lecture06

lecture06 - INTRODUCTION TO NUMERICAL SIMULATION LECTURE 6....

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I NTRODUCTION TO N UMERICAL S IMULATION L ECTURE 6. Solution of Sparse Linear Systems 1 T ODAY S O UTLINE : ± Solution of Sparse Linear Systems General Sparse Factorization Graph Based Approach Sparse Matrix Data Structures ² Scattering S OLUTION OF S PARSE L INEAR S YSTEMS General Sparse Factorization ± Graph Based Approach o Construction Structurally Symmetric Matrices and Graphs Note that structurally symmetric does not imply that the values of the matrix are symmetric (i.e., a 12 a 21 ) ² One Node Per Matrix Row ² One Edge Per Off-Diagonal Pair Can one apply these graph-based techniques to the following matrix? 1 5 0 0 5 1 2 0 0 2 1 0 0 0 3 1 This matrix is not “structurally symmetric” in that there is a zero in a 21 and a non-zero value in a 12 . Can still use this approach, just treat it as if there is a nonzero in the a 21 place and use the graphs to do the analysis; there will be some efficiency loss, but the methods will still work. Thus, this technique may be applied to “mildly” structurally symmetric matrices by assuming there is structural symmetry with some loss of efficiency in treating some of the zeros as if they are non-zeros. X X X X X X X X X X X X X X X X X 1 2 3 4 5
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I NTRODUCTION TO N UMERICAL S IMULATION L ECTURE 6. Solution of Sparse Linear Systems 2 o Markowitz Products Markowitz Products = (Node Degree) 2 M 11 Ö 3 ¯ 3 = 9 (degree 1) 2 = 3 2 = 9 M 22 Ö 2 ¯ 2 = 4 (degree 2) 2 = 2 2 = 4 M 11 Ö 3 ¯ 3 = 9 (degree 3) 2 = 3 2 = 9 M 11 Ö 2 ¯ 2 = 4 (degree 4) 2 = 2 2 = 4 M 11 Ö 2 ¯ 2 = 4 (degree 5) 2 = 2 2 = 4 o Factorization One Step of LU Factorization ± Delete the node associated with pivot row ± “Tie together” the graph edges When node 1 is removed, in the matrix, the non-zero entries in the first row create fill-ins that connect up the other nodes that can be seen in the graph.
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lecture06 - INTRODUCTION TO NUMERICAL SIMULATION LECTURE 6....

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