o oÇ hn3ã@sêddlmZddlmZddlmZmZmZddlm Z dd„Z dd „Z d d „Z d*d d„Z e fdd„Ze fdd„Ze fdd„Ze fdd„Ze fdd„Ze fdd„Ze fdd„Zd+dd„Zd d!„Zd,d#d$„Ze fd%d&„Zd'e dfd(d)„Zd'S)-é)ÚDMNonInvertibleMatrixError)ÚEXé)Ú MatrixErrorÚNonSquareMatrixErrorÚNonInvertibleMatrixError©Ú_iszerocCsH|jr|jS|j|jkr|j |¡ ¡ |j¡S|j | |j¡ ¡¡S)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. )Úis_zero_matrixÚHÚrowsÚcolsÚmultiplyÚinv)ÚM©rúj/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/sympy/matrices/inverse.pyÚ_pinv_full_ranks   rcCs2|jr|jS| ¡\}}t|ƒ}t|ƒ}| |¡S)z¶Subroutine for rank decomposition With rank decompositions, `A` can be decomposed into two full- rank matrices, and each matrix can take pseudoinverse individually. )r r Úrank_decompositionrr)rÚBÚCÚBpÚCprrrÚ_pinv_rank_decompositions   rcCs¶|jr|jS|}|j}zD|j|jkr1| |¡jdd\}}| dd„¡}| |¡ |j¡ |¡WS| |¡jdd\}}| dd„¡}| |¡ |¡ |j¡WStyZtdƒ‚w)zSubroutine using diagonalization This routine can sometimes fail if SymPy's eigenvalue computation is not reliable. T)Ú normalizecSót|ƒrdSd|S©Nrrr©ÚxrrrÚ<óz'_pinv_diagonalization..cSrrrrrrrrCr z[pinv for rank-deficient matrices where diagonalization of A.H*A fails is not supported yet.) r r r r rÚ diagonalizeÚ applyfuncrÚNotImplementedError)rÚAÚAHÚPÚDÚD_pinvrrrÚ_pinv_diagonalization,s&   ÿ ÿÿr)ÚRDcCs<|jr|jS|dkrt|ƒS|dkrt|ƒStdt|ƒƒ‚)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 r*ÚEDzinvalid pinv method %s)r r rr)Ú ValueErrorÚrepr)rÚmethodrrrÚ_pinvLs6r/csj|jstdƒ‚|jdd}| d¡}|dur-|jddd‰t‡‡fdd „tˆjƒDƒƒ}|r3td ƒ‚|S) zfInitial check to see if a matrix is invertible. Raises or returns determinant for use in _inv_ADJ.ú"A Matrix must be square to invert.Ú berkowitz)r.rNT)Úsimplifyc3ó |] }ˆˆ||fƒVqdS©Nr©Ú.0Új©Ú iszerofuncÚokrrÚ ™ó€z%_verify_invertible..ú Matrix det == 0; not invertible.) Ú is_squarerÚdetÚequalsÚrrefÚanyÚranger r)rr9ÚdÚzerorr8rÚ_verify_invertibles  rFcCst||d}| ¡|S)z©Calculates the inverse using the adjugate matrix and a determinant. See Also ======== inv inverse_GE inverse_LU inverse_CH inverse_LDL ©r9)rFÚadjugate)rr9rDrrrÚ_inv_ADJ s rIcs†ddlm}|js tdƒ‚| | ¡| |j¡¡}|jˆddd‰t ‡‡fdd„t ˆjƒDƒƒr5t d ƒ‚|  ˆd d …|jd …f¡S) z™Calculates the inverse using Gaussian elimination. See Also ======== inv inverse_ADJ inverse_LU inverse_CH inverse_LDL r)ÚMatrixr0T)r9r2rc3r3r4rr5©r9Úredrrr;Ær<z_inv_GE..r=N) ÚdenserJr>rÚhstackÚ as_mutableÚeyer rArBrCrÚ_new)rr9rJÚbigrrKrÚ_inv_GE±s rScCs6|jstdƒ‚|jrt||d|j| |j¡tdS)z•Calculates the inverse using LU decomposition. See Also ======== inv inverse_ADJ inverse_GE inverse_CH inverse_LDL r0rG)r>rÚ free_symbolsrFÚLUsolverPr r ©rr9rrrÚ_inv_LUËs   rWcCót||d| | |j¡¡S)z›Calculates the inverse using cholesky decomposition. See Also ======== inv inverse_ADJ inverse_GE inverse_LU inverse_LDL rG)rFÚcholesky_solverPr rVrrrÚ_inv_CHßó rZcCrX)z•Calculates the inverse using LDL decomposition. See Also ======== inv inverse_ADJ inverse_GE inverse_LU inverse_CH rG)rFÚLDLsolverPr rVrrrÚ_inv_LDLðr[r]cCrX)z•Calculates the inverse using QR decomposition. See Also ======== inv inverse_ADJ inverse_GE inverse_CH inverse_LDL rG)rFÚQRsolverPr rVrrrÚ_inv_QRr[r_FcCs0| ¡}|j}|s|jrdS|jr| t¡S|S)z.Try to convert a matrix to a ``DomainMatrix``.N)Úto_DMÚdomainÚis_EXRAWÚ convert_tor)rÚuse_EXÚdMÚKrrrÚ_try_DMs  rgcCs|js|jrdS|j S)z,Check whether to convert to an exact domain.F)Úis_RRÚis_CCÚis_Exact)ÚdomrrrÚ_use_exact_domain s rlTc Cs¼|j\}}|j}||krtdƒ‚t|ƒ}|r| ¡}| |¡}z| ¡\}}Wn ty2tdƒ‚w|r@| |¡}|  ||¡}|rR|jj sJ|  ¡}||  ¡} | S|  ¡|j  |¡} | S)zÌCalculates the inverse using ``DomainMatrix``. See Also ======== inv inverse_ADJ inverse_GE inverse_CH inverse_LDL sympy.polys.matrices.domainmatrix.DomainMatrix.inv r0r=)ÚshaperarrlÚ get_exactrcÚinv_denrrÚ convert_fromÚis_FieldÚto_fieldÚ to_MatrixÚto_sympy) reÚcancelÚmÚnrkÚ use_exactÚ dom_exactÚdMiÚdenÚMirrrÚ_inv_DM+s.   ÿ   þr}cCsVddlm}|jd}|dkr|jdtdS|d|d…d|d…f}|d|d…|dd…f}||dd…d|d…f}||dd…|dd…f}zt|ƒ}Wntyd|jdtdYSw||} | |} || } zt| ƒ} Wnty‡|jdtdYSw| | } ||} | | }|| | }|| | g||ggƒ ¡}|S)z˜Calculates the inverse using BLOCKWISE inversion. See Also ======== inv inverse_ADJ inverse_GE inverse_CH inverse_LDL r)Ú BlockMatrixéÚLU©r.r9Né)Ú&sympy.matrices.expressions.blockmatrixr~rmrr Ú _inv_blockrÚ as_explicit)rr9r~Úir$rrr'ÚD_invÚB_D_iÚBDCÚA_nÚB_nÚdcÚC_nÚD_nÚnnrrrr„Ys6   ÿ  ÿ  r„Nc Cs‚ddlm}m}|jstdƒ‚|r*| ¡}g}|D] }| |j||d¡q||ŽS|dur?|tur?t |dd} | dur>d}n |d vrIt |d d} |durWt ||ƒrUd }nd }|dkr`t | ƒ} n\|d krkt | dd} nQ|d krv|j |d} nF|dkr|j |d} n;|dkrŒ|j|d} n0|dkr—|j|d} n%|d kr¢|j|d} n|dkr­|j|d} n|dkr¸|j|d} ntdƒ‚| | ¡S)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)ÚdiagÚ SparseMatrixr0rNF)rdÚDM)r’ÚDMNCTÚLDLÚGEr“)rurGr€ÚADJÚCHÚQRÚBLOCKzInversion method unrecognized)Úsympy.matricesrr‘r>rÚget_diag_blocksÚappendrr rgÚ isinstancer}Ú inverse_GEÚ inverse_LUÚ inverse_ADJÚ inverse_CHÚ inverse_LDLÚ inverse_QRÚ inverse_BLOCKr,rQ) rr.r9Útry_block_diagrr‘ÚblocksÚrÚblockreÚrvrrrÚ_invsPY €    rª)r*)F)T)Úsympy.polys.matrices.exceptionsrÚsympy.polys.domainsrÚ exceptionsrrrÚ utilitiesr rrr)r/rFrIrSrWrZr]r_rgrlr}r„rªrrrrÚs(   A        .&