scipy.sparse.linalg.spsolve(A, b, permc_spec=None, use_umfpack=True)[source]

Solve the sparse linear system Ax=b, where b may be a vector or a matrix.

A : ndarray or sparse matrix

The square matrix A will be converted into CSC or CSR form

b : ndarray or sparse matrix

The matrix or vector representing the right hand side of the equation. If a vector, b.shape must be (n,) or (n, 1).

permc_spec : str, optional

How to permute the columns of the matrix for sparsity preservation. (default: ‘COLAMD’)

  • NATURAL: natural ordering.
  • MMD_ATA: minimum degree ordering on the structure of A^T A.
  • MMD_AT_PLUS_A: minimum degree ordering on the structure of A^T+A.
  • COLAMD: approximate minimum degree column ordering
use_umfpack : bool, optional

if True (default) then use umfpack for the solution. This is only referenced if b is a vector and scikit-umfpack is installed.

x : ndarray or sparse matrix

the solution of the sparse linear equation. If b is a vector, then x is a vector of size A.shape[1] If b is a matrix, then x is a matrix of size (A.shape[1], b.shape[1])


For solving the matrix expression AX = B, this solver assumes the resulting matrix X is sparse, as is often the case for very sparse inputs. If the resulting X is dense, the construction of this sparse result will be relatively expensive. In that case, consider converting A to a dense matrix and using scipy.linalg.solve or its variants.


>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import spsolve
>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> B = csc_matrix([[2, 0], [-1, 0], [2, 0]], dtype=float)
>>> x = spsolve(A, B)
>>> np.allclose(, B.todense())