oVhBdZddlmZddlmZddlmZmZddlm Z ddl m Z m Z dZ d Zd Zd Zd Zd ZdZdZdZddddZy)z,Functions returning normal forms of matrices) defaultdict) DomainMatrix) DMDomainError DMShapeError)symmetric_residue)QQZZcrt|}tj||j|j}|S)aI Return the Smith Normal Form of a matrix `m` over the ring `domain`. This will only work if the ring is a principal ideal domain. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.polys.matrices.normalforms import smith_normal_form >>> m = DomainMatrix([[ZZ(12), ZZ(6), ZZ(4)], ... [ZZ(3), ZZ(9), ZZ(6)], ... [ZZ(2), ZZ(16), ZZ(14)]], (3, 3), ZZ) >>> print(smith_normal_form(m).to_Matrix()) Matrix([[1, 0, 0], [0, 10, 0], [0, 0, 30]]) )invariant_factorsrdiagdomainshape)minvssmfs P/home/dcms/DCMS/lib/python3.12/site-packages/sympy/polys/matrices/normalforms.pysmith_normal_formrs/$ Q D   D!((AGG 4C Jc|j}|j}|j}|j}t |dD])}t |dD]}||k(r ||||k(ry+t |d|d}t d|D]O}||dz |dz |k(r||||k7s y|j |||||dz |dz d}||k7sOyy)z8 Checks that the matrix is in Smith Normal Form rrFT)rrzeroto_listrangemindiv)rrrrijupperrs ris_smith_normal_formr (sXXF GGE ;;D A 58_uQx AAvQ47d?   a%( #E 1e_ QqS6!A#;$ tAw$ 1Q47AacF1Q3K03ADy rctt|D]8}|||}||z||||zz|||<||z||||zz|||<:yNrlen rrrabcdkes r add_columnsr,Esj3q6]" aDGA#!A$q' /!QA#!A$q' /!Q"rcp|j}|j}|j}t|||dS)a3 Return the tuple of abelian invariants for a matrix `m` (as in the Smith-Normal form) References ========== [1] https://en.wikipedia.org/wiki/Smith_normal_form#Algorithm [2] https://web.archive.org/web/20200331143852/https://sierra.nmsu.edu/morandi/notes/SmithNormalForm.pdf Frfull)rrr_smith_normal_decomp)rrrs rr r Ns2XXF GGE A 6U CCrc|j}|jx\}}}|j}t|||d\}}}t j |||j }t ||||f}t ||||f}|||fS)a Return the Smith-Normal form decomposition of matrix `m`. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.polys.matrices.normalforms import smith_normal_decomp >>> m = DomainMatrix([[ZZ(12), ZZ(6), ZZ(4)], ... [ZZ(3), ZZ(9), ZZ(6)], ... [ZZ(2), ZZ(16), ZZ(14)]], (3, 3), ZZ) >>> a, s, t = smith_normal_decomp(m) >>> assert a == s * m * t Tr.)rr)rrrr0rr to_dense) rrrowscolsrrstrs rsmith_normal_decompr7`s XXF JD$ A%au4HJD!Q   D&% 0 9 9 ;CQvdD\:AQvdD\:A 19rc   !"#$%jsd}t||\" j%j!!%fd}d|vrrd|"| fSyr|"#| $d"#%fd} $%fd}t "Dcgc]}|d%k7s|} }| r9| d%k7r1| ddcd<| d<r#| d#dc#d<#| d<nlt D cgc]} d| %k7s| } } | rF| d%k7r>D]} | | d| dc| d<| | d<r$D]} | | d| dc| d<| | d<t %fdt d Dst %fd t d "DrN||t %fdt d Dr.t %fd t d "DrNfd } dddk7rhj dd} jr d ddz } | jk7r+ddxx| zcc<r#dDcgc]}|| z c}#d<d |vrd}nʼnd d Dcgc]}|d d  }}t|"d z d z f }r|\}}}d gdg"d z zzg|D cgc]} dg| z c} z}d gdg d z zzg|D cgc]} dg| z c} z}tt| #|$|g\#}$}|#z#$|z$#j#$j$n|}ddr$ddg}|j|t t|d z D]}||||d z}}|r݉j||d %k7rʼnrj||\}}}nj!||}j||d}rrj||d}#||d zd dd t#$||d zd dd #||d zd | dd t#$||d zd d| d #||d zdd dd||z||d z<|||<n@n>r0"d kDr #d d #dgz# d kDr$D cgc]} | d d | dgzc} $|ddfz}rt%|#$fSt%|Scc}wcc} wcc}wcc}wcc} wcc} wcc} w)z Return the tuple of abelian invariants for a matrix `m` (as in the Smith-Normal form). If `full=True` then invertible matrices ``s, t`` such that the product ``s, m, t`` is the Smith Normal Form are also returned. zBThe matrix entries must be over a principal ideal domain, but got c t|Dcgc]"}t|Dcgc] }||k(rn c}$c}}Scc}wcc}}wr")r)nrroners reyez!_smith_normal_decomp..eyes8EJ1XN%(;QQD(;NN;NsA; AArctt|dD]8}|||}||z||||zz|||<||z||||zz|||<:y)Nrr#r%s radd_rowsz&_smith_normal_decomp..add_rowssos1Q4y! &A!QAcAad1gIoAaDGcAad1gIoAaDG &rc  dd}td D]} |dk(r j |d|\}}|k(r" d|dd| d sA d|dd| dQ j| |d\}}} j |d|} j||} d|||||  r d||||| |}yNrr)rrgcdexexquo)pivotrr)rr&r'gd_0d_jr?rr/rr3r5rs r clear_columnz*_smith_normal_decomp..clear_columns!Qq$ AtAw$::ad1gu-DAqDyAq!QA.Q1aQB2 ,,uad1g61all1Q47A.ll5!,Aq!QcT2Q1aC#6 rc  dd}td D]} d|k(r j d||\}}|k(r(t d|dd| d sDt d|dd| dW j| d|\}}} j d||} j ||}t d|||||  rt d||||| |}yrA)rrr,rBrC)rDrr)rr&r'rErFrGr4rr/rr6rs r clear_rowz'_smith_normal_decomp..clear_rows!Qq$ AtAw$::ad1gu-DAqDyAq!QA2q11aAr15 ,,uad1g61all1Q47A.ll5!,Aq!Q351aAsSD9 rc34K|]}d|k7ywrNr=.0rrrs r z'_smith_normal_decomp..61qtAw$6rc34K|]}|dk7ywrLr=rMs rrOz'_smith_normal_decomp..rPrQcNt|t|t|dfS)Nr)rr)rr$)rrs rto_domain_matrixz._smith_normal_decomp..to_domain_matrixs#Ac!fc!A$i%8HHrNr.)is_PID ValueErrorrr;ranycanonical_unitis_Fieldr0listmaprextendr$rrBgcdr,tuple)&rrrr/msgr<rHrJrindrrowrTr(elemrr lower_rightrets_smallt_smalls2t2resultr&r'xyr)alphabetar?r4r;r3r5r6rs&`` ` @@@@@@@rr0r0|s ==RSYRZ[oJD$ ;;D **CO Ez s4y#d)+ + I I&(*Dk 5QqT!W_1 5C 5 s1v~CF)QqT!aAi Ai1OAaD!CF)+9Q1aDq99 3q6T> :&)#a&k3q6#ACF  :>C*-c!f+s1v'CFCAK> 6a 6 6 6a 6 6  6a 6 6 6a 6 6I tAw!|  ! !!A$q' * ??AaDG A  ? aDGqLG-.qT2Tq2!Ez&'e,qu, ,";ax*7 %( "D'7#T!V $%g(Fs!s(FFB#T!V $%g(Fs!s(FFB$4q"an EFLAr1bQABA A ADtAwA$q' ds6{1}% A!9fQqSkqAVZZ1%a(D0$ll1a0GAq! 1a(A 1a(+!::a+A.DQ1q5!Q151a!eQ1a8Q1q5!eVQ:1a!eQD5!<Q1q5!QA6%iqs q ) , axabEQqTFNax345CSWAx'51a " V}a""V}m 6 :03 - )G(FN6s6R-*R-8R2 R2 R7) R<' S S0S cptj||\}}}|dk7r||zdk(r d}|dkrdnd}|||fS)a This supports the functions that compute Hermite Normal Form. Explanation =========== Let x, y be the coefficients returned by the extended Euclidean Algorithm, so that x*a + y*b = g. In the algorithms for computing HNF, it is critical that x, y not only satisfy the condition of being small in magnitude -- namely that |x| <= |b|/g, |y| <- |a|/g -- but also that y == 0 when a | b. rrUr)r rB)r&r'rkrlrEs r_gcdexrp"sGhhq!nGAq!Av!a%1* a%BQ a7Nrc |jjs td|j\}}|j j }|}t |dz ddD]}|dk(rn|dz}t |dz ddD]R}|||dk7st||||||\}}}||||z||||z} } t|||||| | T|||} | dkrt|||dddd| } | dk(r|dz }t |dz|D]}|||| z} t|||d| dd!tj|jdd|dfS)a Compute the Hermite Normal Form of DomainMatrix *A* over :ref:`ZZ`. Parameters ========== A : :py:class:`~.DomainMatrix` over domain :ref:`ZZ`. Returns ======= :py:class:`~.DomainMatrix` The HNF of matrix *A*. Raises ====== DMDomainError If the domain of the matrix is not :ref:`ZZ`. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithm 2.4.5.) Matrix must be over domain ZZ.rrUrN) ris_ZZrrto_ddmcopyrrpr,rfrom_rep to_dfm_or_ddm) Arr:r*rruvr)rr5r'qs r_hermite_normal_formr|7s8 88>><== 77DAq  A A 1q5"b !!2 6  Qq1ub"% 2AtAw!|!1a!A$q'21atAw!|QqT!W\1Aq!QA2q1 2 aDG q5 1aQA .A 6 FA 1q5!_ 2aDGqLAq!QAq1 2?!2H  !2 3AqrE ::rc |jjs tdtj|r|dkr tdd}t t }|j\}}||kr td|j}|}|}t|dz ddD]}|dz}t|dz ddD]P} ||| dk7st|||||| \} } } |||| z||| | z}} ||||| | | | | R|||}|dk(r |x|||<}t||\} } } t|D]}| |||z|z|||<|||dk(r||||<t|dz|D]%} ||| |||z}t|| |d| dd'|| z}t|||ftjS)a[ Perform the mod *D* Hermite Normal Form reduction algorithm on :py:class:`~.DomainMatrix` *A*. Explanation =========== If *A* is an $m \times n$ matrix of rank $m$, having Hermite Normal Form $W$, and if *D* is any positive integer known in advance to be a multiple of $\det(W)$, then the HNF of *A* can be computed by an algorithm that works mod *D* in order to prevent coefficient explosion. Parameters ========== A : :py:class:`~.DomainMatrix` over :ref:`ZZ` $m \times n$ matrix, having rank $m$. D : :ref:`ZZ` Positive integer, known to be a multiple of the determinant of the HNF of *A*. Returns ======= :py:class:`~.DomainMatrix` The HNF of matrix *A*. Raises ====== DMDomainError If the domain of the matrix is not :ref:`ZZ`, or if *D* is given but is not in :ref:`ZZ`. DMShapeError If the matrix has more rows than columns. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithm 2.4.8.) rrrz0Modulus D must be positive element of domain ZZ.ctt|D]R}|||} t|| z||||zz|z||||<t|| z||||zz|z||||<Tyr")rr$r) rRrrr&r'r(r)r*r+s radd_columns_mod_Rz8_hermite_normal_form_modulo_D..add_columns_mod_Rss1v FA!QA'QQqT!W)<(A1EAaDG'QQqT!W)<(A1EAaDG Frz2Matrix must have at least as many columns as rows.rUr)rrsrr of_typerdictrrrrrpr,rr2)rxDrWrr:r*rrrryrzr)rr5r'iir{s r_hermite_normal_form_modulo_Drs!Z 88>><== ::a=AENOOF DA 77DAq1uOPP A A A 1q5"b ! Qq1ub"% ;AtAw!| 1a!A$q'21atAw!|QqT!W\1!!Q1aQB:  ; aDG 6OAaDGaA,1a( &B2qzA~AbE!H & Q47a<AaDGq1ua .A!Q1Q47"A 1aQB1 - . a%& Aq62 & / / 11rNF)r check_rankc|jjs td|A|r3|jtj |j dk(r t||St|S)a) Compute the Hermite Normal Form of :py:class:`~.DomainMatrix` *A* over :ref:`ZZ`. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.polys.matrices.normalforms import hermite_normal_form >>> m = DomainMatrix([[ZZ(12), ZZ(6), ZZ(4)], ... [ZZ(3), ZZ(9), ZZ(6)], ... [ZZ(2), ZZ(16), ZZ(14)]], (3, 3), ZZ) >>> print(hermite_normal_form(m).to_Matrix()) Matrix([[10, 0, 2], [0, 15, 3], [0, 0, 2]]) Parameters ========== A : $m \times n$ ``DomainMatrix`` over :ref:`ZZ`. D : :ref:`ZZ`, optional Let $W$ be the HNF of *A*. If known in advance, a positive integer *D* being any multiple of $\det(W)$ may be provided. In this case, if *A* also has rank $m$, then we may use an alternative algorithm that works mod *D* in order to prevent coefficient explosion. check_rank : boolean, optional (default=False) The basic assumption is that, if you pass a value for *D*, then you already believe that *A* has rank $m$, so we do not waste time checking it for you. If you do want this to be checked (and the ordinary, non-modulo *D* algorithm to be used if the check fails), then set *check_rank* to ``True``. Returns ======= :py:class:`~.DomainMatrix` The HNF of matrix *A*. Raises ====== DMDomainError If the domain of the matrix is not :ref:`ZZ`, or if *D* is given but is not in :ref:`ZZ`. DMShapeError If the mod *D* algorithm is used but the matrix has more rows than columns. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithms 2.4.5 and 2.4.8.) rrr) rrsr convert_tor rankrrr|)rxrrs rhermite_normal_formrs\v 88>><==}jALL,<,A,A,Cqwwqz,Q,Q22#A&&r)__doc__ collectionsr domainmatrixr exceptionsrrsympy.ntheory.modularrsympy.polys.domainsr r rr r,r r7r0rpr|rrr=rrrsX2#&33&.:"D$8cL*J;ZU2p!%@'r