\hn3SSKJr SSKJr SSKJrJrJr SSKJ r Sr Sr Sr SS jr \ 4S jr\ 4S jr\ 4S jr\ 4S jr\ 4Sjr\ 4Sjr\ 4SjrSSjrSrSSjr\ 4SjrS\ S4Sjrg))DMNonInvertibleMatrixError)EX) MatrixErrorNonSquareMatrixErrorNonInvertibleMatrixError_iszerocxUR(a UR$URUR:aBURR U5R 5R UR5$URR UR UR5R 55$)aSubroutine for full row or column rank matrices. For full row rank matrices, inverse of ``A * A.H`` Exists. For full column rank matrices, inverse of ``A.H * A`` Exists. This routine can apply for both cases by checking the shape and have small decision. )is_zero_matrixHrowscolsmultiplyinv)Ms N/var/www/auris/envauris/lib/python3.13/site-packages/sympy/matrices/inverse.py_pinv_full_rankrsx ss vvss||A""$--acc22ss||AJJqssO//122cUR(a UR$UR5up[U5n[U5nUR U5$)zSubroutine for rank decomposition With rank decompositions, `A` can be decomposed into two full- rank matrices, and each matrix can take pseudoinverse individually. )r r rank_decompositionrr)rBCBpCps r_pinv_rank_decompositionrsH ss   !DA  B  B ;;r?rclUR(a UR$UnURnURUR:akUR U5R SS9up4UR S5nUR U5R UR5R U5$UR U5R SS9up4UR S5nUR U5R U5R UR5$![a [S5ef=f)zuSubroutine using diagonalization This routine can sometimes fail if SymPy's eigenvalue computation is not reliable. T) normalizec0[U5(aS$SU- $Nrrr xs r'_pinv_diagonalization..< 1+EA+Erc0[U5(aS$SU- $r r r!s rr#r$Cr%rz[pinv for rank-deficient matrices where diagonalization of A.H*A fails is not supported yet.) r r rrr diagonalize applyfuncrNotImplementedError)rAAHPDD_pinvs r_pinv_diagonalizationr/,s ss A BD 66QVV [[^//$/?DA[[!EFF::f%..qss3<**62;;ACC@ @ D! CD DDsBD2A*DD3cUR(a UR$US:Xa [U5$US:Xa [U5$[ S[ U5-5e)aCalculate the Moore-Penrose pseudoinverse of the matrix. The Moore-Penrose pseudoinverse exists and is unique for any matrix. If the matrix is invertible, the pseudoinverse is the same as the inverse. Parameters ========== method : String, optional Specifies the method for computing the pseudoinverse. If ``'RD'``, Rank-Decomposition will be used. If ``'ED'``, Diagonalization will be used. Examples ======== Computing pseudoinverse by rank decomposition : >>> from sympy import Matrix >>> A = Matrix([[1, 2, 3], [4, 5, 6]]) >>> A.pinv() Matrix([ [-17/18, 4/9], [ -1/9, 1/9], [ 13/18, -2/9]]) Computing pseudoinverse by diagonalization : >>> B = A.pinv(method='ED') >>> B.simplify() >>> B Matrix([ [-17/18, 4/9], [ -1/9, 1/9], [ 13/18, -2/9]]) See Also ======== inv pinv_solve References ========== .. [1] https://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse RDEDzinvalid pinv method %s)r r rr/ ValueErrorrepr)rmethods r_pinvr6LsPl ss  ~'** 4$Q''1DL@AArc"^^UR(d [S5eURSS9nURS5nUc;UR SS9Sm[ UU4Sj[ TR555nU(a [S5eU$) zbInitial check to see if a matrix is invertible. Raises or returns determinant for use in _inv_ADJ."A Matrix must be square to invert. berkowitz)r5rT)simplifyc3<># UHnT"TX45v M g7fN).0j iszerofuncoks r %_verify_invertible..s@A:bh'' Matrix det == 0; not invertible.) is_squarerdetequalsrrefanyrangerr)rr@dzerorAs ` @r_verify_invertiblerNs{ ;;"#GHH 55 5 $A 88A;D |vvtv$Q'@rww@@ &'IJJ Hrc:[XS9nUR5U- $)zCalculates the inverse using the adjugate matrix and a determinant. See Also ======== inv inverse_GE inverse_LU inverse_CH inverse_LDL r@)rNadjugate)rr@rLs r_inv_ADJrRs 14A ::# UHnT"TX45v M g7fr<r=)r>r?r@reds rrB_inv_GE..s :/Q:c!$i /rDrEN) denserTrFrhstack as_mutableeyerrIrJrKr_new)rr@rTbigrVs ` @r_inv_GEr^s ;;"#GHH --  166(: ;C ((j4( 8 ;C :%/ :::&'IJJ 66#al# $$rcUR(d [S5eUR(a [XS9 UR UR UR 5[S9$)zuCalculates the inverse using LU decomposition. See Also ======== inv inverse_ADJ inverse_GE inverse_CH inverse_LDL r8rP)rFr free_symbolsrNLUsolver[rr rr@s r_inv_LUrcsE ;;"#GHH~~14 99QUU166]w9 77rch[XS9 URURUR55$)z{Calculates the inverse using cholesky decomposition. See Also ======== inv inverse_ADJ inverse_GE inverse_LU inverse_LDL rP)rNcholesky_solver[rrbs r_inv_CHrfs)q0  AEE!&&M **rch[XS9 URURUR55$)zuCalculates the inverse using LDL decomposition. See Also ======== inv inverse_ADJ inverse_GE inverse_LU inverse_CH rP)rNLDLsolver[rrbs r_inv_LDLris'q0 ::aeeAFFm $$rch[XS9 URURUR55$)zuCalculates the inverse using QR decomposition. See Also ======== inv inverse_ADJ inverse_GE inverse_CH inverse_LDL rP)rNQRsolver[rrbs r_inv_QRrls'q0 99QUU166] ##rFcUR5nURnU(dUR(agUR(aUR[5$U$)z.Try to convert a matrix to a ``DomainMatrix``.N)to_DMdomainis_EXRAW convert_tor)ruse_EXdMKs r_try_DMrus? B A ajj }}R   rcjUR(dUR(agUR(+$)z,Check whether to convert to an exact domain.F)is_RRis_CCis_Exact)doms r_use_exact_domainr{ s!  yyCII<<rcPURup#URnX#:wa [S5e[U5nU(a!UR 5nUR U5nUR 5upxU(a#UR U5nURUW5nU(a?URR(dUR5nXx- R5n U $UR5URRU5- n U $![a [S5ef=f)zCalculates the inverse using ``DomainMatrix``. See Also ======== inv inverse_ADJ inverse_GE inverse_CH inverse_LDL sympy.polys.matrices.domainmatrix.DomainMatrix.inv r8rE)shaperorr{ get_exactrqinv_denrr convert_fromis_Fieldto_field to_Matrixto_sympy) rscancelmnrz use_exact dom_exactdMidenMis r_inv_DMr+s 88DA ))Cv"#GHH"#&IMMO ]]9 %K::<nnS!sI. zz"",,.Ci " " $ I]]_szz2237 7 I! &K&'IJJKs DD%c$SSKJn URSnUS::aURS[S9$USUS-2SUS-24nUSUS-2US-S24nXS-S2SUS-24nXS-S2US-S24n[ U5nXX-n X-n XJ- n [ U 5n U *U -n X-n U *U -nXU *--nU"X/X//5R5nU$![ a URS[S9s$f=f![ a URS[S9s$f=f)zxCalculates the inverse using BLOCKWISE inversion. See Also ======== inv inverse_ADJ inverse_GE inverse_CH inverse_LDL r) BlockMatrixLUr5r@N)&sympy.matrices.expressions.blockmatrixrr}rr _inv_blockr as_explicit)rr@rir*rrr-D_invB_D_iBDCA_nB_ndcC_nD_nnns rrrYseC  ABwuuDWu55 '16'6AE6/A '16'167 A q&'7AF7 A q&'167 A61  GE 'C 'C6o $u*C B #c'C cT'/C sj3*- . : : UR 5n/nUH!nUR URXS95 M# U"U6$UcU[La[USS9n U bSnOUS;a [US S9n Uc[X5(aS nOS nUS:Xa [W 5n OUS :Xa [W SS 9n OUS :XaURUS9n OUS:XaURUS9n OyUS:XaURUS9n OcUS:XaURUS9n OMUS :XaUR!US9n O7US:XaUR#US9n O!US:XaUR%US9n O ['S5eUR)U 5$)aC Return the inverse of a matrix using the method indicated. The default is DM if a suitable domain is found or otherwise GE for dense matrices LDL for sparse matrices. Parameters ========== method : ('DM', 'DMNC', 'GE', 'LU', 'ADJ', 'CH', 'LDL', 'QR') iszerofunc : function, optional Zero-testing function to use. try_block_diag : bool, optional If True then will try to form block diagonal matrices using the method get_diag_blocks(), invert these individually, and then reconstruct the full inverse matrix. 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]]) >>> A = Matrix(A) >>> A.inv('CH') Matrix([ [2/3, 1/3, 1/6], [1/3, 2/3, 1/3], [ 0, 0, 1/2]]) >>> A.inv('ADJ') == A.inv('GE') == A.inv('LU') == A.inv('CH') == A.inv('LDL') == A.inv('QR') True Notes ===== According to the ``method`` keyword, it calls the appropriate method: DM .... Use DomainMatrix ``inv_den`` method DMNC .... Use DomainMatrix ``inv_den`` method without cancellation GE .... inverse_GE(); default for dense matrices LU .... inverse_LU() ADJ ... inverse_ADJ() CH ... inverse_CH() LDL ... inverse_LDL(); default for sparse matrices QR ... inverse_QR() Note, the GE and LU methods may require the matrix to be simplified before it is inverted in order to properly detect zeros during pivoting. In difficult cases a custom zero detection function can be provided by setting the ``iszerofunc`` argument to a function that should return True if its argument is zero. The ADJ routine computes the determinant and uses that to detect singular matrices in addition to testing for zeros on the diagonal. See Also ======== inverse_ADJ inverse_GE inverse_LU inverse_CH inverse_LDL Raises ====== ValueError If the determinant of the matrix is zero. r)diag SparseMatrixr8rF)rrDM)rDMNCTLDLGEr)rrPrADJCHQRBLOCKzInversion method unrecognized)sympy.matricesrrrFrget_diag_blocksappendrr ru isinstancer inverse_GE inverse_LU inverse_ADJ inverse_CH inverse_LDL inverse_QR inverse_BLOCKr3r\) rr5r@try_block_diagrrblocksrblockrsrvs r_invrsr2 ;;"#GHH""$E HHUYYfYD EQx~*/ Qu % >F > ! Qt $~ a & &FF ~ R[ 6  R & 4 \\Z\ 0 4 \\Z\ 0 5 ]]j] 1 4 \\Z\ 0 5 ]]j] 1 4 \\Z\ 0 7  __ _ 3899 66":r)r1)F)T)sympy.polys.matrices.exceptionsrsympy.polys.domainsr exceptionsrrr utilitiesr rrr/r6rNrRr^rcrfrirlrur{rrrr=rrrsF"SS3$$D@>BB&- &#""%4"8("+"#%""$"  ,\%$LGEMr