_Mh{gdZddlmZddlZddlZddlZddlm Z ddl m Z ddlm Z ddl mZddlmZd gZd Zd Zd Zded ZdZ dfdZdgdZdhdZiddddddddddddd d!d"d#d$d%d&d'd(d)d*d+d,d-d.d/d0d1d2d3d4d5id6d7d8d9d:d;dd?d@dAdBdCdDdEdFdGdHdIdJdKdLdMdNdOdPdQdRdSdTdUdVdWdXdYiZ didZZGd[d\Zd]Zd^Zdjd_Zd`Z dkdaZ!dbZ"dcZ#ddZ$dS)lz-Compute the action of the matrix exponential.)warnN)qr)is_pydata_spmatrix)aslinearoperator)IdentityOperator onenormest expm_multiplyc|tj|r5tt |djSt|r0tt |dStj |tj S)Naxis) scipysparseissparsemaxabssumflatrnplinalgnorminfAs b/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/scipy/sparse/linalg/_expm_multiply.py_exact_inf_normrs |Q)3q66::1:%%*+++ A  )3q66::1:%%&&&y~~a(((chtj|r5tt |djSt|r0tt |dStj |dS)Nrr r ) rrrrrrrrrrrrs r _exact_1_normr s |Q$3q66::1:%%*+++ A  $3q66::1:%%&&&y~~a###rct|r&|S|SN)rto_scipy_sparsetracers r_tracer%%s>!  ""((***wwyyrctj|}t|jdks|jd|jdkrt d|jd}|ddg||g}t||dd \}}tj | j ||z}|dd g||g} | || j | zzz } tj | j || z} || |z zS) aEstimate `np.trace(A)` using `3*m3` matrix-vector products. The result is not deterministic. Parameters ---------- A : LinearOperator Linear operator whose trace will be estimated. Has to be square. m3 : int Number of matrix-vector products divided by 3 used to estimate the trace. seed : optional Seed for `numpy.random.default_rng`. Can be provided to obtain deterministic results. Returns ------- trace : LinearOperator.dtype Estimate of the trace Notes ----- This is the Hutch++ algorithm given in [1]_. References ---------- .. [1] Meyer, Raphael A., Cameron Musco, Christopher Musco, and David P. Woodruff. "Hutch++: Optimal Stochastic Trace Estimation." In Symposium on Simplicity in Algorithms (SOSA), pp. 142-155. Society for Industrial and Applied Mathematics, 2021 https://doi.org/10.1137/1.9781611976496.16 z&Expected A to be like a square matrix.g?Teconomic) overwrite_amoder ) rrandom default_rnglenshape ValueErrorchoicermatmatr$conjT) rm3seedrngnSQ_trQAQGrighttrGAGs rtraceestrB-s1D )   % %C 17||qAGBK172;66ABBB  A D$" "E HUZZ\\^ahhuoo5 6 6E 58 rctj|rtj|jd|jd|j}tj|r||jStj||jSt|r7ddl}||jd|jd|jSt|tjj j rt|j|jStj|jd|jd|jS)Nrr dtype)rrreyer1rEasformatformat dia_arrayr isinstancerLinearOperatorrr)routrs r _ident_likerM\s+ |Q =lqwqz171:QWEE <  # # *<<)) )|%%c**33AH=== A  = zz!'!*agajz@@@ Au|*9 : :=qw7777vagaj!'!*AG<<<>> import numpy as np >>> from scipy.sparse import csc_array >>> from scipy.sparse.linalg import expm, expm_multiply >>> A = csc_array([[1, 0], [0, 1]]) >>> A.toarray() array([[1, 0], [0, 1]], dtype=int64) >>> B = np.array([np.exp(-1.), np.exp(-2.)]) >>> B array([ 0.36787944, 0.13533528]) >>> expm_multiply(A, B, start=1, stop=2, num=3, endpoint=True) array([[ 1. , 0.36787944], [ 1.64872127, 0.60653066], [ 2.71828183, 1. ]]) >>> expm(A).dot(B) # Verify 1st timestep array([ 1. , 0.36787944]) >>> expm(1.5*A).dot(B) # Verify 2nd timestep array([ 1.64872127, 0.60653066]) >>> expm(2*A).dot(B) # Verify 3rd timestep array([ 2.71828183, 1. ]) c3K|]}|duV dSr").0args r z expm_multiply..s& ? ?33$; ? ? ? ? ? ?r)traceA)all_expm_multiply_simple_expm_multiply_interval) rBstartstopnumendpointrTXstatuss rr r mssB ? ?5$X"> ? ? ???E !!Qv 6 6 6+Aq%s,4VEEE 6 Hrr*Fc |rtt|jdks|jd|jdkrtd|jd|jdkr td|jd|jdt |}t |t jjj }|jd}t|jdkrd}n5t|jdkr|jd}ntdd } | } |5|rtd d |rt|dnt|}|t|z } || |zz }|rt|nt|} || zdkrd\} }n0d}t!||z|| z|}t#||| |\} }t%|||| | || |S)a  Compute the action of the matrix exponential at a single time point. Parameters ---------- A : transposable linear operator The operator whose exponential is of interest. B : ndarray The matrix to be multiplied by the matrix exponential of A. t : float A time point. traceA : scalar, optional Trace of `A`. If not given the trace is estimated for linear operators, or calculated exactly for sparse matrices. It is used to precondition `A`, thus an approximate trace is acceptable balance : bool Indicates whether or not to apply balancing. Returns ------- F : ndarray :math:`e^{t A} B` Notes ----- This is algorithm (3.2) in Al-Mohy and Higham (2011). r'rr %expected A to be like a square matrixshapes of matrices A  and B  are incompatible*expected B to be like a matrix or a vector>>  %aAr61c7 K KKrc|rt|d}|}|} tj||zt|z } t |D]} t |} t |D]c} |t|| dzzz }|||z}t |}| |z} | |z|t | zkrn|} d| | z} | }| S)z A helper function. Nrer )rprexprqrangerdot)rrXrur|r}r~r{rvrzFetaic1jcoeffc2s rrtrts"!! { A &2a ! !C 1XX   Q  v  Aa1g&Ea A ##BAABw# 2 2222BB !G  Hrr gMrggYt&>gq@H6?ga2U0*c?g3mJ?gtF_?g? gǺ? g;On? g1Zd? g333333? g?g r?gPn?gˡE?gn?gq= ףp?g)\(?g ףp= ?gQ?gQ?gGz@g(\@gq= ףp@gQ@gzG@gp= ף@g{Gz @gRQ @#g@(g@-g@2g!@7g#@cDddlm}|t||zS)a Efficiently estimate the 1-norm of A^p. Parameters ---------- A : ndarray Matrix whose 1-norm of a power is to be computed. p : int Non-negative integer power. 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. rr)scipy.sparse.linalg._onenormestr r)rpruitmax compute_v compute_wr s r_onenormest_matrix_powerras5P;::::: :&q))Q. / //rc2eZdZdZd dZdZdZdZd ZdS) rrab Information about an operator is lazily computed. The information includes the exact 1-norm of the operator, in addition to estimates of 1-norms of powers of the operator. This uses the notation of Computing the Action (2011). This class is specialized enough to probably not be of general interest outside of this module. Nr'r cL||_||_||_i|_||_dS)a Provide the operator and some norm-related information. Parameters ---------- A : linear operator The operator of interest. A_1_norm : float, optional The exact 1-norm of A. ell : int, optional A technical parameter controlling norm estimation quality. scale : int, optional If specified, return the norms of scale*A instead of A. N)_A _A_1_norm_ell_d_scale)selfrrmrnscales r__init__zLazyOperatorNormInfo.__init__s+ !  rc||_dS)z* Set the scale parameter. N)r)rrs r set_scalezLazyOperatorNormInfo.set_scales rc`|jt|j|_|j|jzS)z+ Compute the exact 1-norm. )rr rr)rs ronenormzLazyOperatorNormInfo.onenorms, > !*4733DN{4>))rc||jvr+t|j||j}|d|z z|j|<|j|j|zS)zn Lazily estimate :math:`d_p(A) ~= || A^p ||^(1/p)` where :math:`||.||` is the 1-norm. r*)rrrrr)rrests rdzLazyOperatorNormInfo.dsM DG  *47AtyAACq)DGAJ{471:%%rctt||||dzS)z3 Lazily compute max(d(p), d(p+1)). r )rr)rrs ralphazLazyOperatorNormInfo.alphas,466!99dffQqSkk***r)Nr'r ) __name__ __module__ __qualname____doc__rrrrrrPrrrrrrsn  , ***&&&+++++rrrcttj||t|z S)a A helper function for computing bounds. This is equation (3.10). It measures cost in terms of the number of required matrix products. Parameters ---------- m : int A valid key of _theta. p : int A matrix power. norm_info : LazyOperatorNormInfo Information about 1-norms of related operators. Returns ------- cost_div_m : int Required number of matrix products divided by m. )intrceilr_theta)mrrs r_compute_cost_div_mrs1, rwyq))F1I566 7 77rctj}ttj|}ttj|dz}t fdt ||dzDS)z Compute the largest positive integer p such that p*(p-1) <= m_max + 1. Do this in a slightly dumb way, but safe and not too slow. Parameters ---------- m_max : int A count related to bounds. r c3:K|]}||dz zdzk|VdS)r NrP)rQrm_maxs rrSz!_compute_p_max..s9IIQAqsGuqy4H4Hq4H4H4H4HIIr)rsqrtrfloorrrr)r sqrt_m_maxp_lowp_highs` r_compute_p_maxrsuJ $$ % %E a(( ) )F IIII%vax00III I IIrcP|dkrtdd}d}t||||rhtD]M\}}t t j||z } | || z||zkr|}| }Nn~tdt|dzD]M} t| | dz zdz |dzD].}|tvr#t|| |} | || z||zkr|}| }/Nt|d}||fS)a A helper function for the _expm_multiply_* functions. Parameters ---------- norm_info : LazyOperatorNormInfo Information about norms of certain linear operators of interest. n0 : int Number of columns in the _expm_multiply_* B matrix. tol : float Expected to be :math:`2^{-24}` for single precision or :math:`2^{-53}` for double precision. m_max : int A value related to a bound. ell : int The number of columns used in the 1-norm approximation. This is usually taken to be small, maybe between 1 and 5. Returns ------- best_m : int Related to bounds for error control. best_s : int Amount of scaling. Notes ----- This is code fragment (3.1) in Al-Mohy and Higham (2011). The discussion of default values for m_max and ell is given between the definitions of equation (3.11) and the definition of equation (3.12). r z%expected ell to be a positive integerNr') r2_condition_3_13rritemsrrrrrrr) rryr{rrnbest_mbest_srthetar~rs rrsrss^F Qww@AAA F Fy((**Bs;;    HAuBGI--//%78899A~Q&!8!8  q.//!344 # #A1ac719eAg.. # #;;+Aq)<rct|}d|z|z|dzz}t|t||zz }|||zkS)a] A helper function for the _expm_multiply_* functions. Parameters ---------- A_1_norm : float The precomputed 1-norm of A. n0 : int Number of columns in the _expm_multiply_* B matrix. m_max : int A value related to a bound. ell : int The number of columns used in the 1-norm approximation. This is usually taken to be small, maybe between 1 and 5. Returns ------- value : bool Indicates whether or not the condition has been met. Notes ----- This is condition (3.13) in Al-Mohy and Higham (2011). r'rg)rrrq)rmryrrnp_maxabs rrr1sR8 5 ! !E C%519%A u b5j)))A q1u rc |rtt|jdks|jd|jdkrtd|jd|jdkr td|jd|jdt |} t |t jjj } |jd} t|jdkrd} n5t|jdkr|jd} ntdd } | }|5| rtd d | rt|dnt|}|t| z }ddi}|||d<|||d<tj||fi|\}}t|}|dkrtd|dz }|}|d}||}|f|jz}tj|tj|j|jt}||z }||| zz }| rt'|nt)|}d}t+||z||z|}||zdkrd\}}nt-|| ||\}}t/||||||} t j| r| } n#t5| r| } | |d<||kr|rdSt9||||||||| S||zs|rdSt;||||||||S||zr|rdSt=||||||||St?d)aB Compute the action of the matrix exponential at multiple time points. Parameters ---------- A : transposable linear operator The operator whose exponential is of interest. B : ndarray The matrix 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. traceA : scalar, optional Trace of `A`. If not given the trace is estimated for linear operators, or calculated exactly for sparse matrices. It is used to precondition `A`, thus an approximate trace is acceptable endpoint : bool, optional If True, `stop` is the last time point. Otherwise, it is not included. balance : bool Indicates whether or not to apply balancing. status_only : bool A flag that is set to True for some debugging and testing operations. Returns ------- F : ndarray :math:`e^{t_k A} B` status : int An integer status for testing and debugging. Notes ----- This is algorithm (5.2) in Al-Mohy and Higham (2011). There seems to be a typo, where line 15 of the algorithm should be moved to line 6.5 (between lines 6 and 7). r'rr r`rarbrcrdreNrfrgrhrrjretstepTr[r\z%at least two time points are requiredrDrlrkrozinternal error) rpr0r1r2rMrJrrrrKrrBr%rqrlinspaceempty result_typerEr r rrrsrtrtoarrayrtodense_expm_multiply_interval_core_0_expm_multiply_interval_core_1_expm_multiply_interval_core_2 Exception)!rrXrYrZr[r\rTrv status_onlyrwrxr:ryrzr{r|linspace_kwargssamplesstepnsamplesqht_0t_qX_shaper]rurmrnrr}r~ action_t0s! rrWrWUs^"!! 17||qAGAJ!'!*44@AAAwqzQWQZ------.. . NNE#Au|':'IJJ  A 17||q  QW   WQZEFFF C C ~  K >> +1ab&!DDI |Y''(%%'' I & &(%%'' AaD Avv  611!Qr1ic"66 6!e *  .11!Qr61a.. . a%*  .11!Qr61a.. .()))rc 0|dkrd\} } nC|d|z t||||\} } |dt|D]"} t ||| ||| | || dz<#|dfS)z1 A helper function, for the case q <= s. rrkr*ror )rrrsrrt) rr]rr|rrr{rnryr}r~ks rrrsa BqD!!!!)R#>>> A 1XXGG+AqtQFAFF!A# a4Krc ||z}|jdd} |dzf| z} tj| |j} t |D]} || |z} | | d<d}t d|dzD]}| d}t |}t d|dzD]}||kr4||| |dz zt|z | |<tt||}||| |zz}t | |}||z}||z|t |zkrn|}tj ||z|z|z||| |zz<|dfS)z? A helper function, for the case q > s and q % s == 0. r NrDr r1rrrErrrrqpowr)rr]rr|r}r~rr{r input_shapeK_shapeKrZhigh_prrrrrinf_norm_K_p_1rs rrrs QA'!""+Kzn{*G (((A 1XX,, acF!q!A# , ,A!A ##B1fQh''  v::quuQqsV}},uQxx7AaDc!Qii((! $!01!6!6^+7cOA$6$6666E!B!+Aa!A#gJJ , a4Krc ||z}||z} ||| zz } |jdd} |dzf| z} tj| |j} t | dzD]*}|||z}|| d<d}|| kr|}n| }t d|dzD]}| d}t |}t d|dzD]}||dzkr6||| |dz zt|z | |<|}tt||}||| |zz}t | |}||z}||z|t |zkrn|}tj ||z|z|z||||zz<,|dfS)z> A helper function, for the case q > s and q % s > 0. r NrDrr'r)rr]rr|r}r~rr{rrrrrrrrr effective_drrrrrrrs rrrs QA QA AE A'!""+Kzn{*G (((A 1q5\\,, acF! q55KKKq+a-(( , ,A!A ##B1fQh''   ??quuQqsV}},uQxx7AaDFc!Qii((! $!01!6!6^+7cOA$6$6666E!B!+Aa!A#gJJ , a4Krr")NNNNN)r*NF)NF)r'rFF)rr')NNNNNFF)%rwarningsrnumpyr scipy.linalgrscipy.sparse.linalgscipy.linalg._decomp_qrrscipy.sparse._sputilsrrscipy.sparse.linalg._interfacerrr __all__rr r%rBrMr rVrtrrrrrrrsrrWrrrrPrrrs33&&&&&&444444000000;;;;;;666666  )))$$$,,,,^==="48(,f f f f R?L?L?L?LD    :)  8)  7)  7 ) 7 ) 7 )  7)  7)  7)  7)  G)  G)  G)  G) G!) " G#) $ G%) & G') ) ( D)) * D+) , D-) 0 D1) 2 D3) 4 D5) 6 D7) 8 D9) : D;) < D=) > D?) @ DA) B DC) H CI) J CK) L CM) N CO) ) P CQ) ) Z27)0)0)0)0V>+>+>+>+>+>+>+>+@8882JJJ$7777t!!!H>B@E(-G*G*G*G*T&:!!!!!r