ó ~9­\c@ s"ddlmZmZddlmZmZddlmZmZm Z ddl m Z ddl m Z mZmZmZmZddlmZddlmZddlmZmZmZdd lmZdd lmZdd lmZdd l m!Z!m"Z"dd l#m$Z$ddl%m&Z&m'Z'ddl(m)Z)ddl*m+Z+ddl,m-Z-m.Z.ddl/m0Z0m1Z1defd„ƒYZ2de2fd„ƒYZ3d„Z4d„Z5d„Z6d„Z7d„Z8d„Z9d„Z:d„Z;d„Z<d „Z=d!„Z>d"„Z?d#„Z@d$„ZAd%S(&iÿÿÿÿ(tprint_functiontdivision(tasktQ(tBasictAddtsympify(trange(ttypedtexhaustt conditiontdo_onetunpack(t bottom_up(tsift(t MatrixExprt ZeroMatrixtIdentity(tMatMul(tMatAdd(tMatPow(t Transposet transpose(tTrace(tdett Determinant(t MatrixSlice(tInverse(tMatrixt ShapeError(tretimt BlockMatrixcB sÚeZdZd„Zed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZ d„Z d„Z d „Z d „Z d „Zd „Zd „Zd„Zd„Zed„ƒZed„ƒZd„ZRS(s¯A BlockMatrix is a Matrix composed of other smaller, submatrices The submatrices are stored in a SymPy Matrix object but accessed as part of a Matrix Expression >>> from sympy import (MatrixSymbol, BlockMatrix, symbols, ... Identity, ZeroMatrix, block_collapse) >>> n,m,l = symbols('n m l') >>> X = MatrixSymbol('X', n, n) >>> Y = MatrixSymbol('Y', m ,m) >>> Z = MatrixSymbol('Z', n, m) >>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]]) >>> print(B) Matrix([ [X, Z], [0, Y]]) >>> C = BlockMatrix([[Identity(n), Z]]) >>> print(C) Matrix([[I, Z]]) >>> print(block_collapse(C*B)) Matrix([[X, Z + Z*Y]]) cG sAddlm}tt|ƒ}||Œ}tj||ƒ}|S(Niÿÿÿÿ(tImmutableDenseMatrix(tsympy.matrices.immutableR!tmapRRt__new__(tclstargsR!tmattobj((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR$0s  cC sd}}|j}x6t|jdƒD]!}|||dfjd7}q'Wx6t|jdƒD]!}||d|fjd7}q`W||fS(Nii(tblocksRtshape(tselftnumrowstnumcolstMti((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR*8s  cC s |jjS(N(R)R*(R+((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt blockshapeBscC s |jdS(Ni(R&(R+((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR)FscC s4gt|jdƒD]}|j|dfj^qS(Ni(RR0R)trows(R+R/((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt rowblocksizesJscC s4gt|jdƒD]}|jd|fj^qS(Nii(RR0R)tcols(R+R/((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt colblocksizesNscC sUt|tƒoT|j|jkoT|j|jkoT|j|jkoT|j|jkS(N(t isinstanceR R*R0R2R4(R+tother((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pytstructurally_equalRs cC s=t|tƒr5|j|jkr5t|j|jƒS||S(N(R5R R4R2R)(R+R6((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt _blockmulYscC s:t|tƒr2|j|ƒr2t|j|jƒS||S(N(R5R R7R)(R+R6((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt _blockadd`scC sXg|jD]}t|ƒ^q }t|jd|jd|ƒ}|jƒ}t|ƒS(Nii(R)RRR0R (R+tmatrixtmatricesR.((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt_eval_transposegs"  cC s_|j|jkrOtgt|jdƒD]}t|j||fƒ^q)ŒStdƒ‚dS(Nis+Can't perform trace of irregular blockshape(R2R4RRR0RR)tNotImplementedError(R+R/((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt _eval_traceps 7cC s­|jdkr£|jjƒ\\}}\}}ttj|ƒƒrht|ƒt|||j|ƒSttj|ƒƒr£t|ƒt|||j|ƒSnt|ƒS(Ni(ii( R0R)ttolistRRt invertibleRtIR(R+tAtBtCtD((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt_eval_determinantws!#&cC sŽg|jD]}t|ƒ^q }t|jd|jd|ƒ}g|jD]}t|ƒ^qL}t|jd|jd|ƒ}||fS(Nii(R)RRR0R(R+R:t real_matricest im_matrices((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt as_real_imags " " cC s |jƒS(s Return transpose of matrix. Examples ======== >>> from sympy import MatrixSymbol, BlockMatrix, ZeroMatrix >>> from sympy.abc import l, m, n >>> X = MatrixSymbol('X', n, n) >>> Y = MatrixSymbol('Y', m ,m) >>> Z = MatrixSymbol('Z', n, m) >>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]]) >>> B.transpose() Matrix([ [X.T, 0], [Z.T, Y.T]]) >>> _.transpose() Matrix([ [X, Z], [0, Y]]) (R<(R+((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRŠscC s›x=t|jƒD],\}}||ktkr2Pq||8}qWx=t|jƒD],\}}||ktkrrPqP||8}qPW|j||f||fS(N(t enumerateR2tFalseR4R)(R+R/tjt row_blockR,t col_blockR-((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt_entry¡scC s¬|jd|jdkrtSx‡t|jdƒD]r}xit|jdƒD]T}||kry|j||fj rytS||krL|j||fj rLtSqLWq2WtS(Nii(R0RKRR)t is_Identityt is_ZeroMatrixtTrue(R+R/RL((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRP¯s## cC s|j|jkS(N(R2R4(R+((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pytis_structurally_symmetric»scC sK||krtSt|tƒr5|j|jkr5tStt|ƒj|ƒS(N(RRR5R R)tsupertequals(R+R6((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRU¿s  !(t__name__t __module__t__doc__R$tpropertyR*R0R)R2R4R7R8R9R<R>RFRIRRORPRSRU(((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR s&         tBlockDiagMatrixcB seZdZd„Zed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZ ed„ƒZ dd „Z d „Z d „Z RS( sL A BlockDiagMatrix is a BlockMatrix with matrices only along the diagonal >>> from sympy import MatrixSymbol, BlockDiagMatrix, symbols, Identity >>> n,m,l = symbols('n m l') >>> X = MatrixSymbol('X', n, n) >>> Y = MatrixSymbol('Y', m ,m) >>> BlockDiagMatrix(X, Y) Matrix([ [X, 0], [0, Y]]) cG stjt|ŒS(N(RR$RZ(R%tmats((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR$ÔscC s|jS(N(R&(R+((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pytdiag×scC s”ddlm}|j}gtt|ƒƒD]X}gtt|ƒƒD]9}||kra||nt||j||jƒ^qE^q,}||ƒS(Niÿÿÿÿ(R!(R"R!R&RtlenRR1R3(R+R!R[R/RLtdata((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR)Ûs  ncC s0td„|jDƒƒtd„|jDƒƒfS(Ncs s|]}|jVqdS(N(R1(t.0tblock((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pys æscs s|]}|jVqdS(N(R3(R_R`((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pys çs(tsumR&(R+((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR*äscC st|jƒ}||fS(N(R]R&(R+tn((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR0éscC sg|jD]}|j^q S(N(R&R1(R+R`((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR2îscC sg|jD]}|j^q S(N(R&R3(R+R`((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR4òstignoredcC s&tg|jD]}|jƒ^q ŒS(N(RZR&tinverse(R+texpandR'((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt _eval_inverseöscC skt|tƒrW|j|jkrWtgt|j|jƒD]\}}||^q:ŒStj||ƒSdS(N(R5RZR4R2tzipR&R R8(R+R6tatb((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR8ùs6cC st|tƒr{|j|jkr{|j|jkr{|j|jkr{tgt|j|jƒD]\}}||^q^ŒStj||ƒSdS(N( R5RZR0R2R4RgR&R R9(R+R6RhRi((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR9s 6(RVRWRXR$RYR\R)R*R0R2R4RfR8R9(((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRZÆs     c C s®d„}tttt|titttƒt6ttt ƒt 6tt 6t t 6tt6tttƒt6ƒƒƒƒƒ}||ƒ}t|ddƒ}|dk r¦|ƒS|SdS(s=Evaluates a block matrix expression >>> from sympy import MatrixSymbol, BlockMatrix, symbols, Identity, Matrix, ZeroMatrix, block_collapse >>> n,m,l = symbols('n m l') >>> X = MatrixSymbol('X', n, n) >>> Y = MatrixSymbol('Y', m ,m) >>> Z = MatrixSymbol('Z', n, m) >>> B = BlockMatrix([[X, Z], [ZeroMatrix(m, n), Y]]) >>> print(B) Matrix([ [X, Z], [0, Y]]) >>> C = BlockMatrix([[Identity(n), Z]]) >>> print(C) Matrix([[I, Z]]) >>> print(block_collapse(C*B)) Matrix([[X, Z + Z*Y]]) cS st|tƒo|jtƒS(N(R5RthasR (texpr((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyt ttdoitN(R R R RR t bc_mataddtbc_block_plus_identRt bc_matmultbc_distRRt bc_transposeRt bc_inverseRt bc_unpacktdeblockR tgetattrtNone(RkthasbmtruletresultRn((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pytblock_collapse s "  cC s|jdkr|jdS|S(Nii(ii(ii(R0R)(Rk((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRu0s cC s}t|jd„ƒ}|t}|s)|S|t}|d}x!|dD]}|j|ƒ}qHW|rut|Œ|S|SdS(NcS s t|tƒS(N(R5R (R.((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRl6Rmii(RR&RRRKR9R(RkR&R)t nonblocksR`Ri((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRo5s   c sÖg|jD]}|jr |^q }|s/|Sg|jD]}t|tƒr9|^q9‰ˆrÒt‡fd†ˆDƒƒrÒˆdjrÒtgˆdjD]}t|ƒ^qšŒ}t |t |ƒˆŒj ƒS|S(Nc3 s"|]}|jˆdƒVqdS(iN(R7(R_Ri(R)(sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pys Jsi( R&RPR5R tallRSRZR2RRR]Rn(Rktargtidentstktblock_id((R)sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRpDs%+" &c C s–|jƒ\}}|dkr’tt|ƒtƒr’t|ƒj}tgt|jƒD]6}gt|jƒD]}||||f^qk^qUƒS|S(s Turn a*[X, Y] into [a*X, a*Y] i(t as_coeff_mmulR5R R R)RR1R3(RktfactorR'RCR/RL((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRrRs !JcC s}t|tƒrK|jdjrDd|jdg|jd}}q]|Sn|jƒ\}}d}x|dt|ƒkri|||d!\}}t|tƒrØt|tƒrØ|j|ƒ||<|j|dƒqft|tƒr|jt|ggƒƒ||<|j|dƒqft|tƒr\t|ggƒj|ƒ||<|j|dƒqf|d7}qfWt ||Œj ƒS(Niii( R5RR&t is_Integertas_coeff_matricesR]R R8tpopRRn(RkR„R;R/RBRC((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRq\s&%cC s"tt|jƒjjtƒjƒS(N(R R|RR)t applyfuncRtT(Rk((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRsuscC s5t|ƒ}||kr|Sttt|jƒƒƒS(N(tblockinverse_1x1tblockinverse_2x2Rt reblock_2x2R(Rktexpr2((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRtys  cC sTt|jtƒrP|jjdkrPt|jjdjƒggƒ}t|ƒS|S(Nii(ii(R5RR R0RR)Rd(RkR'((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRŠs$" cC sÌt|jtƒrÄ|jjdkrÄ|jjjƒ\\}}\}}t|||j|j| j||||j|jg|||j|j ||j|||j|jggƒS|SdS(Ni(ii(R5RR R0R)R?RA(RkRBRCRDRE((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyR‹…s $$<@c s4t|tƒ s#|jjtƒ r'|Sd„}|jj|ƒ‰ddlm}yÉ|dt‡fd†tˆj dƒDƒƒgƒ}xƒtdˆj dƒD]k}|ˆ|dfjƒ}x:tdˆj dƒD]"}|j ˆ||fjƒ}q×W|j |ƒ}q¡Wt|ƒSWnt k r/|SXdS(s( Flatten a BlockMatrix of BlockMatrices cS s#t|tƒr|St|ggƒS(N(R5R (tx((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRl”Rmiÿÿÿÿ(Ric3 s)|]}ˆd|fjjdVqdS(iiN(R)R*(R_R/(tbb(sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pys ™siN( R5R R)RjRˆtsympyRRaRR*trow_jointcol_joinR(RCtwrapRtMMtrowR.tcol((RsElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRvs# 5  cC s«t|tƒ s-td„|jjDƒƒ r1|St}||jd||jddd…fƒg||jdd…dfƒ||jdd…dd…fƒggƒS(sC Reblock a BlockMatrix so that it has 2x2 blocks of block matrices cs s|]}|dkVqdS(iN((R_td((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pys ¨siiN(ii(R5R R~R)R*(RCtBM((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pyRŒ¦s -,cC sBd}g}x/|D]'}|j|||fƒ||7}qW|S(s½ Convert sequence of numbers into pairs of low-high pairs >>> from sympy.matrices.expressions.blockmatrix import bounds >>> bounds((1, 10, 50)) [(0, 1), (1, 11), (11, 61)] i(tappend(tsizestlowtrvtsize((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pytbounds°s  c C sTt|ƒ}t|ƒ}tg|D]+}g|D]}t|||ƒ^q/^q"ƒS(s Cut a matrix expression into Blocks >>> from sympy import ImmutableMatrix, blockcut >>> M = ImmutableMatrix(4, 4, range(16)) >>> B = blockcut(M, (1, 3), (1, 3)) >>> type(B).__name__ 'BlockMatrix' >>> ImmutableMatrix(B.blocks[0, 1]) Matrix([[1, 2, 3]]) (RžR R(Rktrowsizestcolsizest rowboundst colboundstrowboundtcolbound((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pytblockcut¾s  N(Bt __future__RRRRRt sympy.coreRRRtsympy.core.compatibilityRtsympy.strategiesRR R R R tsympy.strategies.traverseR tsympy.utilitiesRt"sympy.matrices.expressions.matexprRRRt!sympy.matrices.expressions.matmulRt!sympy.matrices.expressions.mataddRt!sympy.matrices.expressions.matpowRt$sympy.matrices.expressions.transposeRRt sympy.matrices.expressions.traceRt&sympy.matrices.expressions.determinantRRt sympy.matrices.expressions.sliceRt"sympy.matrices.expressions.inverseRtsympy.matricesRRt$sympy.functions.elementary.complexesRRR RZR|RuRoRpRrRqRsRtRŠR‹RvRŒRžR¥(((sElib/python2.7/site-packages/sympy/matrices/expressions/blockmatrix.pytsB(°D &