a Юм.get_colcs^И|И|dИЕИ|И|dИЕИ|И|dИЕ<И|И|dИЕ<dS)Nrr )r┌jr r r┌row_swap2s. z"_row_reduce_list..row_swapcsP||И}t|И|dИГD](}И|И||И||ГИ|<q"dS)z,Does the row op row[i] = a*row[i] - b*row[j]rN)┌range)┌ar┌br┌q┌pйrZisimprr r┌cross_cancel6sz&_row_reduce_list..cross_cancelйrrNrrFT)r┌_dotprodsimpr┌appendr┌ enumerate┌tuple)r┌rowsr┌one┌ iszerofunc┌simpfunc┌normalize_last┌ normalize┌ zero_aboverrrZpiv_rowZpiv_col┌ pivot_cols┌swapsZpivot_offsetZ pivot_valZassumed_nonzeroZnewly_determined┌offset┌valrrr┌rowZpiv_iZpiv_jr rr┌_row_reduce_list s\% " "r,c CsBtt|Г|j|j|j|||||dН \}}}|а|j|j|б||fS)Nйr$r%r&)r,┌listr rr!┌_new) ┌Mr"r#r$r%r&rr'r(r r r┌_row_reduce|s ■r1csВ|jdks|jdkrdStЗfddД|ddЕdfDГГ}И|dГrd|obt|ddЕddЕfИГS|oАt|ddЕddЕfИГS)zжReturns `True` if the matrix is in echelon form. That is, all rows of zeros are at the bottom, and below each leading non-zero in a row are exclusively zeros.rTc3s|]}И|ГVqdSr r )┌.0┌tйr"r r┌ Оєz_is_echelon..rNr)r r┌all┌_is_echelon)r0r"Zzeros_belowr r4rr8Жs"r8FcCs<t|tГr|nt}t|||ddddН\}}}|r8||fS|S)anReturns a matrix row-equivalent to ``M`` that is in echelon form. Note that echelon form of a matrix is *not* unique, however, properties like the row space and the null space are preserved. Examples ======== >>> from sympy import Matrix >>> M = Matrix([[1, 2], [3, 4]]) >>> M.echelon_form() Matrix([ [1, 2], [0, -2]]) TFr-й┌ isinstancer┌ _simplifyr1)r0r"rZwith_pivotsr#r┌pivots┌_r r r┌ _echelon_formЦs r>c s№ddД}t|tГr|nt}|jdks.|jdkr2dS|jdksF|jdkrdЗfddД|DГ}d|vrddS|jdkr╩|jdkr╩Зfd dД|DГ}d|vrЮd |vrЮdS|аб}И|Гr║d|vr║dSИ|Гdur╩dS||ИdН\}}t|И|dddd Н\}} }t| ГS)z Returns the rank of a matrix. Examples ======== >>> from sympy import Matrix >>> from sympy.abc import x >>> m = Matrix([[1, 2], [x, 1 - 1/x]]) >>> m.rank() 2 >>> n = Matrix(3, 3, range(1, 10)) >>> n.rank() 2 csJЗЗfddДЙЗfddДtИjГDГ}ddДt|ГDГ}Иj|ddН|fS)aАPermute columns with complicated elements as far right as they can go. Since the ``sympy`` row reduction algorithms start on the left, having complexity right-shifted speeds things up. Returns a tuple (mat, perm) where perm is a permutation of the columns to perform to shift the complex columns right, and mat is the permuted matrix.cs"tЗfddДИddЕ|fDГГS)Nc3s"|]}И|ГdurdndVqdS)Nrrr )r2┌er4r rr5╧r6zO_rank.._permute_complexity_right..complexity..)┌sumr)r0r"r r┌ complexity╠sz<_rank.._permute_complexity_right..complexitycsg|]}И|Г|fСqSr r )r2r)rAr r┌ ╤r6z<_rank.._permute_complexity_right..cSsg|]\}}|СqSr r )r2rrr r rrB╥r6r)Zorientation)rr┌sortedZpermute)r0r"┌complex┌permr )r0rAr"r┌_permute_complexity_right┬s z(_rank.._permute_complexity_rightrrcsg|]}И|ГСqSr r йr2┌xr4r rrB▀r6z_rank..Fщcsg|]}И|ГСqSr r rGr4r rrBхr6Nr4Tr-)r:rr;r rZdetr1┌len) r0r"rrFr#Zzeros┌drr=r<r r4r┌_rank▓s, rLc Cs<t|tГr|nt}t||||dddН\}}}|r8||f}|S)aЗReturn reduced row-echelon form of matrix and indices of pivot vars. Parameters ========== iszerofunc : Function A function used for detecting whether an element can act as a pivot. ``lambda x: x.is_zero`` is used by default. simplify : Function A function used to simplify elements when looking for a pivot. By default SymPy's ``simplify`` is used. pivots : True or False If ``True``, a tuple containing the row-reduced matrix and a tuple of pivot columns is returned. If ``False`` just the row-reduced matrix is returned. normalize_last : True or False If ``True``, no pivots are normalized to `1` until after all entries above and below each pivot are zeroed. This means the row reduction algorithm is fraction free until the very last step. If ``False``, the naive row reduction procedure is used where each pivot is normalized to be `1` before row operations are used to zero above and below the pivot. Examples ======== >>> from sympy import Matrix >>> from sympy.abc import x >>> m = Matrix([[1, 2], [x, 1 - 1/x]]) >>> m.rref() (Matrix([ [1, 0], [0, 1]]), (0, 1)) >>> rref_matrix, rref_pivots = m.rref() >>> rref_matrix Matrix([ [1, 0], [0, 1]]) >>> rref_pivots (0, 1) Notes ===== The default value of ``normalize_last=True`` can provide significant speedup to row reduction, especially on matrices with symbols. However, if you depend on the form row reduction algorithm leaves entries of the matrix, set ``noramlize_last=False`` T)r%r&r9) r0r"rr<r$r#rr'r=r r r┌_rref°s7 rMN)TTT)TTT)┌typesrZsympy.simplify.simplifyrr;rrZ utilitiesrrZdeterminantrr,r1r8r>rLrMr r r r┌s r F