ó ~9­\c@ sddlmZmZddlZddlmZddlmZmZm Z m Z ddl m Z ddl mZddlmZddlmZdd lmZdd lmZd d lmZd d lmZd dlmZmZdefd„ƒYZdeefd„ƒYZdS(iÿÿÿÿ(tdivisiontprint_functionN(t defaultdict(tCallabletas_intt is_sequencetrange(tDict(tExpr(tS(tAbs(tsqrt(tuniqi(ta2idx(tMatrix(t MatrixBaset ShapeErrort SparseMatrixcB sÓeZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z d „Z d „Zd „Zed„ƒZd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zed„ƒZd„Zd„Zd„Zed„ƒZ d„Z!d„Z"d„Z#d „Z$d!„Z%d"„Z&d#„Z'd$„Z(d%„Z)d&„Z*d'„Z+d(„Z,d)„Z-d*d+„Z.d*d,„Z/ee+d.d.d-ƒZ1ee&d.d.d-ƒZ2RS(/sj A sparse matrix (a matrix with a large number of zero elements). Examples ======== >>> from sympy.matrices import SparseMatrix >>> SparseMatrix(2, 2, range(4)) Matrix([ [0, 1], [2, 3]]) >>> SparseMatrix(2, 2, {(1, 1): 2}) Matrix([ [0, 0], [0, 2]]) See Also ======== sympy.matrices.dense.Matrix cO stj|ƒ}t|ƒdkrnt|dtƒrn|dj|_|dj|_t|djƒ|_|Si|_t|ƒdkrˆt |dƒ|_t |dƒ|_t|dt ƒrI|d}x¶t |jƒD]f}x]t |jƒD]L}|j ||j |ƒ|j |ƒƒƒ}|rò||j||fÈt(t_newR(R!RP((RPR!s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt_diagonal_solveÆsc K s7|jƒ}|jƒ}|j|jƒ}|sf|j}|ddd…f}||}||}n|jddƒ}|dkr|j}n(|dkr¨|j}ntd|ƒ‚|j gt |j ƒD]"} ||dd…| fƒ^qÎŒ} |s*|| dd…dfd} | | } n|j | ƒS(sÚReturn the matrix inverse using Cholesky or LDL (default) decomposition as selected with the ``method`` keyword: 'CH' or 'LDL', respectively. Examples ======== >>> from sympy import SparseMatrix, Matrix >>> A = SparseMatrix([ ... [ 2, -1, 0], ... [-1, 2, -1], ... [ 0, 0, 2]]) >>> A.inv('CH') Matrix([ [2/3, 1/3, 1/6], [1/3, 2/3, 1/3], [ 0, 0, 1/2]]) >>> A.inv(method='LDL') # use of 'method=' is optional Matrix([ [2/3, 1/3, 1/6], [1/3, 2/3, 1/3], [ 0, 0, 1/2]]) >>> A * _ Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]]) iNtmethodtLDLtCHs$Method may be "CH" or "LDL", not %s.(ii( t is_symmetrict as_mutableteyeRRNR8t _LDL_solveRSRJthstackRRR^( R!R tsymtMtItttr1R`tsolveR#RRtscale((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt _eval_inverseÊs(          A cC s|jd„ƒS(NcS s t|ƒS(N(R (tx((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyR\ÿR](t applyfunc(R!((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt _eval_AbsþscC sÁt|tƒs ||j|ƒSi}|jdƒ}xptƒj|jjƒ|jjƒƒD]G}|jj||ƒ|jj||ƒ}|dkr]||| s (RRyR^RR(R!Ru((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt_eval_conjugatescC stt|ƒƒ}tt|ƒƒ}i}t|ƒt|ƒt|jƒkr±xÃt|ƒD]N\}}x?t|ƒD]1\}} |jj|| fdƒ|||fDs (RtminR^(RRRtentries((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt _eval_eyeBsc s`tjjˆŒ}tˆjƒˆjˆjkr:t}nt‡‡fd†ˆjDƒƒp_|S(Nc3 s"|]}ˆ|jˆŒVqdS(N(thas(RR&(tpatternsR!(s4lib/python2.7/site-packages/sympy/matrices/sparse.pys Ns( R R9R“RRRRR/tany(R!R”tzhas((R”R!s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt _eval_hasGs c s?t‡fd†tˆjƒDƒƒs)tStˆjƒˆjkS(Nc3 s%|]}ˆ||fdkVqdS(iN((RR#(R!(s4lib/python2.7/site-packages/sympy/matrices/sparse.pys Qs(tallRRR/RR(R!((R!s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt_eval_is_IdentityPs%cC s,||jj|ƒ}t|jƒƒdkS(Ni(RNRqRtvalues(R!tsimpfunctdiff((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt_eval_is_symmetricUsc sUt|tƒs ||j|ƒSttƒ‰x1|jjƒD] \\}}}|ˆ||ns( RRR^RRRRyRRsRwRR(R!R1R#R$R|RuRA((R~RžR}RŸs4lib/python2.7/site-packages/sympy/matrices/sparse.pyt_eval_matrix_mulYs  " "*%cC sÜt|tƒst|ƒ}ni}xU|jjƒD]D\}}|\}}||krh||j7}n||||f>> from sympy.matrices import SparseMatrix >>> a = SparseMatrix(((1, 2), (3, 4))) >>> a Matrix([ [1, 2], [3, 4]]) >>> a.T Matrix([ [1, 3], [2, 4]]) cS s+i|]!\\}}}|||f“qS(((RR#R$R|((s4lib/python2.7/site-packages/sympy/matrices/sparse.pys ™s (RRyR^RR(R!Ru((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt_eval_transposeˆscC s/g|jjƒD]\}}|js|^qS(N(RRytis_zero(R!R[R'((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt _eval_valuesœscC s|j||iƒS(N(R^(RRR((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt _eval_zerosŸscC s@|jƒ\}}|j|ƒ}|j|ƒ}|jj|ƒS(N(t _LDL_sparseRMR_RNRO(R!RPRFtDtZRQ((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyRf£sc C sð|jƒ}|j|jƒ}|j|j|jƒ}x­tt|ƒƒD]™}x||D]„}||krN|||f|||f>> from sympy.matrices import SparseMatrix >>> m = SparseMatrix(2, 2, lambda i, j: i*2+j) >>> m Matrix([ [0, 1], [2, 3]]) >>> m.applyfunc(lambda i: 2*i) Matrix([ [0, 2], [4, 6]]) s`f` must be callable.N(tcallableR:R®RRytpopR.(R!tftoutR[R'tfv((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyRqøs   cC sddlm}||ƒS(s,Returns an Immutable version of this Matrix.i(tImmutableSparseMatrix(t immutableR·(R!R·((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt as_immutablescC s t|ƒS(sCReturns a mutable version of this matrix. Examples ======== >>> from sympy import ImmutableMatrix >>> X = ImmutableMatrix([[1, 2], [3, 4]]) >>> Y = X.as_mutable() >>> Y[1, 1] = 5 # Can set values in Y >>> Y Matrix([ [1, 2], [3, 5]]) (R0(R!((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyRdscC s}ddlm}m}|jƒs1tdƒ‚n|jƒjƒ}|j|ƒsa|j|ƒrptdƒ‚n|j|ƒS(sö Returns the Cholesky decomposition L of a matrix A such that L * L.T = A A must be a square, symmetric, positive-definite and non-singular matrix Examples ======== >>> from sympy.matrices import SparseMatrix >>> A = SparseMatrix(((25,15,-5),(15,18,0),(-5,0,11))) >>> A.cholesky() Matrix([ [ 5, 0, 0], [ 3, 3, 0], [-1, 1, 3]]) >>> A.cholesky() * A.cholesky().T == A True iÿÿÿÿ(tnantoos:Cholesky decomposition applies only to symmetric matrices.sACholesky decomposition applies only to positive-definite matrices( tsympy.core.numbersRºR»RcRRdRLR“R^(R!RºR»Ri((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytcholesky,s cC sFgtt|jjƒƒdd„ƒD]}t|||fƒ^q%S(s—Returns a column-sorted list of non-zero elements of the matrix. Examples ======== >>> from sympy.matrices import SparseMatrix >>> a=SparseMatrix(((1, 2), (3, 4))) >>> a Matrix([ [1, 2], [3, 4]]) >>> a.CL [(0, 0, 1), (1, 0, 3), (0, 1, 2), (1, 1, 4)] See Also ======== col_op row_list R&cS stt|ƒƒS(N(R=treversed(R[((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyR\`R](tsortedR=RRR6(R!R[((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytcol_listLscC s|j|j|j|jƒS(N(R^RRR(R!((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyR®bscC s°ddlm}m}|jƒs1tdƒ‚n|jƒjƒ\}}|j|ƒs…|j|ƒs…|j|ƒs…|j|ƒr”tdƒ‚n|j|ƒ|j|ƒfS(sý Returns the LDL Decomposition (matrices ``L`` and ``D``) of matrix ``A``, such that ``L * D * L.T == A``. ``A`` must be a square, symmetric, positive-definite and non-singular. This method eliminates the use of square root and ensures that all the diagonal entries of L are 1. Examples ======== >>> from sympy.matrices import SparseMatrix >>> A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11))) >>> L, D = A.LDLdecomposition() >>> L Matrix([ [ 1, 0, 0], [ 3/5, 1, 0], [-1/5, 1/3, 1]]) >>> D Matrix([ [25, 0, 0], [ 0, 9, 0], [ 0, 0, 9]]) >>> L * D * L.T == A True iÿÿÿÿ(RºR»s5LDL decomposition applies only to symmetric matrices.s<LDL decomposition applies only to positive-definite matrices( R¼RºR»RcRRdR©R“R^(R!RºR»RFRª((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytLDLdecompositiones <c C s!gt|jƒD] }g^q}x=|jƒD]/\}}}||kr/||j|ƒq/q/Wt|ƒ}|g|j}|g|j}x†t|jƒD]u}xl||d D]\}x.|||kré||}|||<|}q¼W|||kr³|||<||>> from sympy.matrices import SparseMatrix >>> S = SparseMatrix([ ... [1, 0, 3, 2], ... [0, 0, 1, 0], ... [4, 0, 0, 5], ... [0, 6, 7, 0]]) >>> S.liupc() ([[0], [], [0], [1, 2]], [4, 3, 4, 4]) References ========== Symbolic Sparse Cholesky Factorization using Elimination Trees, Jeroen Van Grondelle (1999) http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582 iÿÿÿÿ(RRR­RCR( R!R)tRR*t_tinftparenttvirtualRk((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytliupcs "     cC s t|jƒS(s2Returns the number of non-zero elements in Matrix.(RR(R!((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytnnz¹scC sFgtt|jjƒƒdd„ƒD]}t|||fƒ^q%S(s–Returns a row-sorted list of non-zero elements of the matrix. Examples ======== >>> from sympy.matrices import SparseMatrix >>> a = SparseMatrix(((1, 2), (3, 4))) >>> a Matrix([ [1, 2], [3, 4]]) >>> a.RL [(0, 0, 1), (0, 1, 2), (1, 0, 3), (1, 1, 4)] See Also ======== row_op col_list R&cS s t|ƒS(N(R=(R[((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyR\ÒR](R¿R=RRR6(R!R[((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyR­½scC sº|jƒ\}}t|ƒ}tj|ƒ}x†t|jƒD]u}xL||D]@}x7||kr||kr||j|ƒ||}qWWqNWttt ||ƒƒƒ||>> from sympy.matrices import SparseMatrix >>> S = SparseMatrix([ ... [1, 0, 3, 2], ... [0, 0, 1, 0], ... [4, 0, 0, 5], ... [0, 6, 7, 0]]) >>> S.row_structure_symbolic_cholesky() [[0], [], [0], [1, 2]] References ========== Symbolic Sparse Cholesky Factorization using Elimination Trees, Jeroen Van Grondelle (1999) http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582 ( RÇRR®tdeepcopyRRRCR=R¿Rs(R!RÂRÅRÄtLrowR[R$((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyRTÔs $cC sm|j|jŒ}|rixN|jD]@}||j|}|rO||j|>> from sympy.matrices import SparseMatrix, Matrix, ones >>> A = Matrix([1, 2, 3]) >>> B = Matrix([2, 3, 4]) >>> S = SparseMatrix(A.row_join(B)) >>> S Matrix([ [1, 2], [2, 3], [3, 4]]) If each line of S represent coefficients of Ax + By and x and y are [2, 3] then S*xy is: >>> r = S*Matrix([2, 3]); r Matrix([ [ 8], [13], [18]]) But let's add 1 to the middle value and then solve for the least-squares value of xy: >>> xy = S.solve_least_squares(Matrix([8, 14, 18])); xy Matrix([ [ 5/3], [10/3]]) The error is given by S*xy - r: >>> S*xy - r Matrix([ [1/3], [1/3], [1/3]]) >>> _.norm().n(2) 0.58 If a different xy is used, the norm will be higher: >>> xy += ones(2, 1)/10 >>> (S*xy - r).norm().n(2) 1.5 R`(RNtinv(R!RPR`Rk((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytsolve_least_squaress6 cC sf|jsN|j|jkr*tdƒ‚qb|j|jkrbtdƒ‚qbn|jd|ƒ|SdS(s–Return solution to self*soln = rhs using given inversion method. For a list of possible inversion methods, see the .inv() docstring. sUnder-determined system.s]For over-determined system, M, having more rows than columns, try M.solve_least_squares(rhs).R`N(t is_squareRRRRÍ(R!RPR`((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyRm<s  sAlternate faster representationN(3t__name__t __module__t__doc__RR4RIRKRSRLR_RoRrRxRR‚RŽt classmethodR’R—R™RR R¢R£R¤R¥R§R¨RfR©RMtpropertyR°RORqR¹RdR½RÀR®RÁRÇRÈR­RTRÌRÎRmR.tRLtCL(((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyRs\ 3 .  "  4              $        ( ,   # 9 R0cB s›eZed„ƒZd„Zd„ZdZd„Zd„Z d„Z d„Z d„Z d„Z d „Zd „Zd „Zd „Zd „Zd„ZRS(cO s ||ŒS(N((RRR ((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyR^PscC sx|j||ƒ}|dk rt|\}}}|rI||j||f>> from sympy.matrices import SparseMatrix, ones >>> M = SparseMatrix(2, 2, {}) >>> M[1] = 1; M Matrix([ [0, 1], [0, 0]]) >>> M[1, 1] = 2; M Matrix([ [0, 1], [0, 2]]) >>> M = SparseMatrix(2, 2, {}) >>> M[:, 1] = [1, 1]; M Matrix([ [0, 1], [0, 1]]) >>> M = SparseMatrix(2, 2, {}) >>> M[1, :] = [[1, 1]]; M Matrix([ [0, 0], [1, 1]]) To replace row r you assign to position r*m where m is the number of columns: >>> M = SparseMatrix(4, 4, {}) >>> m = M.cols >>> M[3*m] = ones(1, m)*2; M Matrix([ [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [2, 2, 2, 2]]) And to replace column c you can assign to position c: >>> M[2] = ones(m, 1)*4; M Matrix([ [0, 0, 4, 0], [0, 0, 4, 0], [0, 0, 4, 0], [2, 2, 4, 2]]) N(t_setitemR.R(R!R&R%RRR#R$((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyRKTs1 cC s |jƒS(N(R®(R!((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyRdscC sªi}t||jƒ}xs|jD]h\}}||kr=q"||krm|j||f|||df>> from sympy.matrices import SparseMatrix >>> M = SparseMatrix([[0, 0], [0, 1]]) >>> M Matrix([ [0, 0], [0, 1]]) >>> M.col_del(0) >>> M Matrix([ [0], [1]]) See Also ======== row_del iN(R RR(R!R[tnewDR#R$((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytcol_del’s  $! c C sY|jdkr@|j|jkr@|jd|jgƒj|ƒS||}}|j|jksktƒ‚n|jƒ}t|tƒsd}|j}x«t |jƒD]W}xNt |jƒD]=}||}|rî||j ||j|f>> from sympy import SparseMatrix, Matrix, ones >>> A = SparseMatrix(ones(3)) >>> A Matrix([ [1, 1, 1], [1, 1, 1], [1, 1, 1]]) >>> B = SparseMatrix.eye(3) >>> B Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]]) >>> C = A.col_join(B); C Matrix([ [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 0, 0], [0, 1, 0], [0, 0, 1]]) >>> C == A.col_join(Matrix(B)) True Joining along columns is the same as appending rows at the end of the matrix: >>> C == A.row_insert(A.rows, Matrix(B)) True ii( RRR^tcol_joinRR®RRR°RRRy( R!R1tAtBR[tbR#R$R'((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyRÚµs&(!     "cC s†xt|jƒD]n}|jj||ftjƒ}|||ƒ}|r_||j||f>> from sympy.matrices import SparseMatrix >>> M = SparseMatrix.eye(3)*2 >>> M[1, 0] = -1 >>> M.col_op(1, lambda v, i: v + 2*M[i, 0]); M Matrix([ [ 2, 4, 0], [-1, 0, 0], [ 0, 0, 2]]) N(RRRR8R R9R³(R!R$R´R#R'R¶((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytcol_opósc C sö||kr||}}n|jƒ}g}x—|D]\}}}||kr||jj||fƒ|j||fƒq5||kr´|jj||fƒ||j||f>> from sympy.matrices import SparseMatrix >>> S = SparseMatrix.eye(3); S[2, 1] = 2 >>> S.col_swap(1, 0); S Matrix([ [0, 1, 0], [1, 0, 0], [2, 0, 1]]) N(RÀRR³RC( R!R#R$RttemptiitjjR'R[((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytcol_swap s     cC s5t|ƒstdƒ‚n|j|t|ƒƒdS(Ns&`value` must be of type list or tuple.(RR:t copyin_matrixR(R!R&R%((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt copyin_list)s cC sö|j|ƒ\}}}}|j}||||}} ||| fkrZtdƒ‚nt|tƒsÂx†t|jƒD]B} x9t|jƒD](} || | f|| || |f>> from sympy.matrices import SparseMatrix >>> M = SparseMatrix.zeros(3); M Matrix([ [0, 0, 0], [0, 0, 0], [0, 0, 0]]) >>> M.fill(1); M Matrix([ [1, 1, 1], [1, 1, 1], [1, 1, 1]]) c s5i|]+}tˆjƒD]}ˆ||f“qqS((RR(RR#R$(R!R'(s4lib/python2.7/site-packages/sympy/matrices/sparse.pys es N(RRRR(R!R%((R!R's4lib/python2.7/site-packages/sympy/matrices/sparse.pytfillHs  cC sªi}t||jƒ}xs|jD]h\}}||kr=q"||krm|j||f||d|f>> from sympy.matrices import SparseMatrix >>> M = SparseMatrix([[0, 0], [0, 1]]) >>> M Matrix([ [0, 0], [0, 1]]) >>> M.row_del(0) >>> M Matrix([[0, 1]]) See Also ======== col_del iN(R RR(R!R[RØR#R$((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytrow_delhs  $! c C sY|jdkr@|j|jkr@|j|jdgƒj|ƒS||}}|j|jksktƒ‚n|jƒ}t|tƒsd}|j}x«t |jƒD]W}xNt |jƒD]=}||}|rî||j |||jf>> from sympy import SparseMatrix, Matrix >>> A = SparseMatrix(((1, 0, 1), (0, 1, 0), (1, 1, 0))) >>> A Matrix([ [1, 0, 1], [0, 1, 0], [1, 1, 0]]) >>> B = SparseMatrix(((1, 0, 0), (0, 1, 0), (0, 0, 1))) >>> B Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]]) >>> C = A.row_join(B); C Matrix([ [1, 0, 1, 1, 0, 0], [0, 1, 0, 0, 1, 0], [1, 1, 0, 0, 0, 1]]) >>> C == A.row_join(Matrix(B)) True Joining at row ends is the same as appending columns at the end of the matrix: >>> C == A.col_insert(A.cols, B) True ii( RRR^trow_joinRR®RRR°RRRy( R!R1RÛRÜR[RÝR#R$R'((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyRî‰s&$!     "cC s†xt|jƒD]n}|jj||ftjƒ}|||ƒ}|r_||j||f>> from sympy.matrices import SparseMatrix >>> M = SparseMatrix.eye(3)*2 >>> M[0, 1] = -1 >>> M.row_op(1, lambda v, j: v + 2*M[0, j]); M Matrix([ [2, -1, 0], [4, 0, 0], [0, 0, 2]]) See Also ======== row zip_row_op col_op N(RRRR8R R9R³(R!R#R´R$R'R¶((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytrow_opÃsc C sö||kr||}}n|jƒ}g}x—|D]\}}}||kr||jj||fƒ|j||fƒq5||kr´|jj||fƒ||j||f>> from sympy.matrices import SparseMatrix >>> S = SparseMatrix.eye(3); S[2, 1] = 2 >>> S.row_swap(1, 0); S Matrix([ [0, 1, 0], [1, 0, 0], [0, 2, 1]]) N(R­RR³RC( R!R#R$RRßRàRáR'R[((s4lib/python2.7/site-packages/sympy/matrices/sparse.pytrow_swapâs     c s#ˆj|‡‡‡fd†ƒdS(síIn-place operation on row ``i`` using two-arg functor whose args are interpreted as ``(self[i, j], self[k, j])``. Examples ======== >>> from sympy.matrices import SparseMatrix >>> M = SparseMatrix.eye(3)*2 >>> M[0, 1] = -1 >>> M.zip_row_op(1, 0, lambda v, u: v + 2*u); M Matrix([ [2, -1, 0], [4, 0, 0], [0, 0, 2]]) See Also ======== row row_op col_op c sˆ|ˆˆ|fƒS(N((R'R$(R´R[R!(s4lib/python2.7/site-packages/sympy/matrices/sparse.pyR\R]N(Rï(R!R#R[R´((R´R[R!s4lib/python2.7/site-packages/sympy/matrices/sparse.pyt zip_row_opsN(RÐRÑRÓR^RKRdR.t__hash__RÙRÚRÞRâRäRãRìRíRîRïRðRñ(((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyR0Os  9  # >     ! :  ( t __future__RRR®t collectionsRtsympy.core.compatibilityRRRRtsympy.core.containersRtsympy.core.exprRtsympy.core.singletonR tsympy.functionsR t(sympy.functions.elementary.miscellaneousR tsympy.utilities.iterablesR tcommonR tdenseRtmatricesRRRR0(((s4lib/python2.7/site-packages/sympy/matrices/sparse.pyts" "ÿÿÿ?