package scipy

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val get_py : string -> Py.Object.t

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

module ArpackError : sig ... end
module ArpackNoConvergence : sig ... end
module LinearOperator : sig ... end
module MatrixRankWarning : sig ... end
module SuperLU : sig ... end
module Arpack : sig ... end
module Dsolve : sig ... end
module Eigen : sig ... end
module Interface : sig ... end
module Isolve : sig ... end
module Iterative : sig ... end
module Linsolve : sig ... end
module Matfuncs : sig ... end
module Utils : sig ... end
val aslinearoperator : Py.Object.t -> Py.Object.t

Return A as a LinearOperator.

'A' may be any of the following types:

  • ndarray
  • matrix
  • sparse matrix (e.g. csr_matrix, lil_matrix, etc.)
  • LinearOperator
  • An object with .shape and .matvec attributes

See the LinearOperator documentation for additional information.

Notes ----- If 'A' has no .dtype attribute, the data type is determined by calling :func:`LinearOperator.matvec()` - set the .dtype attribute to prevent this call upon the linear operator creation.

Examples -------- >>> from scipy.sparse.linalg import aslinearoperator >>> M = np.array([1,2,3],[4,5,6], dtype=np.int32) >>> aslinearoperator(M) <2x3 MatrixLinearOperator with dtype=int32>

val bicg : ?x0:Py.Object.t -> ?tol:Py.Object.t -> ?maxiter:Py.Object.t -> ?m:Py.Object.t -> ?callback:Py.Object.t -> ?atol:Py.Object.t -> a:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> b:[> `Ndarray ] Np.Obj.t -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * int

Use BIConjugate Gradient iteration to solve ``Ax = b``.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real or complex N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` and ``A^T x`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters ---------------- x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

val bicgstab : ?x0:Py.Object.t -> ?tol:Py.Object.t -> ?maxiter:Py.Object.t -> ?m:Py.Object.t -> ?callback:Py.Object.t -> ?atol:Py.Object.t -> a:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> b:[> `Ndarray ] Np.Obj.t -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * int

Use BIConjugate Gradient STABilized iteration to solve ``Ax = b``.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real or complex N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters ---------------- x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

val cg : ?x0:Py.Object.t -> ?tol:Py.Object.t -> ?maxiter:Py.Object.t -> ?m:Py.Object.t -> ?callback:Py.Object.t -> ?atol:Py.Object.t -> a:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> b:[> `Ndarray ] Np.Obj.t -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * int

Use Conjugate Gradient iteration to solve ``Ax = 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. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters ---------------- x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

val cgs : ?x0:Py.Object.t -> ?tol:Py.Object.t -> ?maxiter:Py.Object.t -> ?m:Py.Object.t -> ?callback:Py.Object.t -> ?atol:Py.Object.t -> a:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> b:[> `Ndarray ] Np.Obj.t -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * int

Use Conjugate Gradient Squared iteration to solve ``Ax = b``.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real-valued N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters ---------------- x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

val eigs : ?k:int -> ?m:[ `Arr of [> `ArrayLike ] Np.Obj.t | `LinearOperator of Py.Object.t ] -> ?sigma:Py.Object.t -> ?which:[ `LM | `SM | `LR | `SR | `LI | `SI ] -> ?v0:[> `Ndarray ] Np.Obj.t -> ?ncv:int -> ?maxiter:int -> ?tol:float -> ?return_eigenvectors:bool -> ?minv:[ `Arr of [> `ArrayLike ] Np.Obj.t | `LinearOperator of Py.Object.t ] -> ?oPinv:[ `Arr of [> `ArrayLike ] Np.Obj.t | `LinearOperator of Py.Object.t ] -> ?oPpart:Py.Object.t -> a:[ `Arr of [> `ArrayLike ] Np.Obj.t | `LinearOperator of Py.Object.t ] -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t

Find k eigenvalues and eigenvectors of the square matrix A.

Solves ``A * xi = wi * xi``, the standard eigenvalue problem for wi eigenvalues with corresponding eigenvectors xi.

If M is specified, solves ``A * xi = wi * M * xi``, the generalized eigenvalue problem for wi eigenvalues with corresponding eigenvectors xi

Parameters ---------- A : ndarray, sparse matrix or LinearOperator An array, sparse matrix, or LinearOperator representing the operation ``A * x``, where A is a real or complex square matrix. k : int, optional The number of eigenvalues and eigenvectors desired. `k` must be smaller than N-1. It is not possible to compute all eigenvectors of a matrix. M : ndarray, sparse matrix or LinearOperator, optional An array, sparse matrix, or LinearOperator representing the operation M*x for the generalized eigenvalue problem

A * x = w * M * x.

M must represent a real, symmetric matrix if A is real, and must represent a complex, hermitian matrix if A is complex. For best results, the data type of M should be the same as that of A. Additionally:

If `sigma` is None, M is positive definite

If sigma is specified, M is positive semi-definite

If sigma is None, eigs requires an operator to compute the solution of the linear equation ``M * x = b``. This is done internally via a (sparse) LU decomposition for an explicit matrix M, or via an iterative solver for a general linear operator. Alternatively, the user can supply the matrix or operator Minv, which gives ``x = Minv * b = M^-1 * b``. sigma : real or complex, optional Find eigenvalues near sigma using shift-invert mode. This requires an operator to compute the solution of the linear system ``A - sigma * M * x = b``, where M is the identity matrix if unspecified. This is computed internally via a (sparse) LU decomposition for explicit matrices A & M, or via an iterative solver if either A or M is a general linear operator. Alternatively, the user can supply the matrix or operator OPinv, which gives ``x = OPinv * b = A - sigma * M^-1 * b``. For a real matrix A, shift-invert can either be done in imaginary mode or real mode, specified by the parameter OPpart ('r' or 'i'). Note that when sigma is specified, the keyword 'which' (below) refers to the shifted eigenvalues ``w'i`` where:

If A is real and OPpart == 'r' (default), ``w'i = 1/2 * 1/(w[i]-sigma) + 1/(w[i]-conj(sigma))``.

If A is real and OPpart == 'i', ``w'i = 1/2i * 1/(w[i]-sigma) - 1/(w[i]-conj(sigma))``.

If A is complex, ``w'i = 1/(wi-sigma)``.

v0 : ndarray, optional Starting vector for iteration. Default: random ncv : int, optional The number of Lanczos vectors generated `ncv` must be greater than `k`; it is recommended that ``ncv > 2*k``. Default: ``min(n, max(2*k + 1, 20))`` which : str, 'LM' | 'SM' | 'LR' | 'SR' | 'LI' | 'SI', optional Which `k` eigenvectors and eigenvalues to find:

'LM' : largest magnitude

'SM' : smallest magnitude

'LR' : largest real part

'SR' : smallest real part

'LI' : largest imaginary part

'SI' : smallest imaginary part

When sigma != None, 'which' refers to the shifted eigenvalues w'i (see discussion in 'sigma', above). ARPACK is generally better at finding large values than small values. If small eigenvalues are desired, consider using shift-invert mode for better performance. maxiter : int, optional Maximum number of Arnoldi update iterations allowed Default: ``n*10`` tol : float, optional Relative accuracy for eigenvalues (stopping criterion) The default value of 0 implies machine precision. return_eigenvectors : bool, optional Return eigenvectors (True) in addition to eigenvalues Minv : ndarray, sparse matrix or LinearOperator, optional See notes in M, above. OPinv : ndarray, sparse matrix or LinearOperator, optional See notes in sigma, above. OPpart : 'r' or 'i', optional See notes in sigma, above

Returns ------- w : ndarray Array of k eigenvalues. v : ndarray An array of `k` eigenvectors. ``v:, i`` is the eigenvector corresponding to the eigenvalue wi.

Raises ------ ArpackNoConvergence When the requested convergence is not obtained. The currently converged eigenvalues and eigenvectors can be found as ``eigenvalues`` and ``eigenvectors`` attributes of the exception object.

See Also -------- eigsh : eigenvalues and eigenvectors for symmetric matrix A svds : singular value decomposition for a matrix A

Notes ----- This function is a wrapper to the ARPACK 1_ SNEUPD, DNEUPD, CNEUPD, ZNEUPD, functions which use the Implicitly Restarted Arnoldi Method to find the eigenvalues and eigenvectors 2_.

References ---------- .. 1 ARPACK Software, http://www.caam.rice.edu/software/ARPACK/ .. 2 R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA, 1998.

Examples -------- Find 6 eigenvectors of the identity matrix:

>>> from scipy.sparse.linalg import eigs >>> id = np.eye(13) >>> vals, vecs = eigs(id, k=6) >>> vals array( 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j) >>> vecs.shape (13, 6)

val eigsh : ?k:int -> ?m:Py.Object.t -> ?sigma:Py.Object.t -> ?which:Py.Object.t -> ?v0:Py.Object.t -> ?ncv:Py.Object.t -> ?maxiter:Py.Object.t -> ?tol:Py.Object.t -> ?return_eigenvectors:Py.Object.t -> ?minv:Py.Object.t -> ?oPinv:Py.Object.t -> ?mode:Py.Object.t -> a:[ `Arr of [> `ArrayLike ] Np.Obj.t | `LinearOperator of Py.Object.t ] -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t

Find k eigenvalues and eigenvectors of the real symmetric square matrix or complex hermitian matrix A.

Solves ``A * xi = wi * xi``, the standard eigenvalue problem for wi eigenvalues with corresponding eigenvectors xi.

If M is specified, solves ``A * xi = wi * M * xi``, the generalized eigenvalue problem for wi eigenvalues with corresponding eigenvectors xi.

Parameters ---------- A : ndarray, sparse matrix or LinearOperator A square operator representing the operation ``A * x``, where ``A`` is real symmetric or complex hermitian. For buckling mode (see below) ``A`` must additionally be positive-definite. k : int, optional The number of eigenvalues and eigenvectors desired. `k` must be smaller than N. It is not possible to compute all eigenvectors of a matrix.

Returns ------- w : array Array of k eigenvalues. v : array An array representing the `k` eigenvectors. The column ``v:, i`` is the eigenvector corresponding to the eigenvalue ``wi``.

Other Parameters ---------------- M : An N x N matrix, array, sparse matrix, or linear operator representing the operation ``M @ x`` for the generalized eigenvalue problem

A @ x = w * M @ x.

M must represent a real, symmetric matrix if A is real, and must represent a complex, hermitian matrix if A is complex. For best results, the data type of M should be the same as that of A. Additionally:

If sigma is None, M is symmetric positive definite.

If sigma is specified, M is symmetric positive semi-definite.

In buckling mode, M is symmetric indefinite.

If sigma is None, eigsh requires an operator to compute the solution of the linear equation ``M @ x = b``. This is done internally via a (sparse) LU decomposition for an explicit matrix M, or via an iterative solver for a general linear operator. Alternatively, the user can supply the matrix or operator Minv, which gives ``x = Minv @ b = M^-1 @ b``. sigma : real Find eigenvalues near sigma using shift-invert mode. This requires an operator to compute the solution of the linear system ``A - sigma * M x = b``, where M is the identity matrix if unspecified. This is computed internally via a (sparse) LU decomposition for explicit matrices A & M, or via an iterative solver if either A or M is a general linear operator. Alternatively, the user can supply the matrix or operator OPinv, which gives ``x = OPinv @ b = A - sigma * M^-1 @ b``. Note that when sigma is specified, the keyword 'which' refers to the shifted eigenvalues ``w'i`` where:

if mode == 'normal', ``w'i = 1 / (wi - sigma)``.

if mode == 'cayley', ``w'i = (wi + sigma) / (wi - sigma)``.

if mode == 'buckling', ``w'i = wi / (wi - sigma)``.

(see further discussion in 'mode' below) v0 : ndarray, optional Starting vector for iteration. Default: random ncv : int, optional The number of Lanczos vectors generated ncv must be greater than k and smaller than n; it is recommended that ``ncv > 2*k``. Default: ``min(n, max(2*k + 1, 20))`` which : str 'LM' | 'SM' | 'LA' | 'SA' | 'BE' If A is a complex hermitian matrix, 'BE' is invalid. Which `k` eigenvectors and eigenvalues to find:

'LM' : Largest (in magnitude) eigenvalues.

'SM' : Smallest (in magnitude) eigenvalues.

'LA' : Largest (algebraic) eigenvalues.

'SA' : Smallest (algebraic) eigenvalues.

'BE' : Half (k/2) from each end of the spectrum.

When k is odd, return one more (k/2+1) from the high end. When sigma != None, 'which' refers to the shifted eigenvalues ``w'i`` (see discussion in 'sigma', above). ARPACK is generally better at finding large values than small values. If small eigenvalues are desired, consider using shift-invert mode for better performance. maxiter : int, optional Maximum number of Arnoldi update iterations allowed. Default: ``n*10`` tol : float Relative accuracy for eigenvalues (stopping criterion). The default value of 0 implies machine precision. Minv : N x N matrix, array, sparse matrix, or LinearOperator See notes in M, above. OPinv : N x N matrix, array, sparse matrix, or LinearOperator See notes in sigma, above. return_eigenvectors : bool Return eigenvectors (True) in addition to eigenvalues. This value determines the order in which eigenvalues are sorted. The sort order is also dependent on the `which` variable.

For which = 'LM' or 'SA': If `return_eigenvectors` is True, eigenvalues are sorted by algebraic value.

If `return_eigenvectors` is False, eigenvalues are sorted by absolute value.

For which = 'BE' or 'LA': eigenvalues are always sorted by algebraic value.

For which = 'SM': If `return_eigenvectors` is True, eigenvalues are sorted by algebraic value.

If `return_eigenvectors` is False, eigenvalues are sorted by decreasing absolute value.

mode : string 'normal' | 'buckling' | 'cayley' Specify strategy to use for shift-invert mode. This argument applies only for real-valued A and sigma != None. For shift-invert mode, ARPACK internally solves the eigenvalue problem ``OP * x'i = w'i * B * x'i`` and transforms the resulting Ritz vectors x'i and Ritz values w'i into the desired eigenvectors and eigenvalues of the problem ``A * xi = wi * M * xi``. The modes are as follows:

'normal' : OP = A - sigma * M^-1 @ M, B = M, w'i = 1 / (wi - sigma)

'buckling' : OP = A - sigma * M^-1 @ A, B = A, w'i = wi / (wi - sigma)

'cayley' : OP = A - sigma * M^-1 @ A + sigma * M, B = M, w'i = (wi + sigma) / (wi - sigma)

The choice of mode will affect which eigenvalues are selected by the keyword 'which', and can also impact the stability of convergence (see 2 for a discussion).

Raises ------ ArpackNoConvergence When the requested convergence is not obtained.

The currently converged eigenvalues and eigenvectors can be found as ``eigenvalues`` and ``eigenvectors`` attributes of the exception object.

See Also -------- eigs : eigenvalues and eigenvectors for a general (nonsymmetric) matrix A svds : singular value decomposition for a matrix A

Notes ----- This function is a wrapper to the ARPACK 1_ SSEUPD and DSEUPD functions which use the Implicitly Restarted Lanczos Method to find the eigenvalues and eigenvectors 2_.

References ---------- .. 1 ARPACK Software, http://www.caam.rice.edu/software/ARPACK/ .. 2 R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA, 1998.

Examples -------- >>> from scipy.sparse.linalg import eigsh >>> identity = np.eye(13) >>> eigenvalues, eigenvectors = eigsh(identity, k=6) >>> eigenvalues array(1., 1., 1., 1., 1., 1.) >>> eigenvectors.shape (13, 6)

val expm : [> `ArrayLike ] Np.Obj.t -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t

Compute the matrix exponential using Pade approximation.

Parameters ---------- A : (M,M) array_like or sparse matrix 2D Array or Matrix (sparse or dense) to be exponentiated

Returns ------- expA : (M,M) ndarray Matrix exponential of `A`

Notes ----- This is algorithm (6.1) which is a simplification of algorithm (5.1).

.. versionadded:: 0.12.0

References ---------- .. 1 Awad H. Al-Mohy and Nicholas J. Higham (2009) 'A New Scaling and Squaring Algorithm for the Matrix Exponential.' SIAM Journal on Matrix Analysis and Applications. 31 (3). pp. 970-989. ISSN 1095-7162

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import expm >>> A = csc_matrix([1, 0, 0], [0, 2, 0], [0, 0, 3]) >>> A.todense() matrix([1, 0, 0], [0, 2, 0], [0, 0, 3], dtype=int64) >>> Aexp = expm(A) >>> Aexp <3x3 sparse matrix of type '<class 'numpy.float64'>' with 3 stored elements in Compressed Sparse Column format> >>> Aexp.todense() matrix([ 2.71828183, 0. , 0. ], [ 0. , 7.3890561 , 0. ], [ 0. , 0. , 20.08553692])

val expm_multiply : ?start:[ `Bool of bool | `S of string | `I of int | `F of float ] -> ?stop:[ `Bool of bool | `S of string | `I of int | `F of float ] -> ?num:int -> ?endpoint:bool -> a:Py.Object.t -> b:[> `Ndarray ] Np.Obj.t -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t

Compute the action of the matrix exponential of A on B.

Parameters ---------- A : transposable linear operator The operator whose exponential is of interest. B : ndarray The matrix or vector to be multiplied by the matrix exponential of A. start : scalar, optional The starting time point of the sequence. stop : scalar, optional The end time point of the sequence, unless `endpoint` is set to False. In that case, the sequence consists of all but the last of ``num + 1`` evenly spaced time points, so that `stop` is excluded. Note that the step size changes when `endpoint` is False. num : int, optional Number of time points to use. endpoint : bool, optional If True, `stop` is the last time point. Otherwise, it is not included.

Returns ------- expm_A_B : ndarray The result of the action :math:`e^

_k A

B`.

Notes ----- The optional arguments defining the sequence of evenly spaced time points are compatible with the arguments of `numpy.linspace`.

The output ndarray shape is somewhat complicated so I explain it here. The ndim of the output could be either 1, 2, or 3. It would be 1 if you are computing the expm action on a single vector at a single time point. It would be 2 if you are computing the expm action on a vector at multiple time points, or if you are computing the expm action on a matrix at a single time point. It would be 3 if you want the action on a matrix with multiple columns at multiple time points. If multiple time points are requested, expm_A_B0 will always be the action of the expm at the first time point, regardless of whether the action is on a vector or a matrix.

References ---------- .. 1 Awad H. Al-Mohy and Nicholas J. Higham (2011) 'Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators.' SIAM Journal on Scientific Computing, 33 (2). pp. 488-511. ISSN 1064-8275 http://eprints.ma.man.ac.uk/1591/

.. 2 Nicholas J. Higham and Awad H. Al-Mohy (2010) 'Computing Matrix Functions.' Acta Numerica, 19. 159-208. ISSN 0962-4929 http://eprints.ma.man.ac.uk/1451/

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import expm, expm_multiply >>> A = csc_matrix([1, 0], [0, 1]) >>> A.todense() matrix([1, 0], [0, 1], dtype=int64) >>> B = np.array(np.exp(-1.), np.exp(-2.)) >>> B array( 0.36787944, 0.13533528) >>> expm_multiply(A, B, start=1, stop=2, num=3, endpoint=True) array([ 1. , 0.36787944], [ 1.64872127, 0.60653066], [ 2.71828183, 1. ]) >>> expm(A).dot(B) # Verify 1st timestep array( 1. , 0.36787944) >>> expm(1.5*A).dot(B) # Verify 2nd timestep array( 1.64872127, 0.60653066) >>> expm(2*A).dot(B) # Verify 3rd timestep array( 2.71828183, 1. )

val factorized : [> `Ndarray ] Np.Obj.t -> Py.Object.t

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.)

val gcrotmk : ?x0:[> `Ndarray ] Np.Obj.t -> ?tol:Py.Object.t -> ?maxiter:int -> ?m:[ `PyObject of Py.Object.t | `Spmatrix of [> `Spmatrix ] Np.Obj.t ] -> ?callback:Py.Object.t -> ?m':int -> ?k:int -> ?cu:Py.Object.t -> ?discard_C:bool -> ?truncate:[ `Oldest | `Smallest ] -> ?atol:Py.Object.t -> a:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> b:[> `Ndarray ] Np.Obj.t -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * int

Solve a matrix equation using flexible GCROT(m,k) algorithm.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real or complex N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1). x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is `tol`.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : int, optional Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator, optional Preconditioner for A. The preconditioner should approximate the inverse of A. gcrotmk is a 'flexible' algorithm and the preconditioner can vary from iteration to iteration. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function, optional User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector. m : int, optional Number of inner FGMRES iterations per each outer iteration. Default: 20 k : int, optional Number of vectors to carry between inner FGMRES iterations. According to 2_, good values are around m. Default: m CU : list of tuples, optional List of tuples ``(c, u)`` which contain the columns of the matrices C and U in the GCROT(m,k) algorithm. For details, see 2_. The list given and vectors contained in it are modified in-place. If not given, start from empty matrices. The ``c`` elements in the tuples can be ``None``, in which case the vectors are recomputed via ``c = A u`` on start and orthogonalized as described in 3_. discard_C : bool, optional Discard the C-vectors at the end. Useful if recycling Krylov subspaces for different linear systems. truncate : 'oldest', 'smallest', optional Truncation scheme to use. Drop: oldest vectors, or vectors with smallest singular values using the scheme discussed in 1,2. See 2_ for detailed comparison. Default: 'oldest'

Returns ------- x : array or matrix The solution found. info : int Provides convergence information:

* 0 : successful exit * >0 : convergence to tolerance not achieved, number of iterations

References ---------- .. 1 E. de Sturler, ''Truncation strategies for optimal Krylov subspace methods'', SIAM J. Numer. Anal. 36, 864 (1999). .. 2 J.E. Hicken and D.W. Zingg, ''A simplified and flexible variant of GCROT for solving nonsymmetric linear systems'', SIAM J. Sci. Comput. 32, 172 (2010). .. 3 M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti, ''Recycling Krylov subspaces for sequences of linear systems'', SIAM J. Sci. Comput. 28, 1651 (2006).

val gmres : ?x0:Py.Object.t -> ?tol:Py.Object.t -> ?restart:Py.Object.t -> ?maxiter:Py.Object.t -> ?m:Py.Object.t -> ?callback:Py.Object.t -> ?restrt:Py.Object.t -> ?atol:Py.Object.t -> ?callback_type:Py.Object.t -> a:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> b:[> `Ndarray ] Np.Obj.t -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * int

Use Generalized Minimal RESidual iteration to solve ``Ax = b``.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real or complex N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : int Provides convergence information: * 0 : successful exit * >0 : convergence to tolerance not achieved, number of iterations * <0 : illegal input or breakdown

Other parameters ---------------- x0 : array, matrix Starting guess for the solution (a vector of zeros by default). tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. restart : int, optional Number of iterations between restarts. Larger values increase iteration cost, but may be necessary for convergence. Default is 20. maxiter : int, optional Maximum number of iterations (restart cycles). Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator Inverse of the preconditioner of A. M should approximate the inverse of A and be easy to solve for (see Notes). Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. By default, no preconditioner is used. callback : function User-supplied function to call after each iteration. It is called as `callback(args)`, where `args` are selected by `callback_type`. callback_type : 'x', 'pr_norm', 'legacy', optional Callback function argument requested:

  • ``x``: current iterate (ndarray), called on every restart
  • ``pr_norm``: relative (preconditioned) residual norm (float), called on every inner iteration
  • ``legacy`` (default): same as ``pr_norm``, but also changes the meaning of 'maxiter' to count inner iterations instead of restart cycles. restrt : int, optional DEPRECATED - use `restart` instead.

See Also -------- LinearOperator

Notes ----- A preconditioner, P, is chosen such that P is close to A but easy to solve for. The preconditioner parameter required by this routine is ``M = P^-1``. The inverse should preferably not be calculated explicitly. Rather, use the following template to produce M::

# Construct a linear operator that computes P^-1 * x. import scipy.sparse.linalg as spla M_x = lambda x: spla.spsolve(P, x) M = spla.LinearOperator((n, n), M_x)

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import gmres >>> A = csc_matrix([3, 2, 0], [1, -1, 0], [0, 5, 1], dtype=float) >>> b = np.array(2, 4, -1, dtype=float) >>> x, exitCode = gmres(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True

val inv : [> `ArrayLike ] Np.Obj.t -> [> `ArrayLike ] Np.Obj.t

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`

Notes ----- This computes the sparse inverse of `A`. If the inverse of `A` is expected to be non-sparse, it will likely be faster to convert `A` to dense and use scipy.linalg.inv.

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import inv >>> A = csc_matrix([1., 0.], [1., 2.]) >>> Ainv = inv(A) >>> Ainv <2x2 sparse matrix of type '<class 'numpy.float64'>' with 3 stored elements in Compressed Sparse Column format> >>> A.dot(Ainv) <2x2 sparse matrix of type '<class 'numpy.float64'>' with 2 stored elements in Compressed Sparse Column format> >>> A.dot(Ainv).todense() matrix([ 1., 0.], [ 0., 1.])

.. versionadded:: 0.12.0

val lgmres : ?x0:[> `Ndarray ] Np.Obj.t -> ?tol:Py.Object.t -> ?maxiter:int -> ?m:[ `PyObject of Py.Object.t | `Spmatrix of [> `Spmatrix ] Np.Obj.t ] -> ?callback:Py.Object.t -> ?inner_m:int -> ?outer_k:int -> ?outer_v:Py.Object.t -> ?store_outer_Av:bool -> ?prepend_outer_v:bool -> ?atol:Py.Object.t -> a:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> b:[> `Ndarray ] Np.Obj.t -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * int

Solve a matrix equation using the LGMRES algorithm.

The LGMRES algorithm 1_ 2_ is designed to avoid some problems in the convergence in restarted GMRES, and often converges in fewer iterations.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real or complex N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1). x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is `tol`.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : int, optional Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator, optional Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function, optional User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector. inner_m : int, optional Number of inner GMRES iterations per each outer iteration. outer_k : int, optional Number of vectors to carry between inner GMRES iterations. According to 1_, good values are in the range of 1...3. However, note that if you want to use the additional vectors to accelerate solving multiple similar problems, larger values may be beneficial. outer_v : list of tuples, optional List containing tuples ``(v, Av)`` of vectors and corresponding matrix-vector products, used to augment the Krylov subspace, and carried between inner GMRES iterations. The element ``Av`` can be `None` if the matrix-vector product should be re-evaluated. This parameter is modified in-place by `lgmres`, and can be used to pass 'guess' vectors in and out of the algorithm when solving similar problems. store_outer_Av : bool, optional Whether LGMRES should store also A*v in addition to vectors `v` in the `outer_v` list. Default is True. prepend_outer_v : bool, optional Whether to put outer_v augmentation vectors before Krylov iterates. In standard LGMRES, prepend_outer_v=False.

Returns ------- x : array or matrix The converged solution. info : int Provides convergence information:

  • 0 : successful exit
  • >0 : convergence to tolerance not achieved, number of iterations
  • <0 : illegal input or breakdown

Notes ----- The LGMRES algorithm 1_ 2_ is designed to avoid the slowing of convergence in restarted GMRES, due to alternating residual vectors. Typically, it often outperforms GMRES(m) of comparable memory requirements by some measure, or at least is not much worse.

Another advantage in this algorithm is that you can supply it with 'guess' vectors in the `outer_v` argument that augment the Krylov subspace. If the solution lies close to the span of these vectors, the algorithm converges faster. This can be useful if several very similar matrices need to be inverted one after another, such as in Newton-Krylov iteration where the Jacobian matrix often changes little in the nonlinear steps.

References ---------- .. 1 A.H. Baker and E.R. Jessup and T. Manteuffel, 'A Technique for Accelerating the Convergence of Restarted GMRES', SIAM J. Matrix Anal. Appl. 26, 962 (2005). .. 2 A.H. Baker, 'On Improving the Performance of the Linear Solver restarted GMRES', PhD thesis, University of Colorado (2003).

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import lgmres >>> A = csc_matrix([3, 2, 0], [1, -1, 0], [0, 5, 1], dtype=float) >>> b = np.array(2, 4, -1, dtype=float) >>> x, exitCode = lgmres(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True

val lobpcg : ?b:[ `PyObject of Py.Object.t | `Spmatrix of [> `Spmatrix ] Np.Obj.t ] -> ?m:[ `PyObject of Py.Object.t | `Spmatrix of [> `Spmatrix ] Np.Obj.t ] -> ?y:[ `Ndarray of [> `Ndarray ] Np.Obj.t | `PyObject of Py.Object.t ] -> ?tol:[ `Bool of bool | `S of string | `I of int | `F of float ] -> ?maxiter:int -> ?largest:bool -> ?verbosityLevel:int -> ?retLambdaHistory:bool -> ?retResidualNormsHistory:bool -> a:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> x:[ `Ndarray of [> `Ndarray ] Np.Obj.t | `PyObject of Py.Object.t ] -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * Py.Object.t * Py.Object.t

Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG)

LOBPCG is a preconditioned eigensolver for large symmetric positive definite (SPD) generalized eigenproblems.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The symmetric linear operator of the problem, usually a sparse matrix. Often called the 'stiffness matrix'. X : ndarray, float32 or float64 Initial approximation to the ``k`` eigenvectors (non-sparse). If `A` has ``shape=(n,n)`` then `X` should have shape ``shape=(n,k)``. B : dense matrix, sparse matrix, LinearOperator, optional The right hand side operator in a generalized eigenproblem. By default, ``B = Identity``. Often called the 'mass matrix'. M : dense matrix, sparse matrix, LinearOperator, optional Preconditioner to `A`; by default ``M = Identity``. `M` should approximate the inverse of `A`. Y : ndarray, float32 or float64, optional n-by-sizeY matrix of constraints (non-sparse), sizeY < n The iterations will be performed in the B-orthogonal complement of the column-space of Y. Y must be full rank. tol : scalar, optional Solver tolerance (stopping criterion). The default is ``tol=n*sqrt(eps)``. maxiter : int, optional Maximum number of iterations. The default is ``maxiter = 20``. largest : bool, optional When True, solve for the largest eigenvalues, otherwise the smallest. verbosityLevel : int, optional Controls solver output. The default is ``verbosityLevel=0``. retLambdaHistory : bool, optional Whether to return eigenvalue history. Default is False. retResidualNormsHistory : bool, optional Whether to return history of residual norms. Default is False.

Returns ------- w : ndarray Array of ``k`` eigenvalues v : ndarray An array of ``k`` eigenvectors. `v` has the same shape as `X`. lambdas : list of ndarray, optional The eigenvalue history, if `retLambdaHistory` is True. rnorms : list of ndarray, optional The history of residual norms, if `retResidualNormsHistory` is True.

Notes ----- If both ``retLambdaHistory`` and ``retResidualNormsHistory`` are True, the return tuple has the following format ``(lambda, V, lambda history, residual norms history)``.

In the following ``n`` denotes the matrix size and ``m`` the number of required eigenvalues (smallest or largest).

The LOBPCG code internally solves eigenproblems of the size ``3m`` on every iteration by calling the 'standard' dense eigensolver, so if ``m`` is not small enough compared to ``n``, it does not make sense to call the LOBPCG code, but rather one should use the 'standard' eigensolver, e.g. numpy or scipy function in this case. If one calls the LOBPCG algorithm for ``5m > n``, it will most likely break internally, so the code tries to call the standard function instead.

It is not that ``n`` should be large for the LOBPCG to work, but rather the ratio ``n / m`` should be large. It you call LOBPCG with ``m=1`` and ``n=10``, it works though ``n`` is small. The method is intended for extremely large ``n / m``, see e.g., reference 28 in https://arxiv.org/abs/0705.2626

The convergence speed depends basically on two factors:

1. How well relatively separated the seeking eigenvalues are from the rest of the eigenvalues. One can try to vary ``m`` to make this better.

2. How well conditioned the problem is. This can be changed by using proper preconditioning. For example, a rod vibration test problem (under tests directory) is ill-conditioned for large ``n``, so convergence will be slow, unless efficient preconditioning is used. For this specific problem, a good simple preconditioner function would be a linear solve for `A`, which is easy to code since A is tridiagonal.

References ---------- .. 1 A. V. Knyazev (2001), Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method. SIAM Journal on Scientific Computing 23, no. 2, pp. 517-541. http://dx.doi.org/10.1137/S1064827500366124

.. 2 A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov (2007), Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in hypre and PETSc. https://arxiv.org/abs/0705.2626

.. 3 A. V. Knyazev's C and MATLAB implementations: https://bitbucket.org/joseroman/blopex

Examples --------

Solve ``A x = lambda x`` with constraints and preconditioning.

>>> import numpy as np >>> from scipy.sparse import spdiags, issparse >>> from scipy.sparse.linalg import lobpcg, LinearOperator >>> n = 100 >>> vals = np.arange(1, n + 1) >>> A = spdiags(vals, 0, n, n) >>> A.toarray() array([ 1., 0., 0., ..., 0., 0., 0.], [ 0., 2., 0., ..., 0., 0., 0.], [ 0., 0., 3., ..., 0., 0., 0.], ..., [ 0., 0., 0., ..., 98., 0., 0.], [ 0., 0., 0., ..., 0., 99., 0.], [ 0., 0., 0., ..., 0., 0., 100.])

Constraints:

>>> Y = np.eye(n, 3)

Initial guess for eigenvectors, should have linearly independent columns. Column dimension = number of requested eigenvalues.

>>> X = np.random.rand(n, 3)

Preconditioner in the inverse of A in this example:

>>> invA = spdiags(1./vals, 0, n, n)

The preconditiner must be defined by a function:

>>> def precond( x ): ... return invA @ x

The argument x of the preconditioner function is a matrix inside `lobpcg`, thus the use of matrix-matrix product ``@``.

The preconditioner function is passed to lobpcg as a `LinearOperator`:

>>> M = LinearOperator(matvec=precond, matmat=precond, ... shape=(n, n), dtype=float)

Let us now solve the eigenvalue problem for the matrix A:

>>> eigenvalues, _ = lobpcg(A, X, Y=Y, M=M, largest=False) >>> eigenvalues array(4., 5., 6.)

Note that the vectors passed in Y are the eigenvectors of the 3 smallest eigenvalues. The results returned are orthogonal to those.

val lsmr : ?damp:float -> ?atol:Py.Object.t -> ?btol:Py.Object.t -> ?conlim:float -> ?maxiter:int -> ?show:bool -> ?x0:[> `Ndarray ] Np.Obj.t -> a:[ `Arr of [> `ArrayLike ] Np.Obj.t | `LinearOperator of Py.Object.t ] -> b:[> `Ndarray ] Np.Obj.t -> unit -> Py.Object.t * int * int * float * float * float * float * float

Iterative solver for least-squares problems.

lsmr solves the system of linear equations ``Ax = b``. If the system is inconsistent, it solves the least-squares problem ``min ||b - Ax||_2``. A is a rectangular matrix of dimension m-by-n, where all cases are allowed: m = n, m > n, or m < n. B is a vector of length m. The matrix A may be dense or sparse (usually sparse).

Parameters ---------- A : matrix, sparse matrix, ndarray, LinearOperator Matrix A in the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` and ``A^H x`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array_like, shape (m,) Vector b in the linear system. damp : float Damping factor for regularized least-squares. `lsmr` solves the regularized least-squares problem::

min ||(b) - ( A )x|| ||(0) (damp*I) ||_2

where damp is a scalar. If damp is None or 0, the system is solved without regularization. atol, btol : float, optional Stopping tolerances. `lsmr` continues iterations until a certain backward error estimate is smaller than some quantity depending on atol and btol. Let ``r = b - Ax`` be the residual vector for the current approximate solution ``x``. If ``Ax = b`` seems to be consistent, ``lsmr`` terminates when ``norm(r) <= atol * norm(A) * norm(x) + btol * norm(b)``. Otherwise, lsmr terminates when ``norm(A^H r) <= atol * norm(A) * norm(r)``. If both tolerances are 1.0e-6 (say), the final ``norm(r)`` should be accurate to about 6 digits. (The final x will usually have fewer correct digits, depending on ``cond(A)`` and the size of LAMBDA.) If `atol` or `btol` is None, a default value of 1.0e-6 will be used. Ideally, they should be estimates of the relative error in the entries of A and B respectively. For example, if the entries of `A` have 7 correct digits, set atol = 1e-7. This prevents the algorithm from doing unnecessary work beyond the uncertainty of the input data. conlim : float, optional `lsmr` terminates if an estimate of ``cond(A)`` exceeds `conlim`. For compatible systems ``Ax = b``, conlim could be as large as 1.0e+12 (say). For least-squares problems, `conlim` should be less than 1.0e+8. If `conlim` is None, the default value is 1e+8. Maximum precision can be obtained by setting ``atol = btol = conlim = 0``, but the number of iterations may then be excessive. maxiter : int, optional `lsmr` terminates if the number of iterations reaches `maxiter`. The default is ``maxiter = min(m, n)``. For ill-conditioned systems, a larger value of `maxiter` may be needed. show : bool, optional Print iterations logs if ``show=True``. x0 : array_like, shape (n,), optional Initial guess of x, if None zeros are used.

.. versionadded:: 1.0.0 Returns ------- x : ndarray of float Least-square solution returned. istop : int istop gives the reason for stopping::

istop = 0 means x=0 is a solution. If x0 was given, then x=x0 is a solution. = 1 means x is an approximate solution to A*x = B, according to atol and btol. = 2 means x approximately solves the least-squares problem according to atol. = 3 means COND(A) seems to be greater than CONLIM. = 4 is the same as 1 with atol = btol = eps (machine precision) = 5 is the same as 2 with atol = eps. = 6 is the same as 3 with CONLIM = 1/eps. = 7 means ITN reached maxiter before the other stopping conditions were satisfied.

itn : int Number of iterations used. normr : float ``norm(b-Ax)`` normar : float ``norm(A^H (b - Ax))`` norma : float ``norm(A)`` conda : float Condition number of A. normx : float ``norm(x)``

Notes -----

.. versionadded:: 0.11.0

References ---------- .. 1 D. C.-L. Fong and M. A. Saunders, 'LSMR: An iterative algorithm for sparse least-squares problems', SIAM J. Sci. Comput., vol. 33, pp. 2950-2971, 2011. https://arxiv.org/abs/1006.0758 .. 2 LSMR Software, https://web.stanford.edu/group/SOL/software/lsmr/

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import lsmr >>> A = csc_matrix([1., 0.], [1., 1.], [0., 1.], dtype=float)

The first example has the trivial solution `0, 0`

>>> b = np.array(0., 0., 0., dtype=float) >>> x, istop, itn, normr = lsmr(A, b):4 >>> istop 0 >>> x array( 0., 0.)

The stopping code `istop=0` returned indicates that a vector of zeros was found as a solution. The returned solution `x` indeed contains `0., 0.`. The next example has a non-trivial solution:

>>> b = np.array(1., 0., -1., dtype=float) >>> x, istop, itn, normr = lsmr(A, b):4 >>> istop 1 >>> x array( 1., -1.) >>> itn 1 >>> normr 4.440892098500627e-16

As indicated by `istop=1`, `lsmr` found a solution obeying the tolerance limits. The given solution `1., -1.` obviously solves the equation. The remaining return values include information about the number of iterations (`itn=1`) and the remaining difference of left and right side of the solved equation. The final example demonstrates the behavior in the case where there is no solution for the equation:

>>> b = np.array(1., 0.01, -1., dtype=float) >>> x, istop, itn, normr = lsmr(A, b):4 >>> istop 2 >>> x array( 1.00333333, -0.99666667) >>> A.dot(x)-b array( 0.00333333, -0.00333333, 0.00333333) >>> normr 0.005773502691896255

`istop` indicates that the system is inconsistent and thus `x` is rather an approximate solution to the corresponding least-squares problem. `normr` contains the minimal distance that was found.

val lsqr : ?damp:float -> ?atol:Py.Object.t -> ?btol:Py.Object.t -> ?conlim:float -> ?iter_lim:int -> ?show:bool -> ?calc_var:bool -> ?x0:[> `Ndarray ] Np.Obj.t -> a:[ `Arr of [> `ArrayLike ] Np.Obj.t | `LinearOperator of Py.Object.t ] -> b:[> `Ndarray ] Np.Obj.t -> unit -> Py.Object.t * int * int * float * float * float * float * float * float * Py.Object.t

Find the least-squares solution to a large, sparse, linear system of equations.

The function solves ``Ax = b`` or ``min ||b - Ax||^2`` or ``min ||Ax - b||^2 + d^2 ||x||^2``.

The matrix A may be square or rectangular (over-determined or under-determined), and may have any rank.

::

1. Unsymmetric equations -- solve A*x = b

2. Linear least squares -- solve A*x = b in the least-squares sense

3. Damped least squares -- solve ( A )*x = ( b ) ( damp*I ) ( 0 ) in the least-squares sense

Parameters ---------- A : sparse matrix, ndarray, LinearOperator Representation of an m-by-n matrix. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` and ``A^T x`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array_like, shape (m,) Right-hand side vector ``b``. damp : float Damping coefficient. atol, btol : float, optional Stopping tolerances. If both are 1.0e-9 (say), the final residual norm should be accurate to about 9 digits. (The final x will usually have fewer correct digits, depending on cond(A) and the size of damp.) conlim : float, optional Another stopping tolerance. lsqr terminates if an estimate of ``cond(A)`` exceeds `conlim`. For compatible systems ``Ax = b``, `conlim` could be as large as 1.0e+12 (say). For least-squares problems, conlim should be less than 1.0e+8. Maximum precision can be obtained by setting ``atol = btol = conlim = zero``, but the number of iterations may then be excessive. iter_lim : int, optional Explicit limitation on number of iterations (for safety). show : bool, optional Display an iteration log. calc_var : bool, optional Whether to estimate diagonals of ``(A'A + damp^2*I)^

1

}

``. x0 : array_like, shape (n,), optional Initial guess of x, if None zeros are used.

.. versionadded:: 1.0.0

Returns ------- x : ndarray of float The final solution. istop : int Gives the reason for termination. 1 means x is an approximate solution to Ax = b. 2 means x approximately solves the least-squares problem. itn : int Iteration number upon termination. r1norm : float ``norm(r)``, where ``r = b - Ax``. r2norm : float ``sqrt( norm(r)^2 + damp^2 * norm(x)^2 )``. Equal to `r1norm` if ``damp == 0``. anorm : float Estimate of Frobenius norm of ``Abar = [A]; [damp*I]``. acond : float Estimate of ``cond(Abar)``. arnorm : float Estimate of ``norm(A'*r - damp^2*x)``. xnorm : float ``norm(x)`` var : ndarray of float If ``calc_var`` is True, estimates all diagonals of ``(A'A)^

1

}

`` (if ``damp == 0``) or more generally ``(A'A + damp^2*I)^

1

}

``. This is well defined if A has full column rank or ``damp > 0``. (Not sure what var means if ``rank(A) < n`` and ``damp = 0.``)

Notes ----- LSQR uses an iterative method to approximate the solution. The number of iterations required to reach a certain accuracy depends strongly on the scaling of the problem. Poor scaling of the rows or columns of A should therefore be avoided where possible.

For example, in problem 1 the solution is unaltered by row-scaling. If a row of A is very small or large compared to the other rows of A, the corresponding row of ( A b ) should be scaled up or down.

In problems 1 and 2, the solution x is easily recovered following column-scaling. Unless better information is known, the nonzero columns of A should be scaled so that they all have the same Euclidean norm (e.g., 1.0).

In problem 3, there is no freedom to re-scale if damp is nonzero. However, the value of damp should be assigned only after attention has been paid to the scaling of A.

The parameter damp is intended to help regularize ill-conditioned systems, by preventing the true solution from being very large. Another aid to regularization is provided by the parameter acond, which may be used to terminate iterations before the computed solution becomes very large.

If some initial estimate ``x0`` is known and if ``damp == 0``, one could proceed as follows:

1. Compute a residual vector ``r0 = b - A*x0``. 2. Use LSQR to solve the system ``A*dx = r0``. 3. Add the correction dx to obtain a final solution ``x = x0 + dx``.

This requires that ``x0`` be available before and after the call to LSQR. To judge the benefits, suppose LSQR takes k1 iterations to solve A*x = b and k2 iterations to solve A*dx = r0. If x0 is 'good', norm(r0) will be smaller than norm(b). If the same stopping tolerances atol and btol are used for each system, k1 and k2 will be similar, but the final solution x0 + dx should be more accurate. The only way to reduce the total work is to use a larger stopping tolerance for the second system. If some value btol is suitable for A*x = b, the larger value btol*norm(b)/norm(r0) should be suitable for A*dx = r0.

Preconditioning is another way to reduce the number of iterations. If it is possible to solve a related system ``M*x = b`` efficiently, where M approximates A in some helpful way (e.g. M - A has low rank or its elements are small relative to those of A), LSQR may converge more rapidly on the system ``A*M(inverse)*z = b``, after which x can be recovered by solving M*x = z.

If A is symmetric, LSQR should not be used!

Alternatives are the symmetric conjugate-gradient method (cg) and/or SYMMLQ. SYMMLQ is an implementation of symmetric cg that applies to any symmetric A and will converge more rapidly than LSQR. If A is positive definite, there are other implementations of symmetric cg that require slightly less work per iteration than SYMMLQ (but will take the same number of iterations).

References ---------- .. 1 C. C. Paige and M. A. Saunders (1982a). 'LSQR: An algorithm for sparse linear equations and sparse least squares', ACM TOMS 8(1), 43-71. .. 2 C. C. Paige and M. A. Saunders (1982b). 'Algorithm 583. LSQR: Sparse linear equations and least squares problems', ACM TOMS 8(2), 195-209. .. 3 M. A. Saunders (1995). 'Solution of sparse rectangular systems using LSQR and CRAIG', BIT 35, 588-604.

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import lsqr >>> A = csc_matrix([1., 0.], [1., 1.], [0., 1.], dtype=float)

The first example has the trivial solution `0, 0`

>>> b = np.array(0., 0., 0., dtype=float) >>> x, istop, itn, normr = lsqr(A, b):4 The exact solution is x = 0 >>> istop 0 >>> x array( 0., 0.)

The stopping code `istop=0` returned indicates that a vector of zeros was found as a solution. The returned solution `x` indeed contains `0., 0.`. The next example has a non-trivial solution:

>>> b = np.array(1., 0., -1., dtype=float) >>> x, istop, itn, r1norm = lsqr(A, b):4 >>> istop 1 >>> x array( 1., -1.) >>> itn 1 >>> r1norm 4.440892098500627e-16

As indicated by `istop=1`, `lsqr` found a solution obeying the tolerance limits. The given solution `1., -1.` obviously solves the equation. The remaining return values include information about the number of iterations (`itn=1`) and the remaining difference of left and right side of the solved equation. The final example demonstrates the behavior in the case where there is no solution for the equation:

>>> b = np.array(1., 0.01, -1., dtype=float) >>> x, istop, itn, r1norm = lsqr(A, b):4 >>> istop 2 >>> x array( 1.00333333, -0.99666667) >>> A.dot(x)-b array( 0.00333333, -0.00333333, 0.00333333) >>> r1norm 0.005773502691896255

`istop` indicates that the system is inconsistent and thus `x` is rather an approximate solution to the corresponding least-squares problem. `r1norm` contains the norm of the minimal residual that was found.

val minres : ?x0:Py.Object.t -> ?shift:Py.Object.t -> ?tol:Py.Object.t -> ?maxiter:Py.Object.t -> ?m:Py.Object.t -> ?callback:Py.Object.t -> ?show:Py.Object.t -> ?check:Py.Object.t -> a:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> b:[> `Ndarray ] Np.Obj.t -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * int

Use MINimum RESidual iteration to solve Ax=b

MINRES minimizes norm(A*x - b) for a real symmetric matrix A. Unlike the Conjugate Gradient method, A can be indefinite or singular.

If shift != 0 then the method solves (A - shift*I)x = b

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real symmetric N-by-N matrix of the linear system Alternatively, ``A`` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters ---------------- x0 : array, matrix Starting guess for the solution. tol : float Tolerance to achieve. The algorithm terminates when the relative residual is below `tol`. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

References ---------- Solution of sparse indefinite systems of linear equations, C. C. Paige and M. A. Saunders (1975), SIAM J. Numer. Anal. 12(4), pp. 617-629. https://web.stanford.edu/group/SOL/software/minres/

This file is a translation of the following MATLAB implementation: https://web.stanford.edu/group/SOL/software/minres/minres-matlab.zip

val norm : ?ord:[ `Fro | `PyObject of Py.Object.t ] -> ?axis:[ `T2_tuple_of_ints of Py.Object.t | `I of int ] -> x:Py.Object.t -> unit -> Py.Object.t

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.

Parameters ---------- x : a sparse matrix Input sparse matrix. ord : non-zero int, inf, -inf, 'fro', optional Order of the norm (see table under ``Notes``). inf means numpy's `inf` object. axis : nt, 2-tuple of ints, None, optional If `axis` is an integer, it specifies the axis of `x` along which to compute the vector norms. If `axis` is a 2-tuple, it specifies the axes that hold 2-D matrices, and the matrix norms of these matrices are computed. If `axis` is None then either a vector norm (when `x` is 1-D) or a matrix norm (when `x` is 2-D) is returned.

Returns ------- n : float or ndarray

Notes ----- Some of the ord are not implemented because some associated functions like, _multi_svd_norm, are not yet available for sparse matrix.

This docstring is modified based on numpy.linalg.norm. https://github.com/numpy/numpy/blob/master/numpy/linalg/linalg.py

The following norms can be calculated:

===== ============================ ord norm for sparse matrices ===== ============================ None Frobenius norm 'fro' Frobenius norm inf max(sum(abs(x), axis=1)) -inf min(sum(abs(x), axis=1)) 0 abs(x).sum(axis=axis) 1 max(sum(abs(x), axis=0)) -1 min(sum(abs(x), axis=0)) 2 Not implemented -2 Not implemented other Not implemented ===== ============================

The Frobenius norm is given by 1_:

:math:`||A||_F = \sum_{i,j} abs(a_{i,j})^2^

/2

`

References ---------- .. 1 G. H. Golub and C. F. Van Loan, *Matrix Computations*, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

Examples -------- >>> from scipy.sparse import * >>> import numpy as np >>> from scipy.sparse.linalg import norm >>> a = np.arange(9) - 4 >>> a array(-4, -3, -2, -1, 0, 1, 2, 3, 4) >>> b = a.reshape((3, 3)) >>> b array([-4, -3, -2], [-1, 0, 1], [ 2, 3, 4])

>>> b = csr_matrix(b) >>> norm(b) 7.745966692414834 >>> norm(b, 'fro') 7.745966692414834 >>> norm(b, np.inf) 9 >>> norm(b, -np.inf) 2 >>> norm(b, 1) 7 >>> norm(b, -1) 6

val onenormest : ?t:int -> ?itmax:int -> ?compute_v:bool -> ?compute_w:bool -> a: [ `Ndarray of [> `Ndarray ] Np.Obj.t | `Other_linear_operator of Py.Object.t ] -> unit -> float * [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t

Compute a lower bound of the 1-norm of a sparse matrix.

Parameters ---------- A : ndarray or other linear operator A linear operator that can be transposed and that can produce matrix products. t : int, optional A positive parameter controlling the tradeoff between accuracy versus time and memory usage. Larger values take longer and use more memory but give more accurate output. itmax : int, optional Use at most this many iterations. compute_v : bool, optional Request a norm-maximizing linear operator input vector if True. compute_w : bool, optional Request a norm-maximizing linear operator output vector if True.

Returns ------- est : float An underestimate of the 1-norm of the sparse matrix. v : ndarray, optional The vector such that ||Av||_1 == est*||v||_1. It can be thought of as an input to the linear operator that gives an output with particularly large norm. w : ndarray, optional The vector Av which has relatively large 1-norm. It can be thought of as an output of the linear operator that is relatively large in norm compared to the input.

Notes ----- This is algorithm 2.4 of 1.

In 2 it is described as follows. 'This algorithm typically requires the evaluation of about 4t matrix-vector products and almost invariably produces a norm estimate (which is, in fact, a lower bound on the norm) correct to within a factor 3.'

.. versionadded:: 0.13.0

References ---------- .. 1 Nicholas J. Higham and Francoise Tisseur (2000), 'A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra.' SIAM J. Matrix Anal. Appl. Vol. 21, No. 4, pp. 1185-1201.

.. 2 Awad H. Al-Mohy and Nicholas J. Higham (2009), 'A new scaling and squaring algorithm for the matrix exponential.' SIAM J. Matrix Anal. Appl. Vol. 31, No. 3, pp. 970-989.

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import onenormest >>> A = csc_matrix([1., 0., 0.], [5., 8., 2.], [0., -1., 0.], dtype=float) >>> A.todense() matrix([ 1., 0., 0.], [ 5., 8., 2.], [ 0., -1., 0.]) >>> onenormest(A) 9.0 >>> np.linalg.norm(A.todense(), ord=1) 9.0

val qmr : ?x0:Py.Object.t -> ?tol:Py.Object.t -> ?maxiter:Py.Object.t -> ?m1:Py.Object.t -> ?m2:Py.Object.t -> ?callback:Py.Object.t -> ?atol:Py.Object.t -> a:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> b:[> `Ndarray ] Np.Obj.t -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * int

Use Quasi-Minimal Residual iteration to solve ``Ax = b``.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real-valued N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` and ``A^T x`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters ---------------- x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M1 : sparse matrix, dense matrix, LinearOperator Left preconditioner for A. M2 : sparse matrix, dense matrix, LinearOperator Right preconditioner for A. Used together with the left preconditioner M1. The matrix M1*A*M2 should have better conditioned than A alone. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

See Also -------- LinearOperator

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import qmr >>> A = csc_matrix([3, 2, 0], [1, -1, 0], [0, 5, 1], dtype=float) >>> b = np.array(2, 4, -1, dtype=float) >>> x, exitCode = qmr(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True

val spilu : ?drop_tol:float -> ?fill_factor:float -> ?drop_rule:string -> ?permc_spec:Py.Object.t -> ?diag_pivot_thresh:Py.Object.t -> ?relax:Py.Object.t -> ?panel_size:Py.Object.t -> ?options:Py.Object.t -> a:[> `Ndarray ] Np.Obj.t -> unit -> Py.Object.t

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.)

val splu : ?permc_spec:string -> ?diag_pivot_thresh:float -> ?relax:int -> ?panel_size:int -> ?options:Py.Object.t -> a:[> `Spmatrix ] Np.Obj.t -> unit -> Py.Object.t

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.)

val spsolve : ?permc_spec:string -> ?use_umfpack:bool -> a:[> `ArrayLike ] Np.Obj.t -> b:[> `ArrayLike ] Np.Obj.t -> unit -> [> `ArrayLike ] Np.Obj.t

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

val spsolve_triangular : ?lower:bool -> ?overwrite_A:bool -> ?overwrite_b:bool -> ?unit_diagonal:bool -> a:[> `Spmatrix ] Np.Obj.t -> b:Py.Object.t -> unit -> Py.Object.t

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

val svds : ?k:int -> ?ncv:int -> ?tol:float -> ?which:[ `LM | `SM ] -> ?v0:[> `Ndarray ] Np.Obj.t -> ?maxiter:int -> ?return_singular_vectors:[ `Bool of bool | `S of string ] -> ?solver:string -> a:[ `LinearOperator of Py.Object.t | `Spmatrix of [> `Spmatrix ] Np.Obj.t ] -> unit -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t * [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t

Compute the largest or smallest k singular values/vectors for a sparse matrix. The order of the singular values is not guaranteed.

Parameters ---------- A : sparse matrix, LinearOperator Array to compute the SVD on, of shape (M, N) k : int, optional Number of singular values and vectors to compute. Must be 1 <= k < min(A.shape). ncv : int, optional The number of Lanczos vectors generated ncv must be greater than k+1 and smaller than n; it is recommended that ncv > 2*k Default: ``min(n, max(2*k + 1, 20))`` tol : float, optional Tolerance for singular values. Zero (default) means machine precision. which : str, 'LM' | 'SM', optional Which `k` singular values to find:

  • 'LM' : largest singular values
  • 'SM' : smallest singular values

.. versionadded:: 0.12.0 v0 : ndarray, optional Starting vector for iteration, of length min(A.shape). Should be an (approximate) left singular vector if N > M and a right singular vector otherwise. Default: random

.. versionadded:: 0.12.0 maxiter : int, optional Maximum number of iterations.

.. versionadded:: 0.12.0 return_singular_vectors : bool or str, optional

  • True: return singular vectors (True) in addition to singular values.

.. versionadded:: 0.12.0

  • 'u': only return the u matrix, without computing vh (if N > M).
  • 'vh': only return the vh matrix, without computing u (if N <= M).

.. versionadded:: 0.16.0 solver : str, optional Eigenvalue solver to use. Should be 'arpack' or 'lobpcg'. Default: 'arpack'

Returns ------- u : ndarray, shape=(M, k) Unitary matrix having left singular vectors as columns. If `return_singular_vectors` is 'vh', this variable is not computed, and None is returned instead. s : ndarray, shape=(k,) The singular values. vt : ndarray, shape=(k, N) Unitary matrix having right singular vectors as rows. If `return_singular_vectors` is 'u', this variable is not computed, and None is returned instead.

Notes ----- This is a naive implementation using ARPACK or LOBPCG as an eigensolver on A.H * A or A * A.H, depending on which one is more efficient.

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import svds, eigs >>> A = csc_matrix([1, 0, 0], [5, 0, 2], [0, -1, 0], [0, 0, 3], dtype=float) >>> u, s, vt = svds(A, k=2) >>> s array( 2.75193379, 5.6059665 ) >>> np.sqrt(eigs(A.dot(A.T), k=2)0).real array( 5.6059665 , 2.75193379)

val use_solver : ?kwargs:(string * Py.Object.t) list -> unit -> Py.Object.t

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.

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