o GZh< @sddlmZmZddlmZddlmZmZmZddl m Z m Z ddl m Z mZddlmZddlmZddlmZmZmZmZmZmZmZdd lmZdd lmZdd lm Z dd l!m"Z#d dl$m%Z%d dl&m'Z'd dl(m)Z)d dl*m*Z*d dl+m,Z,d dl-m.Z.m/Z/m0Z0m1Z1Gddde'e Z2e 3e e2fe2ddZ4ddZ5ddZ6ddZ7ddZ8d d!Z9d"d#Z:d$d%Z;d&d'Zd*d+Z?d,d-Z@e@ed<d.S)/)askQ) handlers_dict)BasicsympifyS)mulMul)NumberIntegerDummyadjoint)rm_idunpacktypedflattenexhaustdo_onenew)NonInvertibleMatrixError) MatrixBase)sympy_deprecation_warning)validate_matmul_integer)Inverse) MatrixExpr)MatPow transpose)PermutationMatrix) ZeroMatrixIdentityGenericIdentity OneMatrixcseZdZdZdZeZddddddZedd Z e d d Z d$d d Z ddZ ddZfddZddZddZddZddZddZddZd%d d!Zd"d#ZZS)&MatMula A product of matrix expressions Examples ======== >>> from sympy import MatMul, MatrixSymbol >>> A = MatrixSymbol('A', 5, 4) >>> B = MatrixSymbol('B', 4, 3) >>> C = MatrixSymbol('C', 3, 6) >>> MatMul(A, B, C) A*B*C TFN)evaluatecheck_sympifycs|sjSttfdd|}|rttt|}tjg|R}|\}}|dur3tdddd|dur;t ||s?|S|rF |S|S)Ncs j|kSN)identity)iclsP/var/www/auris/lib/python3.10/site-packages/sympy/matrices/expressions/matmul.py0 z MatMul.__new__..zaPassing check to MatMul is deprecated and the check argument will be removed in a future version.z1.11z,remove-check-argument-from-matrix-operations)Zdeprecated_since_versionZactive_deprecations_targetF) r+listfiltermaprr__new__as_coeff_matricesrvalidate _evaluate)r.r'r(r)argsobjfactormatricesr/r-r0r6*s(  zMatMul.__new__cCst|Sr*) canonicalize)r.exprr/r/r0r9JszMatMul._evaluatecCs$dd|jD}|dj|djfS)NcSg|]}|jr|qSr/ is_Matrix.0argr/r/r0 Pz MatMul.shape..r)r:rowscols)selfr=r/r/r0shapeNsz MatMul.shapec  sjddlm}ddlm|\}}t|dkr"||d||fSdgt|ddgt|d}|d<|d<dd} |d| tdt|D]}t|<qNt |ddD] \}} | j dd||<q_fd d t |D}t |} t fd d |Drd }||| gtdddgt||R} t dd |Dsd}|r| S| S)Nr)SumImmutableMatrixrrHcss d} td|V|d7}q)NrTzi_%ir )counterr/r/r0fbs zMatMul._entry..fdummy_generatorcs,g|]\}}|j||ddqS)r)rR)_entryrDr,rE)rRindicesr/r0rFos,z!MatMul._entry..c3s|]}|VqdSr*hasrDvrNr/r0 qz MatMul._entry..Tcss|] }t|ttfVqdSr*) isinstancer intrXr/r/r0rZysF)Zsympy.concrete.summationsrMZsympy.matrices.immutablerOr7lengetrangenext enumeraterLr fromiteranyzipdoit) rKr,jexpandkwargsrMcoeffr=Z ind_rangesrQrEZ expr_in_sumresultr/)rOrRrUr0rSSs6      z MatMul._entrycCsBdd|jD}dd|jD}t|}|jdurtd||fS)NcSg|]}|js|qSr/rArDxr/r/r0rF~rGz,MatMul.as_coeff_matrices..cSr@r/rArmr/r/r0rFrGFz3noncommutative scalars in MatMul are not supported.)r:r is_commutativeNotImplementedError)rKZscalarsr=rjr/r/r0r7}s  zMatMul.as_coeff_matricescCs|\}}|t|fSr*)r7r&rKrjr=r/r/r0 as_coeff_mmuls  zMatMul.as_coeff_mmulc s tt|jdi|}||SNr/)superr&rhr9)rKriexpanded __class__r/r0rhs z MatMul.expandcCs4|\}}t|gdd|dddDRS)aTransposition of matrix multiplication. 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[1] https://en.wikipedia.org/wiki/Transpose cSg|]}t|qSr/rrCr/r/r0rFz*MatMul._eval_transpose..NrH)r7r&rfrqr/r/r0_eval_transposes zMatMul._eval_transposecCs"tdd|jdddDS)NcSrxr/rrCr/r/r0rFryz(MatMul._eval_adjoint..rH)r&r:rfrKr/r/r0 _eval_adjoints"zMatMul._eval_adjointcCs4|\}}|dkrddlm}|||SdS)Nr)trace)rrr}rf)rKr<mmulr}r/r/r0 _eval_traces  zMatMul._eval_tracecCs<ddlm}|\}}t|}||jttt||S)Nr) Determinant)Z&sympy.matrices.expressions.determinantrr7 only_squaresrIr r3r5)rKrr<r=Zsquare_matricesr/r/r0_eval_determinants  zMatMul._eval_determinantcCs>tdd|jDrtdd|jdddDSt|S)Ncss |] }t|tr|jVqdSr*)r\r is_squarerCr/r/r0rZz'MatMul._eval_inverse..css*|]}t|tr |n|dVqdS)rHN)r\rinverserCr/r/r0rZs  rH)allr:r&rfrr{r/r/r0 _eval_inverseszMatMul._eval_inversec s@dd}|rtfdd|jD}n|j}tt|}|S)NdeepTc3s |] }|jdiVqdSrs)rfrChintsr/r0rZrzMatMul.doit..)r_tupler:r>r&)rKrrr:r?r/rr0rfs  z MatMul.doitc sjddjD}ddjD}|r1t|}t|}|r1|r1t||kr1tdfdd|D||gS)NcSr@r/rormr/r/r0rFrGz#MatMul.args_cnc..cSrlr/rrmr/r/r0rFrGz"repeated commutative arguments: %scs$g|]}tj|dkr|qSr)r3r:count)rDcir{r/r0rFs$)r:r^set ValueError)rKZcsetwarnriZcoeff_cZcoeff_ncZclenr/r{r0args_cncszMatMul.args_cncc sddlmfddt|jD}g}|D]U}|jd|}|j|dd}|r0t|}nt|jd}|rHtfddt|D}nt|jd}|j| } | D]} | || || | qYq|S)Nr Transposecsg|] \}}|r|qSr/rVrTrnr/r0rFz8MatMul._eval_derivative_matrix_lines..cs"g|] }|jr |n|qSr/)rBrf)rDr,rr/r0rFs"r) r rrbr:r&rcr#rLreversed_eval_derivative_matrix_linesZ append_firstZ append_secondappend) rKrnZ with_x_indlinesindZ left_argsZ right_argsZ right_matZleft_revdr,r/)rrnr0rs&     z$MatMul._eval_derivative_matrix_lines)T)FT)__name__ __module__ __qualname____doc__Z is_MatMulr$r+r6 classmethodr9propertyrLrSr7rrrhrzr|rrrrfrr __classcell__r/r/rvr0r&s*    *   r&cGs(|ddkr |dd}ttg|RS)Nrr)rr&r:r/r/r0newmuls  rcCs>tdd|jDrdd|jD}t|dj|djS|S)Ncss"|] }|jp |jo |jVqdSr*)is_zerorBZ is_ZeroMatrixrCr/r/r0rZszany_zeros..cSr@r/rArCr/r/r0rFrGzany_zeros..rrH)rdr:r"rIrJ)rr=r/r/r0 any_zeross rcCstdd|jDs |Sg}|jd}|jddD]}t|ttfr/t|ttfr/||}q|||}q||t|S)a Merge explicit MatrixBase arguments >>> from sympy import MatrixSymbol, Matrix, MatMul, pprint >>> from sympy.matrices.expressions.matmul import merge_explicit >>> A = MatrixSymbol('A', 2, 2) >>> B = Matrix([[1, 1], [1, 1]]) >>> C = Matrix([[1, 2], [3, 4]]) >>> X = MatMul(A, B, C) >>> pprint(X) [1 1] [1 2] A*[ ]*[ ] [1 1] [3 4] >>> pprint(merge_explicit(X)) [4 6] A*[ ] [4 6] >>> X = MatMul(B, A, C) >>> pprint(X) [1 1] [1 2] [ ]*A*[ ] [1 1] [3 4] >>> pprint(merge_explicit(X)) [1 1] [1 2] [ ]*A*[ ] [1 1] [3 4] css|]}t|tVqdSr*)r\rrCr/r/r0rZr[z!merge_explicit..rrN)rdr:r\rr rr&)matmulnewargslastrEr/r/r0merge_explicits    rcCs:|\}}tdd|}||krt|g|jRS|S)z Remove Identities from a MatMul This is a modified version of sympy.strategies.rm_id. This is necessary because MatMul may contain both MatrixExprs and Exprs as args. See Also ======== sympy.strategies.rm_id cSs |jduS)NT)Z is_Identityrr/r/r0r16r2zremove_ids..)rrrrr:)rr<r~rkr/r/r0 remove_ids's rcCs(|\}}|dkrt|g|RS|SNr)r7r)rr<r=r/r/r0factor_in_front<s rc Cs |\}}|dg}tdt|D]}|d}||}t|trJt|jtrJ|jj}t|}t||| dkrJ|d| t |j dg}qt|trt|jtr|jj} t| }t| ||||krt |j d} | |d<t|||D]} | || <q{q|j dks|j dkr| |qt|t r|j\} } n|tj} } t|t r|j\}}n|tj}}| |kr| |}t | |jdd|d<qt|tsz|}Wn tyd}Ynw|dur| |kr| |}t | |jdd|d<q| |qt|g|RS)aCombine consecutive powers with the same base into one, e.g. $$A \times A^2 \Rightarrow A^3$$ This also cancels out the possible matrix inverses using the knowledgebase of :class:`~.Inverse`, e.g., $$ Y \times X \times X^{-1} \Rightarrow Y $$ rrrHNF)r)r7r`r^r\rrEr&r:r3r#rLrrrrZOnerfrrrr)rr<r:new_argsr,ABZBargslZAargsr+rgZA_baseZA_expZB_baseZB_expZnew_expZ B_base_invr/r/r0combine_powersBsZ              rc Cs|j}t|}|dkr |S|dg}td|D],}|d}||}t|tr>t|tr>|jd}|jd}t|||d<q||qt|S)zGRefine products of permutation matrices as the products of cycles. rrrH)r:r^r`r\r!rr&) rr:rrkr,rrZcycle_1Zcycle_2r/r/r0combine_permutationss      rcCs|\}}|dg}|ddD]/}|d}t|tr!t|ts'||q||t|jd|jd||jd9}qt|g|RS)zj Combine products of OneMatrix e.g. OneMatrix(2, 3) * OneMatrix(3, 4) -> 3 * OneMatrix(2, 4) rrNrH)r7r\r%rpoprLr)rr<r:rrrr/r/r0combine_one_matricess   rcs|jtdkr?ddlm}djr'djr'|fdddjDSdjr?djr?|fdddjDS|S)zr Simplify MatMul expressions but distributing rational term to MatMul. e.g. 2*(A+B) -> 2*A + 2*B rr)MatAddrcsg|] }t|dqSrr&rfrDmatrr/r0rFrz$distribute_monom..csg|] }td|qS)rrrrr/r0rFr)r:r^ZmataddrZ is_MatAddZ is_Rational)rrr/rr0distribute_monoms  rcCs|dkSrr/rr/r/r0r1sr1cGsp|dj|djkrtdg}d}t|D]\}}|j||jkr5|t|||d|d}q|S)z'factor matrices only if they are squarerrHz!Invalid matrices being multipliedr)rIrJ RuntimeErrorrbrr&rf)r=outstartr,Mr/r/r0rsrcCsg}g}|jD]}|jr||q||q|d}|ddD]4}||jkr9tt||r9t|jd}q"|| krOtt ||rOt|jd}q"|||}q"||t |S)z >>> from sympy import MatrixSymbol, Q, assuming, refine >>> X = MatrixSymbol('X', 2, 2) >>> expr = X * X.T >>> print(expr) X*X.T >>> with assuming(Q.orthogonal(X)): ... print(refine(expr)) I rrN) r:rBrTrrZ orthogonalr#rL conjugateZunitaryr&)r?Z assumptionsrZexprargsr:rrEr/r/r0 refine_MatMuls       rN)AZsympy.assumptions.askrrZsympy.assumptions.refinerZ sympy.corerrrZsympy.core.mulrr Zsympy.core.numbersr r Zsympy.core.symbolr Zsympy.functionsrZsympy.strategiesrrrrrrrZsympy.matrices.exceptionsrZsympy.matrices.matrixbaserZsympy.utilities.exceptionsrZ!sympy.matrices.expressions._shaperr8rrZmatexprrZmatpowrr Z permutationr!Zspecialr"r#r$r%r&Zregister_handlerclassrrrrrrrrrrulesr>rrr/r/r/r0sJ   $         V*? 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