# Sparse linear algebra (scipy.sparse.linalg)¶

## Abstract linear operators¶

 LinearOperator(*args, **kwargs) Common interface for performing matrix vector products Return A as a LinearOperator.

## Matrix Operations¶

 Compute the inverse of a sparse matrix 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 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. spsolve_triangular(A, b[, lower, …]) Solve the equation A x = b for x, assuming A is a triangular matrix. Return a function 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, M, callback, atol]) Use BIConjugate Gradient iteration to solve Ax = b. bicgstab(A, b[, x0, tol, maxiter, M, …]) Use BIConjugate Gradient STABilized iteration to solve Ax = b. cg(A, b[, x0, tol, maxiter, M, callback, atol]) Use Conjugate Gradient iteration to solve Ax = b. cgs(A, b[, x0, tol, maxiter, M, callback, atol]) Use Conjugate Gradient Squared iteration to solve Ax = b. gmres(A, b[, x0, tol, restart, maxiter, M, …]) Use Generalized Minimal RESidual iteration to solve Ax = b. lgmres(A, b[, x0, tol, maxiter, M, …]) Solve a matrix equation using the LGMRES algorithm. minres(A, b[, x0, shift, tol, maxiter, M, …]) Use MINimum RESidual iteration to solve Ax=b qmr(A, b[, x0, tol, maxiter, M1, M2, …]) Use Quasi-Minimal Residual iteration to solve Ax = b. gcrotmk(A, b[, x0, tol, maxiter, M, …]) Solve a matrix equation using flexible GCROT(m,k) algorithm.

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)

Singular values problems:

 svds(A[, k, ncv, tol, which, v0, maxiter, …]) Compute the largest or smallest 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. LU factorization of a sparse matrix.

## Exceptions¶

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