5[gmdZddlZddlZddlmZmZddlmZm Z ddl m Z ddl m Z mZddlmZddlmZddlZGd d eZGd d eZGd deZGddej2j4ZGddej2j4ZGddeZ d#dZdZdZ dZ!dZ"dZ#dZ$dZ%dZ&dZ'dZ(d Z)d$d!Z*d"Z+y)%zR Matrix functions that use Pade approximation with inverse scaling and squaring. N) SqrtmError _sqrtm_triu)schurrsf2csf)funm)svdvalssolve_triangular)LinearOperator) onenormestc eZdZy)LogmRankWarningN__name__ __module__ __qualname__Y/var/www/html/bid-api/venv/lib/python3.12/site-packages/scipy/linalg/_matfuncs_inv_ssq.pyr r rr c eZdZy)LogmExactlySingularWarningNrrrrrrrrrc eZdZy)LogmNearlySingularWarningNrrrrrrrrrc eZdZy) LogmErrorNrrrrrrrrrc eZdZy)FractionalMatrixPowerErrorNrrrrrr"rrrc.eZdZdZdZdZdZdZdZy)_MatrixM1PowerOperatorz< A representation of the linear operator (A - I)^p. c|jdk7s|jd|jdk7r td|dks|t|k7r td||_||_|j|_|j|_y)Nrz%expected A to be like a square matrixz'expected p to be a non-negative integer)ndimshape ValueErrorint_A_p)selfAps r__init__z_MatrixM1PowerOperator.__init__,sq 66Q;!''!* 2DE E q5AQKFG GFF WW rcvt|jD] }|jj||z }"|SNranger(r'dotr)xis r_matvecz_MatrixM1PowerOperator._matvec6/twwA A"A rcvt|jD] }|j|j|z }"|Sr.)r0r(r1r'r2s r_rmatvecz_MatrixM1PowerOperator._rmatvec;s/twwAdgg"A rcvt|jD] }|jj||z }"|Sr.r/)r)Xr4s r_matmatz_MatrixM1PowerOperator._matmat@r6rcVt|jj|jSr.)rr'Tr()r)s r_adjointz_MatrixM1PowerOperator._adjointEs%dggii99rN) rrr__doc__r,r5r8r;r>rrrrr's    :rrc6tt||||||S)a Efficiently estimate the 1-norm of (A - I)^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. )titmax compute_v compute_w)r r)r*r+rArBrCrDs r_onenormest_m1_powerrEJs'J ,Q2u Y HHrcttj|jtjz dtjzz S)a Compute the scalar unwinding number. Uses Eq. (5.3) in [1]_, and should be equal to (z - log(exp(z)) / (2 pi i). Note that this definition differs in sign from the original definition in equations (5, 6) in [2]_. The sign convention is justified in [3]_. Parameters ---------- z : complex A complex number. Returns ------- unwinding_number : integer The scalar unwinding number of z. References ---------- .. [1] Nicholas J. Higham and Lijing lin (2011) "A Schur-Pade Algorithm for Fractional Powers of a Matrix." SIAM Journal on Matrix Analysis and Applications, 32 (3). pp. 1056-1078. ISSN 0895-4798 .. [2] Robert M. Corless and David J. Jeffrey, "The unwinding number." Newsletter ACM SIGSAM Bulletin Volume 30, Issue 2, June 1996, Pages 28-35. .. [3] Russell Bradford and Robert M. Corless and James H. Davenport and David J. Jeffrey and Stephen M. Watt, "Reasoning about the elementary functions of complex analysis" Annals of Mathematics and Artificial Intelligence, 36: 303-318, 2002. r!)r&npceilimagpi)zs r_unwindkrLss2H rww1RUU734 55rc|dkst||k7r td|dk(r|dz S|dk(rtj|dz S|}tj|tj dz k\rtj|}|dz }|dz }tj|}d|z}t d|D]}tj|}|d|zz}!||z }|S)a Computes r = a^(1 / (2^k)) - 1. This is algorithm (2) of [1]_. The purpose is to avoid a danger of subtractive cancellation. For more computational efficiency it should probably be cythonized. Parameters ---------- a : complex A complex number. k : integer A nonnegative integer. Returns ------- r : complex The value r = a^(1 / (2^k)) - 1 computed with less cancellation. Notes ----- The algorithm as formulated in the reference does not handle k=0 or k=1 correctly, so these are special-cased in this implementation. This function is intended to not allow `a` to belong to the closed negative real axis, but this constraint is relaxed. References ---------- .. [1] Awad H. Al-Mohy (2012) "A more accurate Briggs method for the logarithm", Numerical Algorithms, 59 : 393--402. rz expected a nonnegative integer kr"r!)r&r%rGsqrtanglerJr0)akk_hatz0rjs r_briggs_helper_functionrVsD 1uA! ;<<Av1u awwqzA~ 88A;"%%!) # AEE U GGAJ Eq%A AQU A! Frc||k(r||z||dz zz}|St||z t||zdz kDr|||z||zz z||z z }|S||z ||zz }tj|}tj|}tj|}|tj|dz ||zzz} t ||z } | r||tj dz| zzz} n||z} dtj| z||z z } | | z}|S)aO Compute a superdiagonal entry of a fractional matrix power. This is Eq. (5.6) in [1]_. Parameters ---------- l1 : complex A diagonal entry of the matrix. l2 : complex A diagonal entry of the matrix. t12 : complex A superdiagonal entry of the matrix. p : float A fractional power. Returns ------- f12 : complex A superdiagonal entry of the fractional matrix power. Notes ----- Care has been taken to return a real number if possible when all of the inputs are real numbers. References ---------- .. [1] Nicholas J. Higham and Lijing lin (2011) "A Schur-Pade Algorithm for Fractional Powers of a Matrix." SIAM Journal on Matrix Analysis and Applications, 32 (3). pp. 1056-1078. ISSN 0895-4798 r"r!?)absrGlogarctanhexprLrJsinh) l1l2t12r+f12rKlog_l1log_l2 arctanh_ztmp_atmp_utmp_btmp_cs r!_fractional_power_superdiag_entryris)F RxAgQqS !" J! R"WBG q( (b!eA&'273 J"Wb !JJqM bffacFVO455&) RUURZ%%778E MEBGGEN"b2g.em Jrc||k(r||z }|St||z t||zdz kDr6|tj|tj|z z||z z }|S||z ||zz }ttj|tj|z }|r:|dztj|tj dz|zzz||z z }|S|dztj|z||z z }|S)a Compute a superdiagonal entry of a matrix logarithm. This is like Eq. (11.28) in [1]_, except the determination of whether l1 and l2 are sufficiently far apart has been modified. Parameters ---------- l1 : complex A diagonal entry of the matrix. l2 : complex A diagonal entry of the matrix. t12 : complex A superdiagonal entry of the matrix. Returns ------- f12 : complex A superdiagonal entry of the matrix logarithm. Notes ----- Care has been taken to return a real number if possible when all of the inputs are real numbers. References ---------- .. [1] Nicholas J. Higham (2008) "Functions of Matrices: Theory and Computation" ISBN 978-0-898716-46-7 r!rX)rYrGrZrLr[rJ)r^r_r`rarKus r_logm_superdiag_entryrl sB RxBh J R"WBG q( (RVVBZ"&&*,-b9 J "Wb ! RVVBZ"&&*, - 'RZZ]RUU2XaZ78BGDC J'BJJqM)R"W5C Jrcxt|jdk7s|jd|jdk7r td|j\}}|}d}tj|}tj ||k7r t dtjtj|dz ddkDrNtj|}|dz }tjtj|dz ddkDrNt|D] }t|}|}d}t|dd z} t|d d z} t| | } d } d D]}| |ks |} n| ||kDrt|d d z} t|ddz} t| | dkrCtfddD}|dkr|} nvdz dkr|dkr|dz }t|}|dz }|t|ddz}t| |}t|}dD]}||ks |} n| nt|}|dz }| |tj|z }tdtj|D}|rt|D]}|||f}t!||}||||f<tj"| }t|dz D]8}|||f}||dz|dzf}|||dzf}t%||||}||||dzf<:tj&|tj(|s t d||| fS)az A helper function for inverse scaling and squaring for Pade approximation. Parameters ---------- T0 : (N, N) array_like upper triangular Matrix involved in inverse scaling and squaring. theta : indexable The values theta[1] .. theta[7] must be available. They represent bounds related to Pade approximation, and they depend on the matrix function which is being computed. For example, different values of theta are required for matrix logarithm than for fractional matrix power. Returns ------- R : (N, N) array_like upper triangular Composition of zero or more matrix square roots of T0, minus I. s : non-negative integer Number of square roots taken. m : positive integer The degree of the Pade approximation. Notes ----- This subroutine appears as a chunk of lines within a couple of published algorithms; for example it appears as lines 4--35 in algorithm (3.1) of [1]_, and as lines 3--34 in algorithm (4.1) of [2]_. The instances of 'goto line 38' in algorithm (3.1) of [1]_ probably mean 'goto line 36' and have been interpreted accordingly. References ---------- .. [1] Nicholas J. Higham and Lijing Lin (2013) "An Improved Schur-Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives." .. [2] Awad H. Al-Mohy and Nicholas J. Higham (2012) "Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm." SIAM Journal on Scientific Computing, 34 (4). C152-C169. ISSN 1095-7197 r!rr"*expected an upper triangular square matrixz%Diagonal entries of T must be nonzeroinitial?gUUUUUU?N)r"r!g?c34K|]}|ks |ywr.r).0r4a3thetas r z+_inverse_squaring_helper..sB12q>Qs )rtrurrr|r{皙?)r|rrc3\K|]$}|jdkDxs|jdk7&ywrNrealrIrwr3s rrzz+_inverse_squaring_helper..)N+Qqvvz8QVVq[8+*,zR is not upper triangular)lenr$r%rGdiag count_nonzero ExceptionmaxabsoluterNr0rrEminidentityallrVexp2ri array_equaltriu)T0rynr=s0tmp_diagr4srQd2d3a2md4j1d5a4etaRhas_principal_branchrUrPrTr+r^r_r`rarxs ` @r_inverse_squaring_helperr8s_\ 288}RXXa[BHHQK7EFF 88DAq A BwwqzH !Q&?@@ &&X\*B 7%( B778$ a &&X\*B 7%( B 2Y N A A a # ,B a # ,B RB A  q>A  ) r6%a+4B !!Q 'C 0 R[ q>BBBBQwa58#AQNQ !!Q 'C 0 R["bkAeAh =  N Q3 )< BKKNAN"''"+NNqA1a4A'1-AAadG GGQBKqsAAqDBAaC1HBQ!V*C3BCCCAa1fI  >>!RWWQZ (344 a7Nrc|dkr tdd|cxkrdkstdtd|dk(r| S|dzdk(r|dz}| |zdd|zdz zz S|dzdk(r|dz dz}| |z dd|zdzzz Std|)Nr"zexpected a positive integer iexpected -1 < t < 1r!rzunnexpected value of i, i = )r%r)r4rArUs r_fractional_power_pade_constantrs1u899 JQJ.// .//Avr Q! FQ1!a=)) Q! UqLQ1!a=))6qc:;;rc~|dkst||k7r tdd|cxkrdkstdtdtj|}t |j dk7s|j d|j dk7r td|j \}}tj |}|td|z|z}td|zdz ddD] }|t||z}t||z|}"||z}tj|tj|s td|S) a Evaluate the Pade approximation of a fractional matrix power. Evaluate the degree-m Pade approximation of R to the fractional matrix power t using the continued fraction in bottom-up fashion using algorithm (4.1) in [1]_. Parameters ---------- R : (N, N) array_like Upper triangular matrix whose fractional power to evaluate. t : float Fractional power between -1 and 1 exclusive. m : positive integer Degree of Pade approximation. Returns ------- U : (N, N) array_like The degree-m Pade approximation of R to the fractional power t. This matrix will be upper triangular. References ---------- .. [1] Nicholas J. Higham and Lijing lin (2011) "A Schur-Pade Algorithm for Fractional Powers of a Matrix." SIAM Journal on Matrix Analysis and Applications, 32 (3). pp. 1056-1078. ISSN 0895-4798 r"zexpected a positive integer mrrr!rrnU is not upper triangular) r&r%rGasarrayrr$rrr0r rrr) rrArridentYrUrhsUs r_fractional_power_paders(> 1uA! 899 JQJ.// .// 1 A 177|qAGGAJ!''!*4EFF 77DAq KKNE +AaC 33A 1Q37Ar "1!Q77 UQY ,#  A >>!RWWQZ (344 Hrc.dddddddd}|j\}}|}tj|}tj|tj|rtj||z}nt ||\}}} t | || }tj|} t d | D} t|d d D]} | |kr|j|}| s|tj| z} || z|tj|<t|d z D]8}|||f}||d z|d zf}|||d zf}t|||| }||||d zf<:tj|tj|s td |S) a Compute a fractional power of an upper triangular matrix. The fractional power is restricted to fractions -1 < t < 1. This uses algorithm (3.1) of [1]_. The Pade approximation itself uses algorithm (4.1) of [2]_. Parameters ---------- T : (N, N) array_like Upper triangular matrix whose fractional power to evaluate. t : float Fractional power between -1 and 1 exclusive. Returns ------- X : (N, N) array_like The fractional power of the matrix. References ---------- .. [1] Nicholas J. Higham and Lijing Lin (2013) "An Improved Schur-Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives." .. [2] Nicholas J. Higham and Lijing lin (2011) "A Schur-Pade Algorithm for Fractional Powers of a Matrix." SIAM Journal on Matrix Analysis and Applications, 32 (3). pp. 1056-1078. ISSN 0895-4798 gǖ2>gמYb?gUN@?g?W[?gX9v?r}gB`"?)r"r!rtrur{r|rrc3\K|]$}|jdkDxs|jdk7&ywrrrs rrzz/_remainder_matrix_power_triu..As)"Mf166A:#<1#>!RWWQZ (344 Hrctj|}t|jdk7s|jd|jdk7r t d|j\}}tj |tj |rd}|}nltj|rGt|\}}tj |tj |s t||\}}nt|d\}}tj|}tj||k7r tdtj|r-tj|dkr|jt}t!||}|?tj"|j$}|j'|j'|S|S) a{ Compute the fractional power of a matrix, for fractions -1 < t < 1. This uses algorithm (3.1) of [1]_. The Pade approximation itself uses algorithm (4.1) of [2]_. Parameters ---------- A : (N, N) array_like Matrix whose fractional power to evaluate. t : float Fractional power between -1 and 1 exclusive. Returns ------- X : (N, N) array_like The fractional power of the matrix. References ---------- .. [1] Nicholas J. Higham and Lijing Lin (2013) "An Improved Schur-Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives." .. [2] Nicholas J. Higham and Lijing lin (2011) "A Schur-Pade Algorithm for Fractional Powers of a Matrix." SIAM Journal on Matrix Analysis and Applications, 32 (3). pp. 1056-1078. ISSN 0895-4798 r!rr"zinput must be a square arrayNcomplexoutputz`cannot use inverse scaling and squaring to find the fractional matrix power of a singular matrix)rGrrr$r%rr isrealobjrrrrrrastyperr conjugater=r1)r*rArZr=T_diagrZHs r_remainder_matrix_powerrTsf@ 1 A 177|qAGGAJ!''!*4788 77DAq ~~a$   <<?8DAq>>!RWWQZ0q!}19-DAqWWQZF 1$(CD D  ||A266&>A- HHW  %Q*A} \\!_  uuQx||Brc tj|}t|jdk7s|jd|jdk7r t d|t |k(r)tj j|t |St|}|dr|d|dz }|tj|z }|tj|z }||d|z zz| |zkr!t tj|}| n t tj|}| t| }tj j||}|j|S|dkr6tj|} | jtj | S|tj|z }t tj|}| t#| fdd\}} tj j||}|j|S#tj j$rYwxYw) z Compute the fractional power of a matrix. See the fractional_matrix_power docstring in matfuncs.py for more info. r!rr"expected a square matrixrct|Sr.)pow)r3bs rz*_fractional_matrix_power..s C1IrF)disp)rGrrr$r%r&linalg matrix_powerrfloorrHrr1 LinAlgError empty_likefillnanr) r*r+rk2p1p2rPrQr:infors @r_fractional_matrix_powerrs 1 A 177|qAGGAJ!''!*4344CF{yy%%aQ00 A uqTAbE\ !_ ^ q2v 2#( *BHHQK AABGGAJAA '1-A &&q!,A558O  1u MM!  rvv !_   q-E:4 II " "1a (uuQxyy$$   s/g3kb?gj+ݓ?gz):˯?gMb?g|?5^?g;On?grh|?gjt?gjt?gQ?grh|?gw/?g?5^I ?gOn?g+?zinternal errorrsc3\K|]$}|jdkDxs|jdk7&ywrrrs rrzz_logm_triu..rrr)rGrrr$r%rrrrrrscipyspecialp_rootsrrr zeros_likezipr rrrZrr0rlrr)r=rr keep_it_realrryrrrnodesweightsrralphabetarr4r^r_r`s r _logm_triursBF 1 A 177|qAGGAJ!''!*4EFF 77DAqWWQZF<<?Frvvfb'AQ'FL  XXg  0E 'r51GAq!]]**1-NE7 JJE {{qdgmmt3()) #+ EGmG KKNE aA7E* t ed1fneAg 66+OA N"''"+NN!#rwwr{ 3"//!  qsAAqDBAaC1HBQ!V*C-b"c:Aa1fI  >>!RWWQZ (344 Hrcd}tjtj|}tj|dk(r`d}t j |t d|s|j}|jd}t|D]}|||fr ||||f<|Stj||krd}t j |td|S)Ng#B ;rz*The logm input matrix is exactly singular.rt) stacklevelz-The logm input matrix may be nearly singular.) rGrranywarningswarnrcopyr$r0r)r=inplacetri_epsabs_diagexact_singularity_msgrr4near_singularity_msgs r)_logm_force_nonsingular_triangular_matrixr4sG{{2771:&H vvh!m L +-GTUVA GGAJqAQT7!!Q$ H 7" #N *,ERST Hrcftj|}t|jdk7s|jd|jdk7r t dt |j jtjrtj|t}tj|} tj|tj|rXt|}tjtj|ddkr|j!t"}t%|S|rGt'|\}}tj|tj|s t)||\}}nt'|d \}}t|d }t%|}tj*|j,}|j/|j/|S#t0t2f$r9tj4|}|j7tj8|cYSwxYw) a^ Compute the matrix logarithm. See the logm docstring in matfuncs.py for more info. Notes ----- In this function we look at triangular matrices that are similar to the input matrix. If any diagonal entry of such a triangular matrix is exactly zero then the original matrix is singular. The matrix logarithm does not exist for such matrices, but in such cases we will pretend that the diagonal entries that are zero are actually slightly positive by an ad-hoc amount, in the interest of returning something more useful than NaN. This will cause a warning. r!rr"r)dtyperorprrT)r)rGrrr$r% issubclassrtypeintegerfloatrrrrrrrrrrrrr=r1rrrrr)r*rr=rrrr:s r_logmrHs" 1 A 177|qAGGAJ!''!*4344!'',, + JJq &<<?L >>!RWWQZ (9!rs@5'29* k       %%  !6!6 :^:H27&HR$6N4n5p,^FR<$/ dL ^HV/d` F (.r