\hx#@SSKJrJrJr SSKJr SrSrSrS Sjr g) )flattenconnected_componentsstrongly_connected_components)NonSquareMatrixErrorcUR(d[e[UR5n[ UR 5R 55n[X45$)a$Returns the list of connected vertices of the graph when a square matrix is viewed as a weighted graph. Examples ======== >>> from sympy import Matrix >>> A = Matrix([ ... [66, 0, 0, 68, 0, 0, 0, 0, 67], ... [0, 55, 0, 0, 0, 0, 54, 53, 0], ... [0, 0, 0, 0, 1, 2, 0, 0, 0], ... [86, 0, 0, 88, 0, 0, 0, 0, 87], ... [0, 0, 10, 0, 11, 12, 0, 0, 0], ... [0, 0, 20, 0, 21, 22, 0, 0, 0], ... [0, 45, 0, 0, 0, 0, 44, 43, 0], ... [0, 35, 0, 0, 0, 0, 34, 33, 0], ... [76, 0, 0, 78, 0, 0, 0, 0, 77]]) >>> A.connected_components() [[0, 3, 8], [1, 6, 7], [2, 4, 5]] Notes ===== Even if any symbolic elements of the matrix can be indeterminate to be zero mathematically, this only takes the account of the structural aspect of the matrix, so they will considered to be nonzero. ) is_squarerrangerowssortedtodokkeysr)MVEs L/var/www/auris/envauris/lib/python3.13/site-packages/sympy/matrices/graph.py_connected_componentsrsB: ;;"" aff Aqwwy~~ A  ''cUR(d[e[USS5nUbUR5$[ UR 5n[ UR5R55n[X#45$)ajReturns the list of strongly connected vertices of the graph when a square matrix is viewed as a weighted graph. Examples ======== >>> from sympy import Matrix >>> A = Matrix([ ... [44, 0, 0, 0, 43, 0, 45, 0, 0], ... [0, 66, 62, 61, 0, 68, 0, 60, 67], ... [0, 0, 22, 21, 0, 0, 0, 20, 0], ... [0, 0, 12, 11, 0, 0, 0, 10, 0], ... [34, 0, 0, 0, 33, 0, 35, 0, 0], ... [0, 86, 82, 81, 0, 88, 0, 80, 87], ... [54, 0, 0, 0, 53, 0, 55, 0, 0], ... [0, 0, 2, 1, 0, 0, 0, 0, 0], ... [0, 76, 72, 71, 0, 78, 0, 70, 77]]) >>> A.strongly_connected_components() [[0, 4, 6], [2, 3, 7], [1, 5, 8]] _repN) r rgetattrsccr r r r rr)rreprrs r_strongly_connected_componentsr+sc* ;;"" !VT "C wwy aff Aqwwy~~ A (! 00rcSSKJn SSKJn SSKJn UR 5nU"[U55nU"U5n/nUHnURXU45 M U"U6n Xi4$)a>Decomposes a square matrix into block diagonal form only using the permutations. Explanation =========== The decomposition is in a form of $A = P^{-1} B P$ where $P$ is a permutation matrix and $B$ is a block diagonal matrix. Returns ======= P, B : PermutationMatrix, BlockDiagMatrix *P* is a permutation matrix for the similarity transform as in the explanation. And *B* is the block diagonal matrix of the result of the permutation. If you would like to get the diagonal blocks from the BlockDiagMatrix, see :meth:`~sympy.matrices.expressions.blockmatrix.BlockDiagMatrix.get_diag_blocks`. Examples ======== >>> from sympy import Matrix, pprint >>> A = Matrix([ ... [66, 0, 0, 68, 0, 0, 0, 0, 67], ... [0, 55, 0, 0, 0, 0, 54, 53, 0], ... [0, 0, 0, 0, 1, 2, 0, 0, 0], ... [86, 0, 0, 88, 0, 0, 0, 0, 87], ... [0, 0, 10, 0, 11, 12, 0, 0, 0], ... [0, 0, 20, 0, 21, 22, 0, 0, 0], ... [0, 45, 0, 0, 0, 0, 44, 43, 0], ... [0, 35, 0, 0, 0, 0, 34, 33, 0], ... [76, 0, 0, 78, 0, 0, 0, 0, 77]]) >>> P, B = A.connected_components_decomposition() >>> pprint(P) PermutationMatrix((1 3)(2 8 5 7 4 6)) >>> pprint(B) [[66 68 67] ] [[ ] ] [[86 88 87] 0 0 ] [[ ] ] [[76 78 77] ] [ ] [ [55 54 53] ] [ [ ] ] [ 0 [45 44 43] 0 ] [ [ ] ] [ [35 34 33] ] [ ] [ [0 1 2 ]] [ [ ]] [ 0 0 [10 11 12]] [ [ ]] [ [20 21 22]] >>> P = P.as_explicit() >>> B = B.as_explicit() >>> P.T*B*P == A True Notes ===== This problem corresponds to the finding of the connected components of a graph, when a matrix is viewed as a weighted graph. r Permutation)BlockDiagMatrixPermutationMatrix) sympy.combinatorics.permutationsr&sympy.matrices.expressions.blockmatrixr&sympy.matrices.expressions.permutationr rrappend) rrrr iblockspPblocksbBs r#_connected_components_decompositionr+MsgL=FH$$&GGG$%A!A F  a1g A 4Krc<SSKJn SSKJn SSKJn UR 5nU(d[[U55nU"[U55nU"U5n/nUH4n /n UHn U RX U 45 M URU 5 M6 U"U5n X|4$)a Decomposes a square matrix into block triangular form only using the permutations. Explanation =========== The decomposition is in a form of $A = P^{-1} B P$ where $P$ is a permutation matrix and $B$ is a block diagonal matrix. Parameters ========== lower : bool Makes $B$ lower block triangular when ``True``. Otherwise, makes $B$ upper block triangular. Returns ======= P, B : PermutationMatrix, BlockMatrix *P* is a permutation matrix for the similarity transform as in the explanation. And *B* is the block triangular matrix of the result of the permutation. Examples ======== >>> from sympy import Matrix, pprint >>> A = Matrix([ ... [44, 0, 0, 0, 43, 0, 45, 0, 0], ... [0, 66, 62, 61, 0, 68, 0, 60, 67], ... [0, 0, 22, 21, 0, 0, 0, 20, 0], ... [0, 0, 12, 11, 0, 0, 0, 10, 0], ... [34, 0, 0, 0, 33, 0, 35, 0, 0], ... [0, 86, 82, 81, 0, 88, 0, 80, 87], ... [54, 0, 0, 0, 53, 0, 55, 0, 0], ... [0, 0, 2, 1, 0, 0, 0, 0, 0], ... [0, 76, 72, 71, 0, 78, 0, 70, 77]]) A lower block triangular decomposition: >>> P, B = A.strongly_connected_components_decomposition() >>> pprint(P) PermutationMatrix((8)(1 4 3 2 6)(5 7)) >>> pprint(B) [[44 43 45] [0 0 0] [0 0 0] ] [[ ] [ ] [ ] ] [[34 33 35] [0 0 0] [0 0 0] ] [[ ] [ ] [ ] ] [[54 53 55] [0 0 0] [0 0 0] ] [ ] [ [0 0 0] [22 21 20] [0 0 0] ] [ [ ] [ ] [ ] ] [ [0 0 0] [12 11 10] [0 0 0] ] [ [ ] [ ] [ ] ] [ [0 0 0] [2 1 0 ] [0 0 0] ] [ ] [ [0 0 0] [62 61 60] [66 68 67]] [ [ ] [ ] [ ]] [ [0 0 0] [82 81 80] [86 88 87]] [ [ ] [ ] [ ]] [ [0 0 0] [72 71 70] [76 78 77]] >>> P = P.as_explicit() >>> B = B.as_explicit() >>> P.T * B * P == A True An upper block triangular decomposition: >>> P, B = A.strongly_connected_components_decomposition(lower=False) >>> pprint(P) PermutationMatrix((0 1 5 7 4 3 2 8 6)) >>> pprint(B) [[66 68 67] [62 61 60] [0 0 0] ] [[ ] [ ] [ ] ] [[86 88 87] [82 81 80] [0 0 0] ] [[ ] [ ] [ ] ] [[76 78 77] [72 71 70] [0 0 0] ] [ ] [ [0 0 0] [22 21 20] [0 0 0] ] [ [ ] [ ] [ ] ] [ [0 0 0] [12 11 10] [0 0 0] ] [ [ ] [ ] [ ] ] [ [0 0 0] [2 1 0 ] [0 0 0] ] [ ] [ [0 0 0] [0 0 0] [44 43 45]] [ [ ] [ ] [ ]] [ [0 0 0] [0 0 0] [34 33 35]] [ [ ] [ ] [ ]] [ [0 0 0] [0 0 0] [54 53 55]] >>> P = P.as_explicit() >>> B = B.as_explicit() >>> P.T * B * P == A True rr) BlockMatrixr) r!rr"r-r#r rlistreversedrr$) rlowerrr-r r%r&r'r acolsr)r*s r,_strongly_connected_components_decompositionr3sD=BH--/G x()GG$%A!A D A KKQ$  D  DA 4KrN)T) sympy.utilities.iterablesrrr exceptionsrrrr+r3rrr7s-AA,"(J1DSltr