U Zʹj<@sddlmZddlmZmZmZddlmZddlm Z m Z ddddZ d dd d d Z d dZ ddZddZddZddZddZddZddddZdddd Zddd!d"Zd#d$Zd%d&Zd'd(Zd)d*Zd+S),)ZZ)SDM sdm_irref sdm_rref_den)DDM) ddm_irref ddm_irref_denauto)methodc Cst||dd\}}t||\}}|dkr>t|}t|\}}nf|dkrbt|\}}}t||}nB|dkr|jdd\} } t| \}}}t||}ntd|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_fmtZMfM_rrefpivotsM_rref_fden_Mrr#X/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/polys/matrices/rref.py_dm_rref%sr%T) keep_domainr c Cs<t||dd\}}t||\}}|dkr8t|\}}}n|dkrtt|\}}|r|j|jkr|jdd\} }|r|d|dfj}q|jj}n |}|jj}n|dkr|j dd\} } t| \} }}|r| j|jkrt| |}|jj}n&| }|r |d|dfj}n|jj}nt d|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 rr rrrr) rrrrrdomainZ clear_denomselementonerr) rr&r rrrr rrr!r"ZM_rref_rr#r#r$ _dm_rref_denYs4       r*cCsL|jj}||krn2|dkr$|}n |dkr6|}ntd|||fS)z?Convert a matrix to the given format and return the old format.densesparsezUnknown format: )repfmtZto_denseZ to_sparser)rr.rr#r#r$rs  rcCs |jjdkrt|St|SdS)z:Compute RREF using Gauss-Jordan elimination with division.r,N)r-r._dm_rref_GJ_sparse_dm_rref_GJ_denserr#r#r$rs rcCs |jjdkrt|St|SdS)z:Compute RREF using fraction-free Gauss-Jordan elimination.r,N)r-r._dm_rref_den_FF_sparse_dm_rref_den_FF_denser1r#r#r$rs rcCs6t|j\}}}t||j|j}t|}|||fS)zACompute RREF using sparse Gauss-Jordan elimination with division.)rr-rshaper'tuplefrom_rep)rM_rref_drr! M_rref_sdmr#r#r$r/sr/cCsT|jjp|jj}|j}t||d}t||j|j}t |}| | |fS)z@Compute RREF using dense Gauss-Jordan elimination with division.)Z_partial_pivot) r'is_RRis_CCr-to_ddmcopyrrr4r5r6 to_dfm_or_ddm)rZ partial_pivotddmr M_rref_ddmr#r#r$r0s  r0cCs<t|j|j\}}}t||j|j}t|}||||fSzACompute RREF using sparse fraction-free Gauss-Jordan elimination.)rr-r'rr4r5r6)rr7r rr8r#r#r$r2sr2cCsJ|j}t||j\}}t||j|j}t|}|| ||fSr@) r-r;r<rr'rr4r5r6r=)rr>r rr?r#r#r$r3s r3Fr cCs|dkr0|dr*|dtd }d}qd}nvd}|j}|jrNt||d}nX|jrbt||d}nD|jsn|jrxd}d}n.|j r|j j dkr|sd}d}n|rd}nd}||fS) z3Choose the fastest method for computing RREF for M.r Z_denseNr+r,r r r) endswithlenr'Zis_ZZ_dm_rref_choose_method_ZZZis_QQ_dm_rref_choose_method_QQr9r:Zis_EXr-r.)rr r rKr#r#r$rs*   rc Cst|\}}}|td|dkr$dSt|\}}tdd|Ddd}tj}|D]&} t|| }|d|krPdSqP|dkrd Sd Sd S) z5Choose the fastest method for computing RREF over QQ.r cSsg|] }|qSr# bit_length).0nr#r#r$ /sz-_dm_rref_choose_method_QQ..default2rrN)_dm_row_densitymin_dm_QQ_numers_denomsmaxrr)ZlcmrI) rr densityr!ncolsnumersdenomsZ numer_bitsZ denom_lcmdr#r#r$rDs    rDc Csd}t|\}}}|dkr.||dkr*dSdS|dkr:dS|d||krNdSt|}tdd|Dd d }td d ||}d|||d|} || krdSdSd S) z5Choose the fastest method for computing RREF over ZZ.i' rGr rrFcSsg|] }|qSr#rHrJer#r#r$rLnsz-_dm_rref_choose_method_ZZ..rMrNgUUUUUU?N)rQ _dm_elementsrT) rr ZPARAMrUnrows_nzrVelementsbitsZwidenessZ max_densityr#r#r$rCFs" rCcCsN|jd}|j}|s&dd|fSt|}ttt||}|||fSdS)aDensity measure for sparse matrices. Defines the "density", ``d`` as the average number of non-zero entries per row except ignoring rows that are fully zero. RREF can ignore fully zero rows so they are excluded. By definition ``d >= 1`` except that we define ``d = 0`` for the zero matrix. Returns ``(density, nrows_nz, ncols)`` where ``nrows_nz`` counts the number of nonzero rows and ``ncols`` is the number of columns. rMrN)r4r-Zto_sdmvaluesrBsummap)rrVZrows_nzr^rUr#r#r$rQ|s  rQcCs|\}}|S)z*Return nonzero elements of a DomainMatrix.)Z to_flat_nz)rr_r!r#r#r$r]s r]cCs,t|}dd|D}dd|D}||fS)zBReturns the numerators and denominators of a DomainMatrix over QQ.cSsg|] }|jqSr#) numeratorr[r#r#r$rLsz(_dm_QQ_numers_denoms..cSsg|] }|jqSr#r r[r#r#r$rLs)r])ZMqr_rWrXr#r#r$rSsrScCs|j}|jr|S|SdS)z.Convert a DomainMatrix to a field if possible.N)r'Zhas_assoc_FieldZto_field)rrEr#r#r$rsrN)Zsympy.polys.domainsrZsympy.polys.matrices.sdmrrrZsympy.polys.matrices.ddmrZsympy.polys.matrices.denserrr%r*rrrr/r0r2r3rrDrCrQr]rSrr#r#r#r$s&  4N  .-6