.
expm(A) Compute the matrix exponential using Pade approximation.
expm_multiply(A, B[, start, stop, num, endpoint]) Compute the action of the matrix exponential of A on B.
factorized(A) Return a fuction for solving a sparse linear system, with A pre-factorized.
gmres(A, b[, x0, tol, restart, maxiter, ...]) Use Generalized Minimal RESidual iteration to solve A x = b.
inv(A) Compute the inverse of a sparse matrix :Parameters: A : (M,M) ndarray or sparse matrix square matrix to be inverted :Returns: Ainv : (M,M) ndarray or sparse matrix inverse of A ..
lgmres(A, b[, x0, tol, maxiter, M, ...]) Solve a matrix equation using the LGMRES algorithm.
lobpcg(A, X[, B, M, Y, tol, maxiter, ...]) Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG) LOBPCG is a preconditioned eigensolver for large symmetric positive definite (SPD) generalized eigenproblems.
lsmr(A, b[, damp, atol, btol, conlim, ...]) Iterative solver for least-squares problems.
lsqr(A, b[, damp, atol, btol, conlim, ...]) Find the least-squares solution to a large, sparse, linear system of equations.
minres(A, b[, x0, shift, tol, maxiter, ...]) Use MINimum RESidual iteration to solve Ax=b MINRES minimizes norm(A*x - b) for a real symmetric matrix A.
norm(x[, ord, axis]) Norm of a sparse matrix This function is able to return one of seven different matrix norms, depending on the value of the ord parameter.
onenormest(A[, t, itmax, compute_v, compute_w]) Compute a lower bound of the 1-norm of a sparse matrix.
qmr(A, b[, x0, tol, maxiter, xtype, M1, M2, ...]) Use Quasi-Minimal Residual iteration to solve A x = b :Parameters: A : {sparse matrix, dense matrix, LinearOperator} The real-valued N-by-N matrix of the linear system.
spilu(A[, drop_tol, fill_factor, drop_rule, ...]) Compute an incomplete LU decomposition for a sparse, square matrix.
splu(A[, permc_spec, diag_pivot_thresh, ...]) Compute the LU decomposition of a sparse, square matrix.
spsolve(A, b[, permc_spec, use_umfpack]) Solve the sparse linear system Ax=b, where b may be a vector or a matrix.
svds(A[, k, ncv, tol, which, v0, maxiter, ...]) Compute the largest k singular values/vectors for a sparse matrix.
use_solver(**kwargs) Select default sparse direct solver to be used.

Classes

LinearOperator(dtype, shape) Common interface for performing matrix vector products Many iterative methods (e.g.
SuperLU LU factorization of a sparse matrix.
Tester Nose test runner.

Exceptions

ArpackError(info[, infodict]) ARPACK error
ArpackNoConvergence(msg, eigenvalues, ...) ARPACK iteration did not converge ..
MatrixRankWarning