Otj<ddlmZddlmZmZmZddlmZddlm Z m Z dddZ ddd d Z d Z d Zd ZdZdZdZdZdddZdddZdddZdZdZdZdZdS))ZZ)SDM sdm_irref sdm_rref_den)DDM) ddm_irref ddm_irref_denauto)methodct||d\}}t||\}}|dkr"t|}t|\}}n|dkr&t |\}}}t||z }nW|dkr?|d\} } t | \}}}t||z }nt d|t||\}} ||fS) a Compute the reduced row echelon form of a ``DomainMatrix``. This function is the implementation of :meth:`DomainMatrix.rref`. Chooses the best algorithm depending on the domain, shape, and sparsity of the matrix as well as things like the bit count in the case of :ref:`ZZ` or :ref:`QQ`. The result is returned over the field associated with the domain of the Matrix. See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.rref The ``DomainMatrix`` method that calls this function. sympy.polys.matrices.rref._dm_rref_den Alternative function for computing RREF with denominator. F denominatorGJFFCDTconvertUnknown method for rref: )_dm_rref_choose_method _dm_to_fmt _to_field _dm_rref_GJ_dm_rref_den_FFclear_denoms_rowwise ValueError) Mr use_fmtold_fmtMfM_rrefpivotsM_rref_fden_Mrs W/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/polys/matrices/rref.py_dm_rrefr'%s&-QEJJJOFGAw''JAw ~~ q\\$R 4 / 2 2#v8$$s* 4&&t&442 / 3 3#v8$$s*=V==>>>67++IFA 6>T) keep_domainr ct||d\}}t||\}}|dkrt|\}}}n |dkr~tt |\}}|rN|j|jkr>|d\} }|r|d|dfj}n|jj}n|}|jj}n|dkr| d\} } t| \} }}|r/| j|jkrt | |z }|jj}n9| }|r|d|dfj}n|jj}ntd|t||\}} |||fS) a Compute the reduced row echelon form of a ``DomainMatrix`` with denominator. This function is the implementation of :meth:`DomainMatrix.rref_den`. Chooses the best algorithm depending on the domain, shape, and sparsity of the matrix as well as things like the bit count in the case of :ref:`ZZ` or :ref:`QQ`. The result is returned over the same domain as the input matrix unless ``keep_domain=False`` in which case the result might be over an associated ring or field domain. See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.rref_den The ``DomainMatrix`` method that calls this function. sympy.polys.matrices.rref._dm_rref Alternative function for computing RREF without denominator. Tr rrrrrr) rrrrrdomain clear_denomselementonerr) rr)r rrr r#r!r"r$r%M_rref_rs r& _dm_rref_denr0Ys(-QDIIIOFGAw''JAw ~~-a00VV 4&y||44&  $8?ah66 --d-;;IAv (Qq \*2m'F-#CC 4&&t&442 / 3 3#v  (8?ah66x((3.F(,CCF (Qq \*2m'=V==>>>67++IFA 3 r(c|jj}||krnH|dkr|}n-|dkr|}nt d|||fS)z?Convert a matrix to the given format and return the old format.densesparsezUnknown format: )repfmtto_dense to_sparser)rr5rs r&rrsheiG#~~  JJLL  KKMM1C11222 g:r(c^|jjdkrt|St|S)z:Compute RREF using Gauss-Jordan elimination with division.r3)r4r5_dm_rref_GJ_sparse_dm_rref_GJ_densers r&rrs-uyH!!$$$ ###r(c^|jjdkrt|St|S)z:Compute RREF using fraction-free Gauss-Jordan elimination.r3)r4r5_dm_rref_den_FF_sparse_dm_rref_den_FF_denser;s r&rrs-uyH%a((($Q'''r(ct|j\}}}t||j|j}t |}|||fS)zACompute RREF using sparse Gauss-Jordan elimination with division.)rr4rshaper+tuplefrom_rep)rM_rref_dr!r$ M_rref_sdms r&r9r9sO#AE**HfaXqw11J 6]]F ::j ! !6 ))r(cP|jjp |jj}|j}t ||}t||j|j}t|}| | |fS)z@Compute RREF using dense Gauss-Jordan elimination with division.)_partial_pivot) r+is_RRis_CCr4to_ddmcopyrrr@rArB to_dfm_or_ddm)r partial_pivotddmr! M_rref_ddms r&r:r:sHN4ahnM %,,..    C s= 9 9 9FS!'18,,J 6]]F ::j..00 1 16 99r(ct|j|j\}}}t||j|j}t |}||||fSzACompute RREF using sparse fraction-free Gauss-Jordan elimination.)rr4r+rr@rArB)rrCr#r!rDs r&r=r=sU(99Hc6Xqw11J 6]]F ::j ! !3 ..r(c0|j}t||j\}}t ||j|j}t|}|| ||fSrP) r4rIrJr r+rr@rArBrK)rrMr#r!rNs r&r>r>sw %,,..    CQX..KCS!'18,,J 6]]F ::j..00 1 13 >>r(Fr cb|dkr3|dr|dtd }d}nvd}nsd}|j}|jrt ||}nQ|jrt ||}n8|js|jrd}d}n%|j r|j j dkr|sd}d}n|rd}nd}||fS) z3Choose the fastest method for computing RREF for M.r _denseNr2r3r rr) endswithlenr+is_ZZ_dm_rref_choose_method_ZZis_QQ_dm_rref_choose_method_QQrGrHis_EXr4r5)rr rrKs r&rrs ??8 $ $ Oc(mm^O,FGGGG H 7 .qkJJJFF W .qkJJJFF W  FGG W g--k-FGG  7?r(ctt|\}}}|td|dz krdSt|\}}td|Dd}tj}|D]5} t j|| }|d|zkrdS6|dkrdSd S) z5Choose the fastest method for computing RREF over QQ.rc6g|]}|S bit_length).0ns r& z-_dm_rref_choose_method_QQ../s 555allnn555r(default2rr)_dm_row_densitymin_dm_QQ_numers_denomsmaxrr.lcmrb) rrdensityr$ncolsnumersdenoms numer_bits denom_lcmds r&rYrYs(**GQQa  t*!,,NFF55f555qAAAJI F9a((    ! !AjL 0 044 1""ttr(c4d}t|\}}}|dkr ||dz krdSdS|dkrdS|d||z zkrdSt|}td|Dd }tdd |z|z }d|||dzzz z|z} || krdSdS) z5Choose the fastest method for computing RREF over ZZ.i' r^rrr]c6g|]}|Sr`rarces r&rez-_dm_rref_choose_method_ZZ..ns 1111 111r(rfrggUUUUUU?)rj _dm_elementsrm) rrPARAMronrows_nzrpelementsbitswideness max_densitys r&rWrWFs$ E /q11GXu"}} U1W  44{{t 1uX~% % %tAH 111111 = = =D1c%i())HuhtQw.//8;Kttr(c|jd}|j}|sdd|fSt |}t t t||z }|||fS)aDensity measure for sparse matrices. Defines the "density", ``d`` as the average number of non-zero entries per row except ignoring rows that are fully zero. RREF can ignore fully zero rows so they are excluded. By definition ``d >= 1`` except that we define ``d = 0`` for the zero matrix. Returns ``(density, nrows_nz, ncols)`` where ``nrows_nz`` counts the number of nonzero rows and ``ncols`` is the number of columns. rfr)r@r4to_sdmvaluesrUsummap)rrprows_nzr}ros r&rjrj|sr GAJEellnn##%%G (!U{w<<c#w''((83%''r(c4|\}}|S)z*Return nonzero elements of a DomainMatrix.) to_flat_nz)rr~r$s r&r{r{s,,..KHa Or(cXt|}d|D}d|D}||fS)zBReturns the numerators and denominators of a DomainMatrix over QQ.cg|] }|j Sr`) numeratorrys r&rez(_dm_QQ_numers_denoms..s , , ,aak , , ,r(cg|] }|j Sr`r rys r&rez(_dm_QQ_numers_denoms..s . . .am . . .r()r{)Mqr~rqrrs r&rlrls?BH , ,8 , , ,F . .X . . .F 6>r(cJ|j}|jr|S|S)z.Convert a DomainMatrix to a field if possible.)r+has_assoc_Fieldto_field)rr[s r&rrs' Azz||r(N)sympy.polys.domainsrsympy.polys.matrices.sdmrrrsympy.polys.matrices.ddmrsympy.polys.matrices.denserr r'r0rrrr9r:r=r>rrYrWrjr{rlrr`r(r&rs<#"""""AAAAAAAAAA((((((????????!11111h$(KKKKK\   "$$$(((***:::///???6;+++++\16*****Z1633333l(((, r(