a kº”hâBã@s¤dZddlmZddlmZddlmZmZddlm Z ddl m Z m Z dd „Z d d „Zd d „Zdd„Zdd„Zdd„Zdd„Zdd„Zdd„Zdddœdd„ZdS)z,Functions returning normal forms of matricesé)Ú defaultdicté)Ú DomainMatrix)Ú DMDomainErrorÚ DMShapeError)Úsymmetric_residue)ÚQQÚZZcCst|ƒ}t ||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]]) )Úinvariant_factorsrÚdiagÚdomainÚshape)ÚmÚinvsÚsmf©rúN/var/www/auris/lib/python3.9/site-packages/sympy/polys/matrices/normalforms.pyÚsmith_normal_formsrcCsì|j}|j}|j}| ¡}t|dƒD]8}t|dƒD]&}||krDq6||||ks6dSq6q&t|d|dƒ}td|ƒD]j}||d|d|kr°||||krædSq|| |||||d|d¡d}||kr|dSq|dS)z8 Checks that the matrix is in Smith Normal Form rrFT)r r ÚzeroÚto_listÚrangeÚminÚdiv)rr r rÚiÚjÚupperÚrrrrÚis_smith_normal_form(s& (rc Csbtt|ƒƒD]P}|||}|||||||||<|||||||||<q dS©N©rÚlen© rrrÚaÚbÚcÚdÚkÚerrrÚ add_columnsEs  r(cCs$|j}|j}| ¡}t|||ddS)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 F©r Úfull)r r rÚ_smith_normal_decomp)rr r rrrr Ns r c Csr|j}|j\}}}| ¡}t|||dd\}}}t |||¡ ¡}t||||fd}t||||fd}|||fS)aå 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))r r )r r rr+rr Úto_dense) rr ÚrowsÚcolsr rÚsÚtrrrrÚsmith_normal_decomp`sr1c s.ˆjsdˆ›}t|ƒ‚|\‰‰ˆj‰ ˆj‰‡‡ fdd„}d|vr\ˆrXd|ˆƒ|ˆƒfSdSˆrp|ˆƒ‰|ˆƒ‰ dd„‰‡‡‡‡‡‡‡ fdd „}‡‡‡‡‡ ‡ fd d „}‡‡ fd d „tˆƒDƒ}|r|dˆ krˆ|dˆdˆd<ˆ|d<ˆr¬ˆ|dˆdˆd<ˆ|d<nŽ‡‡ fdd „tˆƒDƒ}|r¬|dˆ kr¬ˆD](} | |d| d| d<| |d<qNˆr¬ˆ D](} | |d| d| d<| |d<q‚t‡‡ fdd„tdˆƒDƒƒsìt‡‡ fdd„tdˆƒDƒƒrü|ƒ|ƒq¬‡fdd„} ˆdddkr„ˆ ˆdd¡‰ˆjrDdˆdd‰ˆˆjkr„ˆddˆ9<ˆr„‡fdd „ˆdDƒˆd<d|vr”d} nÎdd „ˆdd…Dƒ} t| ˆˆdˆdfˆd} ˆr^| \} }}dgdgˆdgdd „|Dƒ}dgdgˆdgdd „|Dƒ}t t | ˆ|ˆ |gƒƒ\‰}‰ }|ˆ‰ˆ |‰ ˆ  ¡‰ˆ   ¡‰ n| } ˆddr¾ˆddg}|  | ¡tt |ƒdƒD] }||||d}}|r²ˆ ||¡dˆ kr²ˆrꈠ||¡\}}}n ˆ ||¡}ˆ ||¡d}ˆr˜ˆ ||¡d}ˆˆ||ddd|dƒtˆ ||dd|ddƒˆˆ||dd| ddƒtˆ ||ddd| dƒˆˆ||dddddƒ||||d<|||<nqq˜nPˆrüˆdkräˆdd…ˆdg‰ˆdkrüdd „ˆ Dƒ‰ | ˆddf}ˆr"t|ƒˆˆ fSt|ƒSdS)zë 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 cs‡‡‡fdd„tˆƒDƒS)Ncs&g|]‰‡‡‡fdd„tˆƒDƒ‘qS)csg|]}|ˆkrˆnˆ‘qSrr©Ú.0r)rÚonerrrÚ Œóz@_smith_normal_decomp..eye...©r)r3)Únr4r)rrr5Œr6z5_smith_normal_decomp..eye..r7©r8)r4rr9rÚeye‹sz!_smith_normal_decomp..eyerrc Ssftt|dƒƒD]P}|||}|||||||||<|||||||||<qdS)Nrrr!rrrÚadd_rows˜s  z&_smith_normal_decomp..add_rowsc sòˆdd}tdˆƒD]Ö}ˆ|dˆkr,qˆ ˆ|d|¡\}}|ˆkr~ˆˆd|dd| dƒˆr숈d|dd| dƒqˆ |ˆ|d¡\}}}ˆ ˆ|d|¡}ˆ ||¡}ˆˆd||||| ƒˆr興d||||| ƒ|}qdS©Nrr)rrÚgcdexÚexquo© Zpivotrr%rr"r#ÚgZd_0Zd_j)r;r r*rr-r/rrrÚ clear_column s   z*_smith_normal_decomp..clear_columnc sòˆdd}tdˆƒD]Ö}ˆd|ˆkr,qˆ ˆd||¡\}}|ˆkr~tˆd|dd| dƒˆrìtˆd|dd| dƒqˆ |ˆd|¡\}}}ˆ ˆd||¡}ˆ ||¡}tˆd||||| ƒˆrètˆd||||| ƒ|}qdSr<)rrr(r=r>r?)r.r r*rr0rrrÚ clear_row´s   z'_smith_normal_decomp..clear_rowcs g|]}ˆ|dˆkr|‘qS©rrr2©rrrrr5Ér6z(_smith_normal_decomp..cs g|]}ˆd|ˆkr|‘qSrCr)r3rrDrrr5Ïr6c3s|]}ˆd|ˆkVqdS©rNrr2rDrrÚ Ør6z'_smith_normal_decomp..rc3s|]}ˆ|dˆkVqdSrErr2rDrrrFÙr6cst|t|ƒt|dƒfˆdS)Nr)r r )rr )r)r rrÚto_domain_matrixÝsz._smith_normal_decomp..to_domain_matrixcsg|] }|ˆ‘qSrr)r3Úelem)r$rrr5çr6cSsg|]}|dd…‘qS)rNr)r3rrrrr5ìr6Nr)cSsg|]}dg|‘qSrCr©r3Úrowrrrr5ñr6cSsg|]}dg|‘qSrCrrIrrrr5òr6éÿÿÿÿcSs"g|]}|dd…|dg‘qS)rNrrrIrrrr5r6)Zis_PIDÚ ValueErrorrr4rÚanyZcanonical_unitZis_Fieldr+ÚlistÚmaprÚextendr rr=Úgcdr(Útuple)rr r r*Úmsgr:rArBÚindrJrGrZ lower_rightÚretZs_smallZt_smallÚs2Út2Úresultrr"r#ÚxÚyr%ÚalphaÚbetar) r;r$r.r r*rr4r-r/r0rrr+|s° "$&& ÿ    ÿ $$       r+cCsDt ||¡\}}}|dkr:||dkr:d}|dkr6dnd}|||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. rrKr)r r=)r"r#rYrZr@rrrÚ_gcdex"s r]c Cst|jjstdƒ‚|j\}}| ¡ ¡}|}t|dddƒD]}|dkrNqV|d8}t|dddƒD]l}|||dkrft||||||ƒ\}}}||||||||} } t|||||| | ƒqf|||} | dkrt|||ddddƒ| } | dkr|d7}q:t|d|ƒD],}|||| } t|||d| ddƒq&q:t   |  ¡¡dd…|d…fS)aè 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.) úMatrix must be over domain ZZ.rrKrN) r Úis_ZZrr Zto_ddmÚcopyrr]r(rZfrom_repZ to_dfm_or_ddm) ÚArr8r&rrÚuÚvr%rr/r#ÚqrrrÚ_hermite_normal_form7s0   "    rec Csä|jjstdƒ‚t |¡r"|dkr*tdƒ‚dd„}ttƒ}|j\}}||krTtdƒ‚|  ¡}|}|}t |dddƒD]X}|d8}t |dddƒD]n} ||| dkr’t |||||| ƒ\} } } |||| ||| | } }||||| | | | | ƒq’|||}|dkr(||||<}t ||ƒ\} } } t |ƒD]"}| |||||||<q@|||dkr‚||||<t |d|ƒD]4} ||| |||}t || |d| ddƒq|| }qtt |||ftƒ ¡S) 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.) r^rz0Modulus D must be positive element of domain ZZ.c Ssvtt|ƒƒD]d}|||} t|| ||||||ƒ|||<t|| ||||||ƒ|||<q dSr)rr r) rÚRrrr"r#r$r%r&r'rrrÚadd_columns_mod_R¶s *z8_hermite_normal_form_modulo_D..add_columns_mod_Rz2Matrix must have at least as many columns as rows.rKr)r r_rr Zof_typerÚdictr rrrr]r(rr,)raÚDrgÚWrr8r&rfrrrbrcr%rr/r#ÚiirdrrrÚ_hermite_normal_form_modulo_D„s@-  "      rlNF)riÚ check_rankcCsJ|jjstdƒ‚|dur>|r4| t¡ ¡|jdkr>t||ƒSt|ƒSdS)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.) r^Nr) r r_rZ convert_torZrankr rlre)rarirmrrrÚhermite_normal_formÜs ;$ rn)Ú__doc__Ú collectionsrZ domainmatrixrÚ exceptionsrrZsympy.ntheory.modularrZsympy.polys.domainsrr rrr(r r1r+r]rerlrnrrrrÚs      'MX