*h0SSKJr SSKJr SSKJrJr SSKJrJ r J r J r SSK J r SSjrSS jr\ 4S jr\ S S 4S jr\ S 4S jrSrSr\ S SS4Sjrg)) FunctionType)CoercionFailed)ZZQQ)_get_intermediate_simp_iszero _dotprodsimp _simplify)_find_reasonable_pivotTc ^^^UU4Sjn UU4Sjn UUU4Sjn [[5mSup/n/nU T:Ga&X:Ga [U "U 5U SXE5unnnnUHunnUU - nUTUT-U -'M UcU S- n MMURU 5 US:wa"U "U UU -5 URU UU -45 USLaCXnnUTUT-U-'[ UT-U-S-US-T-5HnT"TUU- 5TU'M Un[ U5H=nUU :XaM USLaUU :aMTUT-U -nU"U5(aM2U "UUUU 5 M? U S- n U T:aX:aGM US LacUS La^[ U5HOunnTUT-U-nUTUT-U-'[ UT-U-S-US-T-5HnT"TUU- 5TU'M MQ T[ U5[ U54$) a{Row reduce a flat list representation of a matrix and return a tuple (rref_matrix, pivot_cols, swaps) where ``rref_matrix`` is a flat list, ``pivot_cols`` are the pivot columns and ``swaps`` are any row swaps that were used in the process of row reduction. Parameters ========== mat : list list of matrix elements, must be ``rows`` * ``cols`` in length rows, cols : integer number of rows and columns in flat list representation one : SymPy object represents the value one, from ``Matrix.one`` iszerofunc : determines if an entry can be used as a pivot simpfunc : used to simplify elements and test if they are zero if ``iszerofunc`` returns `None` normalize_last : indicates where all row reduction should happen in a fraction-free manner and then the rows are normalized (so that the pivots are 1), or whether rows should be normalized along the way (like the naive row reduction algorithm) normalize : whether pivot rows should be normalized so that the pivot value is 1 zero_above : whether entries above the pivot should be zeroed. If ``zero_above=False``, an echelon matrix will be returned. c>TUST2$N)icolsmats V/home/mande/repo/quber/.venv/lib/python3.13/site-packages/sympy/matrices/reductions.pyget_col!_row_reduce_list..get_col/s17d7|cp>TUT-US-T-TUT-US-T-sTUT-US-T-&TUT-US-T-&g)Nrr)rjrrs rrow_swap"_row_reduce_list..row_swap2s] $At| $c!D&!a%&> ;AdFAE4< #afa!eT\":rc>X1- T-n[UT-US-T-5HnT"UTU-UTXT--- 5TU'M! g)z,Does the row op row[i] = a*row[i] - b*row[j]rN)range) arbrqprisimprs r cross_cancel&_row_reduce_list..cross_cancel6sP UDLqvAt|,A1SV8aAE l23CF-rrrNrrFT)rr r appendr enumeratetuple)rrowsrone iszerofuncsimpfuncnormalize_last normalize zero_aboverrr#piv_rowpiv_col pivot_colsswaps pivot_offset pivot_valassumed_nonzeronewly_determinedoffsetvalrrr!rowpiv_ipiv_jr"s` ` @r_row_reduce_listr= s\J?4 #< 0EGJ E D.W^,B *J-B * i) .MVS g F),Ct g% &.   qLG '" 1  WlW4 5 LL'<'#9: ; U "qA!C$ O1T6A:>AE4<8s1v 12A9I;Cg~U"sW}c$h()C# Cg 6 1 Y D.W^^)t"3%j1LE5E$J./I&)Cd U" #5:-1EAIt3CDs1v 12AE2 j!5< //rc [[U5URURURXUXES9 upgnUR URURU5Xx4$)Nr-r.r/)r=listr)rr*_new) Mr+r,r-r.r/rr2r3s r _row_reducerC|sU.d1gqvvqvvquu 8CU 66!&&!&&# & 99rc^URS::dURS::ag[U4SjUSS2S455nT"US5(aU=(a [USS2SS24T5$U=(a [USS2SS24T5$)zReturns `True` if the matrix is in echelon form. That is, all rows of zeros are at the bottom, and below each leading non-zero in a row are exclusively zeros.rTc34># UH nT"U5v M g7frr).0tr+s r _is_echelon..s6XjmmXsrNr%)r)rall _is_echelon)rBr+ zeros_belows ` rrKrKs  vv{affk6Qqr1uX66K!D'@{1QU8Z@@  =;qQRy*==rFc t[U[5(aUO[n[XUSSSS9upVnU(aXV4$U$)aBReturns a matrix row-equivalent to ``M`` that is in echelon form. Note that echelon form of a matrix is *not* unique, however, properties like the row space and the null space are preserved. Examples ======== >>> from sympy import Matrix >>> M = Matrix([[1, 2], [3, 4]]) >>> M.echelon_form() Matrix([ [1, 2], [0, -2]]) TFr?) isinstancerr rC)rBr+simplify with_pivotsr,rpivots_s r _echelon_formrSsC &h ==x9H 5UDNC{ Jrc BSn[U[5(aUO[nURS::dURS::agURS::dURS::aUVs/sH oQ"U5PM nnSU;agURS:XadURS:XaTUVs/sH oQ"U5PM nnSU;aSU;agUR 5nU"U5(aSU;agU"U5SLagU"XS9up[ XUSSSS 9upn [U 5$s snfs snf) zReturns 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 c^^UU4Sjn[TR5Vs/sH o2"U5U4PM nn[U5VVs/sHup5UPM nnnTRUSS9U4$s snfs snnf)aPPermute columns with complicated elements as far right as they can go. Since the ``sympy`` row reduction algorithms start on the left, having complexity right-shifted speeds things up. Returns a tuple (mat, perm) where perm is a permutation of the columns to perform to shift the complex columns right, and mat is the permuted matrix.c<>[U4SjTSS2U455$)Nc3>># UHnT"U5cSOSv M g7f)Nrrr)rFer+s rrHO_rank.._permute_complexity_right..complexity..sJ'QJqM1qq8's)sum)rrBr+s r complexity<_rank.._permute_complexity_right..complexitysJ!AqD'JJ Jrr) orientation)rrsortedpermute)rBr+r[rcomplexrperms`` r_permute_complexity_right(_rank.._permute_complexity_rightsj K 05QVV}=}!JqM1%}=#)'?3?!1?3 $F 3T::>3s A'A,rrFN)r+Tr?)rNrr r)rdetrClen) rBr+rOrbr,xzerosdrrRrQs r_rankrjs ;(&h ==x9H  vv{affkvv{affk()*1A* E>vv{qvv{()*1A*  $e"3 EEG a==Ue^ a=E !,QFFCs/LAq v;-+ +s &D$Dc[US5(dgURnURnUR(aU$UR(aUR [ 5$[SU55(dgUR [ 5$![a Us$f=f![a UR [5s$f=f)N_repc38# UHoRv M g7fr) is_Rational)rFrXs rrH_to_DM_ZZ_QQ..s,!Q==!s) hasattrrldomainis_ZZis_QQ convert_torrrJr)rBrepKs r _to_DM_ZZ_QQrws 1f   &&C Aww  >>"% %,!,,, &>>"% %  J  &>>"% % &s$B>B% B"!B"%CCcURnUR(a&URSS9up#nUR5U- nO&UR(aUR 5up$OeUR 5nXT4$)z7Compute the reduced row echelon form of a DomainMatrix.F) keep_domain)rqrrrref_dento_fieldrsrref to_Matrix)dMrvdM_rrefdenrQM_rrefs r_rref_dmrsk Aww!{{u{=f""$s* '')u    F >rc [U5nUb[U5upgO.[U[5(aUnO[n[ XUUSSS9upgn U(aXg4$U$)aAReturn 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) ``iszerofunc`` can correct rounding errors in matrices with float values. In the following example, calling ``rref()`` leads to floating point errors, incorrectly row reducing the matrix. ``iszerofunc= lambda x: abs(x) < 1e-9`` sets sufficiently small numbers to zero, avoiding this error. >>> m = Matrix([[0.9, -0.1, -0.2, 0], [-0.8, 0.9, -0.4, 0], [-0.1, -0.8, 0.6, 0]]) >>> m.rref() (Matrix([ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]]), (0, 1, 2)) >>> m.rref(iszerofunc=lambda x:abs(x)<1e-9) (Matrix([ [1, 0, -0.301369863013699, 0], [0, 1, -0.712328767123288, 0], [0, 0, 0, 0]]), (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 ``normalize_last=False`` T)r.r/)rwrrNrr rC) rBr+rOrQr-r~rr2r,rRs r_rrefr'sfT aB ~"2,Z h - -H H($4A rN)TTT)typesrsympy.polys.polyerrorsrsympy.polys.domainsrr utilitiesrr r r determinantr r=rCrKrSrjrwrrrrrrst1&OO/AEn0d9=+/:& > !(%U8 %CL&<" %\r