5[g}dZddlmcmZddlZddlZgdZe dZ e dZ e dZ ejdkZdZdd Zd Zdd Zd Zd ZdZdZddZddZddZdZy)ac( ====================================================================== Interpolative matrix decomposition (:mod:`scipy.linalg.interpolative`) ====================================================================== .. moduleauthor:: Kenneth L. Ho .. versionadded:: 0.13 .. currentmodule:: scipy.linalg.interpolative An interpolative decomposition (ID) of a matrix :math:`A \in \mathbb{C}^{m \times n}` of rank :math:`k \leq \min \{ m, n \}` is a factorization .. math:: A \Pi = \begin{bmatrix} A \Pi_{1} & A \Pi_{2} \end{bmatrix} = A \Pi_{1} \begin{bmatrix} I & T \end{bmatrix}, where :math:`\Pi = [\Pi_{1}, \Pi_{2}]` is a permutation matrix with :math:`\Pi_{1} \in \{ 0, 1 \}^{n \times k}`, i.e., :math:`A \Pi_{2} = A \Pi_{1} T`. This can equivalently be written as :math:`A = BP`, where :math:`B = A \Pi_{1}` and :math:`P = [I, T] \Pi^{\mathsf{T}}` are the *skeleton* and *interpolation matrices*, respectively. If :math:`A` does not have exact rank :math:`k`, then there exists an approximation in the form of an ID such that :math:`A = BP + E`, where :math:`\| E \| \sim \sigma_{k + 1}` is on the order of the :math:`(k + 1)`-th largest singular value of :math:`A`. Note that :math:`\sigma_{k + 1}` is the best possible error for a rank-:math:`k` approximation and, in fact, is achieved by the singular value decomposition (SVD) :math:`A \approx U S V^{*}`, where :math:`U \in \mathbb{C}^{m \times k}` and :math:`V \in \mathbb{C}^{n \times k}` have orthonormal columns and :math:`S = \mathop{\mathrm{diag}} (\sigma_{i}) \in \mathbb{C}^{k \times k}` is diagonal with nonnegative entries. The principal advantages of using an ID over an SVD are that: - it is cheaper to construct; - it preserves the structure of :math:`A`; and - it is more efficient to compute with in light of the identity submatrix of :math:`P`. Routines ======== Main functionality: .. autosummary:: :toctree: generated/ interp_decomp reconstruct_matrix_from_id reconstruct_interp_matrix reconstruct_skel_matrix id_to_svd svd estimate_spectral_norm estimate_spectral_norm_diff estimate_rank Support functions: .. autosummary:: :toctree: generated/ seed rand References ========== This module uses the ID software package [1]_ by Martinsson, Rokhlin, Shkolnisky, and Tygert, which is a Fortran library for computing IDs using various algorithms, including the rank-revealing QR approach of [2]_ and the more recent randomized methods described in [3]_, [4]_, and [5]_. This module exposes its functionality in a way convenient for Python users. Note that this module does not add any functionality beyond that of organizing a simpler and more consistent interface. We advise the user to consult also the `documentation for the ID package `_. .. [1] P.G. Martinsson, V. Rokhlin, Y. Shkolnisky, M. Tygert. "ID: a software package for low-rank approximation of matrices via interpolative decompositions, version 0.2." http://tygert.com/id_doc.4.pdf. .. [2] H. Cheng, Z. Gimbutas, P.G. Martinsson, V. Rokhlin. "On the compression of low rank matrices." *SIAM J. Sci. Comput.* 26 (4): 1389--1404, 2005. :doi:`10.1137/030602678`. .. [3] E. Liberty, F. Woolfe, P.G. Martinsson, V. Rokhlin, M. Tygert. "Randomized algorithms for the low-rank approximation of matrices." *Proc. Natl. Acad. Sci. U.S.A.* 104 (51): 20167--20172, 2007. :doi:`10.1073/pnas.0709640104`. .. [4] P.G. Martinsson, V. Rokhlin, M. Tygert. "A randomized algorithm for the decomposition of matrices." *Appl. Comput. Harmon. Anal.* 30 (1): 47--68, 2011. :doi:`10.1016/j.acha.2010.02.003`. .. [5] F. Woolfe, E. Liberty, V. Rokhlin, M. Tygert. "A fast randomized algorithm for the approximation of matrices." *Appl. Comput. Harmon. Anal.* 25 (3): 335--366, 2008. :doi:`10.1016/j.acha.2007.12.002`. Tutorial ======== Initializing ------------ The first step is to import :mod:`scipy.linalg.interpolative` by issuing the command: >>> import scipy.linalg.interpolative as sli Now let's build a matrix. For this, we consider a Hilbert matrix, which is well know to have low rank: >>> from scipy.linalg import hilbert >>> n = 1000 >>> A = hilbert(n) We can also do this explicitly via: >>> import numpy as np >>> n = 1000 >>> A = np.empty((n, n), order='F') >>> for j in range(n): ... for i in range(n): ... A[i,j] = 1. / (i + j + 1) Note the use of the flag ``order='F'`` in :func:`numpy.empty`. This instantiates the matrix in Fortran-contiguous order and is important for avoiding data copying when passing to the backend. We then define multiplication routines for the matrix by regarding it as a :class:`scipy.sparse.linalg.LinearOperator`: >>> from scipy.sparse.linalg import aslinearoperator >>> L = aslinearoperator(A) This automatically sets up methods describing the action of the matrix and its adjoint on a vector. Computing an ID --------------- We have several choices of algorithm to compute an ID. These fall largely according to two dichotomies: 1. how the matrix is represented, i.e., via its entries or via its action on a vector; and 2. whether to approximate it to a fixed relative precision or to a fixed rank. We step through each choice in turn below. In all cases, the ID is represented by three parameters: 1. a rank ``k``; 2. an index array ``idx``; and 3. interpolation coefficients ``proj``. The ID is specified by the relation ``np.dot(A[:,idx[:k]], proj) == A[:,idx[k:]]``. From matrix entries ................... We first consider a matrix given in terms of its entries. To compute an ID to a fixed precision, type: >>> eps = 1e-3 >>> k, idx, proj = sli.interp_decomp(A, eps) where ``eps < 1`` is the desired precision. To compute an ID to a fixed rank, use: >>> idx, proj = sli.interp_decomp(A, k) where ``k >= 1`` is the desired rank. Both algorithms use random sampling and are usually faster than the corresponding older, deterministic algorithms, which can be accessed via the commands: >>> k, idx, proj = sli.interp_decomp(A, eps, rand=False) and: >>> idx, proj = sli.interp_decomp(A, k, rand=False) respectively. From matrix action .................. Now consider a matrix given in terms of its action on a vector as a :class:`scipy.sparse.linalg.LinearOperator`. To compute an ID to a fixed precision, type: >>> k, idx, proj = sli.interp_decomp(L, eps) To compute an ID to a fixed rank, use: >>> idx, proj = sli.interp_decomp(L, k) These algorithms are randomized. Reconstructing an ID -------------------- The ID routines above do not output the skeleton and interpolation matrices explicitly but instead return the relevant information in a more compact (and sometimes more useful) form. To build these matrices, write: >>> B = sli.reconstruct_skel_matrix(A, k, idx) for the skeleton matrix and: >>> P = sli.reconstruct_interp_matrix(idx, proj) for the interpolation matrix. The ID approximation can then be computed as: >>> C = np.dot(B, P) This can also be constructed directly using: >>> C = sli.reconstruct_matrix_from_id(B, idx, proj) without having to first compute ``P``. Alternatively, this can be done explicitly as well using: >>> B = A[:,idx[:k]] >>> P = np.hstack([np.eye(k), proj])[:,np.argsort(idx)] >>> C = np.dot(B, P) Computing an SVD ---------------- An ID can be converted to an SVD via the command: >>> U, S, V = sli.id_to_svd(B, idx, proj) The SVD approximation is then: >>> approx = U @ np.diag(S) @ V.conj().T The SVD can also be computed "fresh" by combining both the ID and conversion steps into one command. Following the various ID algorithms above, there are correspondingly various SVD algorithms that one can employ. From matrix entries ................... We consider first SVD algorithms for a matrix given in terms of its entries. To compute an SVD to a fixed precision, type: >>> U, S, V = sli.svd(A, eps) To compute an SVD to a fixed rank, use: >>> U, S, V = sli.svd(A, k) Both algorithms use random sampling; for the deterministic versions, issue the keyword ``rand=False`` as above. From matrix action .................. Now consider a matrix given in terms of its action on a vector. To compute an SVD to a fixed precision, type: >>> U, S, V = sli.svd(L, eps) To compute an SVD to a fixed rank, use: >>> U, S, V = sli.svd(L, k) Utility routines ---------------- Several utility routines are also available. To estimate the spectral norm of a matrix, use: >>> snorm = sli.estimate_spectral_norm(A) This algorithm is based on the randomized power method and thus requires only matrix-vector products. The number of iterations to take can be set using the keyword ``its`` (default: ``its=20``). The matrix is interpreted as a :class:`scipy.sparse.linalg.LinearOperator`, but it is also valid to supply it as a :class:`numpy.ndarray`, in which case it is trivially converted using :func:`scipy.sparse.linalg.aslinearoperator`. The same algorithm can also estimate the spectral norm of the difference of two matrices ``A1`` and ``A2`` as follows: >>> A1, A2 = A**2, A >>> diff = sli.estimate_spectral_norm_diff(A1, A2) This is often useful for checking the accuracy of a matrix approximation. Some routines in :mod:`scipy.linalg.interpolative` require estimating the rank of a matrix as well. This can be done with either: >>> k = sli.estimate_rank(A, eps) or: >>> k = sli.estimate_rank(L, eps) depending on the representation. The parameter ``eps`` controls the definition of the numerical rank. Finally, the random number generation required for all randomized routines can be controlled via :func:`scipy.linalg.interpolative.seed`. To reset the seed values to their original values, use: >>> sli.seed('default') To specify the seed values, use: >>> s = 42 >>> sli.seed(s) where ``s`` must be an integer or array of 55 floats. If an integer, the array of floats is obtained by using ``numpy.random.rand`` with the given integer seed. To simply generate some random numbers, type: >>> arr = sli.rand(n) where ``n`` is the number of random numbers to generate. Remarks ------- The above functions all automatically detect the appropriate interface and work with both real and complex data types, passing input arguments to the proper backend routine. N) estimate_rankestimate_spectral_normestimate_spectral_norm_diff id_to_svd interp_decomprandreconstruct_interp_matrixreconstruct_matrix_from_idreconstruct_skel_matrixseedsvdz9invalid input dtype (input must be float64 or complex128)z4invalid input type (must be array or LinearOperator)zFinterpolative decomposition on 32-bit systems with complex128 is buggylc |jtjk(ry|jtjk(ryt#t $r }t |d}~wwxYw)NFT)dtypenp complex128float64 _DTYPE_ERRORAttributeError _TYPE_ERROR)Aes U/var/www/html/bid-api/venv/lib/python3.12/site-packages/scipy/linalg/interpolative.py_is_realrsK! 77bmm # WW "  !q !s!AAA A AAcXt|tr|dk(rtjy t |dr|t j |t}|jdk7r td|jdks|jdkDr tdtj|y |3tjt jjd y t jj|}tj|jd y ) a Seed the internal random number generator used in this ID package. The generator is a lagged Fibonacci method with 55-element internal state. Parameters ---------- seed : int, sequence, 'default', optional If 'default', the random seed is reset to a default value. If `seed` is a sequence containing 55 floating-point numbers in range [0,1], these are used to set the internal state of the generator. If the value is an integer, the internal state is obtained from `numpy.random.RandomState` (MT19937) with the integer used as the initial seed. If `seed` is omitted (None), ``numpy.random.rand`` is used to initialize the generator. default__len__)r)7zinvalid input sizerzvalues not in range [0,1]Nr) isinstancestr_backend id_srandohasattrrasfortranarrayfloatshape ValueErrorminmax id_srandirandomr RandomState)r staternds rr r s4$!2 y !!!$e4 ;;% 12 2 YY[1_ a89 95! 299>>"-.ii##D)388B<(cptjtj|j |S)a% Generate standard uniform pseudorandom numbers via a very efficient lagged Fibonacci method. This routine is used for all random number generation in this package and can affect ID and SVD results. Parameters ---------- *shape Shape of output array )r!id_srandrprodreshape)r&s rrrs(   RWWU^ , 4 4U ;;r/c<ddlm}t|}t|tj r|dkr|}|rD|rt j||\}}}n^trtt j||\}}}n7|rt j||\}}}nt j||\}}}||dz |fSt|}|rB|rt j||\}}n[trtt j||\}}n5|rt j ||\}}nt j"||\}}|dz |fSt||r|j$\} } |j&} |dkrQ|}|rt j(|| | | \}}}n(trtt j*|| | | \}}}||dz |fSt|}|rt j,| | | |\}}n'trtt j.| | | |\}}|dz |fSt0)aP Compute ID of a matrix. An ID of a matrix `A` is a factorization defined by a rank `k`, a column index array `idx`, and interpolation coefficients `proj` such that:: numpy.dot(A[:,idx[:k]], proj) = A[:,idx[k:]] The original matrix can then be reconstructed as:: numpy.hstack([A[:,idx[:k]], numpy.dot(A[:,idx[:k]], proj)] )[:,numpy.argsort(idx)] or via the routine :func:`reconstruct_matrix_from_id`. This can equivalently be written as:: numpy.dot(A[:,idx[:k]], numpy.hstack([numpy.eye(k), proj]) )[:,np.argsort(idx)] in terms of the skeleton and interpolation matrices:: B = A[:,idx[:k]] and:: P = numpy.hstack([numpy.eye(k), proj])[:,np.argsort(idx)] respectively. See also :func:`reconstruct_interp_matrix` and :func:`reconstruct_skel_matrix`. The ID can be computed to any relative precision or rank (depending on the value of `eps_or_k`). If a precision is specified (`eps_or_k < 1`), then this function has the output signature:: k, idx, proj = interp_decomp(A, eps_or_k) Otherwise, if a rank is specified (`eps_or_k >= 1`), then the output signature is:: idx, proj = interp_decomp(A, eps_or_k) .. This function automatically detects the form of the input parameters and passes them to the appropriate backend. For details, see :func:`_backend.iddp_id`, :func:`_backend.iddp_aid`, :func:`_backend.iddp_rid`, :func:`_backend.iddr_id`, :func:`_backend.iddr_aid`, :func:`_backend.iddr_rid`, :func:`_backend.idzp_id`, :func:`_backend.idzp_aid`, :func:`_backend.idzp_rid`, :func:`_backend.idzr_id`, :func:`_backend.idzr_aid`, and :func:`_backend.idzr_rid`. Parameters ---------- A : :class:`numpy.ndarray` or :class:`scipy.sparse.linalg.LinearOperator` with `rmatvec` Matrix to be factored eps_or_k : float or int Relative error (if `eps_or_k < 1`) or rank (if `eps_or_k >= 1`) of approximation. rand : bool, optional Whether to use random sampling if `A` is of type :class:`numpy.ndarray` (randomized algorithms are always used if `A` is of type :class:`scipy.sparse.linalg.LinearOperator`). Returns ------- k : int Rank required to achieve specified relative precision if `eps_or_k < 1`. idx : :class:`numpy.ndarray` Column index array. proj : :class:`numpy.ndarray` Interpolation coefficients. rLinearOperatorr)scipy.sparse.linalgr6rrrndarrayr!iddp_aid _IS_32BIT _32BIT_ERRORidzp_aididdp_ididzp_idintiddr_aididzr_aididdr_ididzr_idr&rmatveciddp_rididzp_rididdr_rididzr_ridr) reps_or_krr6realepskidxprojmnmatvecas rrrs V3 A;D!RZZ a<C#+#4#4S!#D{$$Q3733$$Q3733r/ct|rtj||dz|\}}}ntj||dz|\}}}|||fS)a Convert ID to SVD. The SVD reconstruction of a matrix with skeleton matrix `B` and ID indices and coefficients `idx` and `proj`, respectively, is:: U, S, V = id_to_svd(B, idx, proj) A = numpy.dot(U, numpy.dot(numpy.diag(S), V.conj().T)) See also :func:`svd`. .. This function automatically detects the matrix data type and calls the appropriate backend. For details, see :func:`_backend.idd_id2svd` and :func:`_backend.idz_id2svd`. Parameters ---------- B : :class:`numpy.ndarray` Skeleton matrix. idx : :class:`numpy.ndarray` Column index array. proj : :class:`numpy.ndarray` Interpolation coefficients. Returns ------- U : :class:`numpy.ndarray` Left singular vectors. S : :class:`numpy.ndarray` Singular values. V : :class:`numpy.ndarray` Right singular vectors. r)rr! idd_id2svd idz_id2svd)rUrMrNUVSs rrrsSD{%%aq$71a%%aq$71a a7Nr/cddlm}|j\}}fd}fd}trt j |||||St j |||||S)a Estimate spectral norm of a matrix by the randomized power method. .. This function automatically detects the matrix data type and calls the appropriate backend. For details, see :func:`_backend.idd_snorm` and :func:`_backend.idz_snorm`. Parameters ---------- A : :class:`scipy.sparse.linalg.LinearOperator` Matrix given as a :class:`scipy.sparse.linalg.LinearOperator` with the `matvec` and `rmatvec` methods (to apply the matrix and its adjoint). its : int, optional Number of power method iterations. Returns ------- float Spectral norm estimate. raslinearoperatorc&j|SNmatvecxrs rrhz&estimate_spectral_norm..matvec xx{r/c&j|SrfrDris rrQz'estimate_spectral_norm..matveca"yy|r/its)r7rdr&rr! idd_snorm idz_snorm)rrprdrOrPrhrQs` rrrsb*5A 77DAq{!!!QSAA!!!QSAAr/c ddlm}||j\}}fd}fd}fd}fd} trt j |||| |||St j |||| |||S)a Estimate spectral norm of the difference of two matrices by the randomized power method. .. This function automatically detects the matrix data type and calls the appropriate backend. For details, see :func:`_backend.idd_diffsnorm` and :func:`_backend.idz_diffsnorm`. Parameters ---------- A : :class:`scipy.sparse.linalg.LinearOperator` First matrix given as a :class:`scipy.sparse.linalg.LinearOperator` with the `matvec` and `rmatvec` methods (to apply the matrix and its adjoint). B : :class:`scipy.sparse.linalg.LinearOperator` Second matrix given as a :class:`scipy.sparse.linalg.LinearOperator` with the `matvec` and `rmatvec` methods (to apply the matrix and its adjoint). its : int, optional Number of power method iterations. Returns ------- float Spectral norm estimate of matrix difference. rrcc&j|Srfrgris rmatvec1z,estimate_spectral_norm_diff..matvec1Grkr/c&j|Srfrmris rmatveca1z-estimate_spectral_norm_diff..matveca1Irnr/c&j|SrfrgrjrUs rmatvec2z,estimate_spectral_norm_diff..matvec2Krkr/c&j|Srfrmrys rmatveca2z-estimate_spectral_norm_diff..matveca2Mrnr/ro)r7rdr&rr! idd_diffsnorm idz_diffsnorm) rrUrprdrOrPrurwrzr|s `` rrr*s25AA 77DAq{%% q(HgwCA A%% q(HgwCA Ar/cddlm}t}ttj rQ|dkr|}|rF|rt j|\}}}ntrtt j|\}}}n|rt j|\}}}nt j|\}}}nt|} | tjkDr%t!d| dtjd|rF|rt j"| \}}}n1trtt j$| \}}}n |rt j&| \}}}nt j(| \}}}nt|rj\} } fd} fd} |dkrL|}|rt j*|| | | | \}}}ntrtt j,|| | | | \}}}n[t|} |rt j.| | | | | \}}}n0trtt j0| | | | | \}}}nt2|||fS) a Compute SVD of a matrix via an ID. An SVD of a matrix `A` is a factorization:: A = numpy.dot(U, numpy.dot(numpy.diag(S), V.conj().T)) where `U` and `V` have orthonormal columns and `S` is nonnegative. The SVD can be computed to any relative precision or rank (depending on the value of `eps_or_k`). See also :func:`interp_decomp` and :func:`id_to_svd`. .. This function automatically detects the form of the input parameters and passes them to the appropriate backend. For details, see :func:`_backend.iddp_svd`, :func:`_backend.iddp_asvd`, :func:`_backend.iddp_rsvd`, :func:`_backend.iddr_svd`, :func:`_backend.iddr_asvd`, :func:`_backend.iddr_rsvd`, :func:`_backend.idzp_svd`, :func:`_backend.idzp_asvd`, :func:`_backend.idzp_rsvd`, :func:`_backend.idzr_svd`, :func:`_backend.idzr_asvd`, and :func:`_backend.idzr_rsvd`. Parameters ---------- A : :class:`numpy.ndarray` or :class:`scipy.sparse.linalg.LinearOperator` Matrix to be factored, given as either a :class:`numpy.ndarray` or a :class:`scipy.sparse.linalg.LinearOperator` with the `matvec` and `rmatvec` methods (to apply the matrix and its adjoint). eps_or_k : float or int Relative error (if `eps_or_k < 1`) or rank (if `eps_or_k >= 1`) of approximation. rand : bool, optional Whether to use random sampling if `A` is of type :class:`numpy.ndarray` (randomized algorithms are always used if `A` is of type :class:`scipy.sparse.linalg.LinearOperator`). Returns ------- U : :class:`numpy.ndarray` Left singular vectors. S : :class:`numpy.ndarray` Singular values. V : :class:`numpy.ndarray` Right singular vectors. rr5rzApproximation rank z exceeds min(A.shape) =  c&j|Srfrgris rrhzsvd..matvecs88A; r/c&j|Srfrmris rrQzsvd..matvecas99Q< r/)r7r6rrrr8r! iddp_asvdr:r; idzp_asvdiddp_svdidzp_svdr?r(r&r' iddr_asvd idzr_asvdiddr_svdidzr_svd iddp_rsvd idzp_rsvd iddr_rsvd idzr_rsvdr)rrIrr6rJrKr_r`rarLrOrPrhrQs` rr r Ws-^3 A;D!RZZ a<C&00a8GAq! **&00a8GAq!&//Q7GAq!&//Q7GAq!H A3qww< #6qc:%%(\N!"566&00A6GAq! **&00A6GAq!&//15GAq!&//15GAq! A~ &ww1  a<C",,S!QH1a&&",,S!QH1aH A",,Q7FAF1a&&",,Q7FAF1a a7Nr/cddlm}t|}t|tj rK|rt j||}nt j||}|dk(rt|j}|St||rM|j\}}|j}|rt j||||St j||||St)an Estimate matrix rank to a specified relative precision using randomized methods. The matrix `A` can be given as either a :class:`numpy.ndarray` or a :class:`scipy.sparse.linalg.LinearOperator`, with different algorithms used for each case. If `A` is of type :class:`numpy.ndarray`, then the output rank is typically about 8 higher than the actual numerical rank. .. This function automatically detects the form of the input parameters and passes them to the appropriate backend. For details, see :func:`_backend.idd_estrank`, :func:`_backend.idd_findrank`, :func:`_backend.idz_estrank`, and :func:`_backend.idz_findrank`. Parameters ---------- A : :class:`numpy.ndarray` or :class:`scipy.sparse.linalg.LinearOperator` Matrix whose rank is to be estimated, given as either a :class:`numpy.ndarray` or a :class:`scipy.sparse.linalg.LinearOperator` with the `rmatvec` method (to apply the matrix adjoint). eps : float Relative error for numerical rank definition. Returns ------- int Estimated matrix rank. rr5)r7r6rrrr8r! idd_estrank idz_estrankr(r&rD idd_findrank idz_findrankr)rrKr6rJrankrOrPrQs rrrs:3 A;D!RZZ ''Q/D''Q/D 19qwwrs@bH 76  UV NO 56 [[5  !')T<$DN!6H$4N%4P&RBD*AZk\2r/