package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Return a function for solving a sparse linear system, with A pre-factorized.

Parameters ---------- A : (N, N) array_like Input.

Returns ------- solve : callable To solve the linear system of equations given in `A`, the `solve` callable should be passed an ndarray of shape (N,).

Examples -------- >>> from scipy.sparse.linalg import factorized >>> A = np.array([ 3. , 2. , -1. ], ... [ 2. , -2. , 4. ], ... [-1. , 0.5, -1. ]) >>> solve = factorized(A) # Makes LU decomposition. >>> rhs1 = np.array(1, -2, 0) >>> solve(rhs1) # Uses the LU factors. array( 1., -2., -2.)

Compute an incomplete LU decomposition for a sparse, square matrix.

The resulting object is an approximation to the inverse of `A`.

Parameters ---------- A : (N, N) array_like Sparse matrix to factorize drop_tol : float, optional Drop tolerance (0 <= tol <= 1) for an incomplete LU decomposition. (default: 1e-4) fill_factor : float, optional Specifies the fill ratio upper bound (>= 1.0) for ILU. (default: 10) drop_rule : str, optional Comma-separated string of drop rules to use. Available rules: ``basic``, ``prows``, ``column``, ``area``, ``secondary``, ``dynamic``, ``interp``. (Default: ``basic,area``)

See SuperLU documentation for details.

Remaining other options Same as for `splu`

Returns ------- invA_approx : scipy.sparse.linalg.SuperLU Object, which has a ``solve`` method.

See also -------- splu : complete LU decomposition

Notes ----- To improve the better approximation to the inverse, you may need to increase `fill_factor` AND decrease `drop_tol`.

This function uses the SuperLU library.

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import spilu >>> A = csc_matrix([1., 0., 0.], [5., 0., 2.], [0., -1., 0.], dtype=float) >>> B = spilu(A) >>> x = np.array(1., 2., 3., dtype=float) >>> B.solve(x) array( 1. , -3. , -1.5) >>> A.dot(B.solve(x)) array( 1., 2., 3.) >>> B.solve(A.dot(x)) array( 1., 2., 3.)

Compute the LU decomposition of a sparse, square matrix.

Parameters ---------- A : sparse matrix Sparse matrix to factorize. Should be in CSR or CSC format. 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

diag_pivot_thresh : float, optional Threshold used for a diagonal entry to be an acceptable pivot. See SuperLU user's guide for details 1_ relax : int, optional Expert option for customizing the degree of relaxing supernodes. See SuperLU user's guide for details 1_ panel_size : int, optional Expert option for customizing the panel size. See SuperLU user's guide for details 1_ options : dict, optional Dictionary containing additional expert options to SuperLU. See SuperLU user guide 1_ (section 2.4 on the 'Options' argument) for more details. For example, you can specify ``options=dict(Equil=False, IterRefine='SINGLE'))`` to turn equilibration off and perform a single iterative refinement.

Returns ------- invA : scipy.sparse.linalg.SuperLU Object, which has a ``solve`` method.

See also -------- spilu : incomplete LU decomposition

Notes ----- This function uses the SuperLU library.

References ---------- .. 1 SuperLU http://crd.lbl.gov/~xiaoye/SuperLU/

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import splu >>> A = csc_matrix([1., 0., 0.], [5., 0., 2.], [0., -1., 0.], dtype=float) >>> B = splu(A) >>> x = np.array(1., 2., 3., dtype=float) >>> B.solve(x) array( 1. , -3. , -1.5) >>> A.dot(B.solve(x)) array( 1., 2., 3.) >>> B.solve(A.dot(x)) array( 1., 2., 3.)

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

Parameters ---------- 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.

Returns ------- 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.shape1 If b is a matrix, then x is a matrix of size (A.shape1, b.shape1)

Notes ----- 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.

Examples -------- >>> 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(A.dot(x).todense(), B.todense()) True

Solve the equation `A x = b` for `x`, assuming A is a triangular matrix.

Parameters ---------- A : (M, M) sparse matrix A sparse square triangular matrix. Should be in CSR format. b : (M,) or (M, N) array_like Right-hand side matrix in `A x = b` lower : bool, optional Whether `A` is a lower or upper triangular matrix. Default is lower triangular matrix. overwrite_A : bool, optional Allow changing `A`. The indices of `A` are going to be sorted and zero entries are going to be removed. Enabling gives a performance gain. Default is False. overwrite_b : bool, optional Allow overwriting data in `b`. Enabling gives a performance gain. Default is False. If `overwrite_b` is True, it should be ensured that `b` has an appropriate dtype to be able to store the result. unit_diagonal : bool, optional If True, diagonal elements of `a` are assumed to be 1 and will not be referenced.

.. versionadded:: 1.4.0

Returns ------- x : (M,) or (M, N) ndarray Solution to the system `A x = b`. Shape of return matches shape of `b`.

Raises ------ LinAlgError If `A` is singular or not triangular. ValueError If shape of `A` or shape of `b` do not match the requirements.

Notes ----- .. versionadded:: 0.19.0

Examples -------- >>> from scipy.sparse import csr_matrix >>> from scipy.sparse.linalg import spsolve_triangular >>> A = csr_matrix([3, 0, 0], [1, -1, 0], [2, 0, 1], dtype=float) >>> B = np.array([2, 0], [-1, 0], [2, 0], dtype=float) >>> x = spsolve_triangular(A, B) >>> np.allclose(A.dot(x), B) True

Select default sparse direct solver to be used.

Parameters ---------- useUmfpack : bool, optional Use UMFPACK over SuperLU. Has effect only if scikits.umfpack is installed. Default: True assumeSortedIndices : bool, optional Allow UMFPACK to skip the step of sorting indices for a CSR/CSC matrix. Has effect only if useUmfpack is True and scikits.umfpack is installed. Default: False

Notes ----- The default sparse solver is umfpack when available (scikits.umfpack is installed). This can be changed by passing useUmfpack = False, which then causes the always present SuperLU based solver to be used.

Umfpack requires a CSR/CSC matrix to have sorted column/row indices. If sure that the matrix fulfills this, pass ``assumeSortedIndices=True`` to gain some speed.