# 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¶

 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.

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