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 Arpack : sig ... end
val eigs : ?k:int -> ?m:[ `LinearOperator of Py.Object.t | `Arr of [> `ArrayLike ] Np.Obj.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:[ `LinearOperator of Py.Object.t | `Arr of [> `ArrayLike ] Np.Obj.t ] -> ?oPinv:[ `LinearOperator of Py.Object.t | `Arr of [> `ArrayLike ] Np.Obj.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 lobpcg : ?b:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> ?m:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `PyObject of Py.Object.t ] -> ?y:[ `Ndarray of [> `Ndarray ] Np.Obj.t | `PyObject of Py.Object.t ] -> ?tol:[ `F of float | `I of int | `S of string | `Bool of bool ] -> ?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 svds : ?k:int -> ?ncv:int -> ?tol:float -> ?which:[ `LM | `SM ] -> ?v0:[> `Ndarray ] Np.Obj.t -> ?maxiter:int -> ?return_singular_vectors:[ `S of string | `Bool of bool ] -> ?solver:string -> a:[ `Spmatrix of [> `Spmatrix ] Np.Obj.t | `LinearOperator of Py.Object.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)

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