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# A let mi be the index of the rst nonzero element of

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Unformatted text preview: e pro le form of forward and backward substitution is i; 1 X zi = bi ; lik zk i=1:n (11a) k=mi yi = zi=di xi = yi ; iX +mi k=i+1 lkixk i=1:n i = n : ;1 : 1 (11b) (11c) { Sums in (10, 11) are zero when a lower index exceeds an upper { L can be stored by rows (cf. Section 1.2) Zeros within the pro le become nonzero and are stored { The inner products in (10a,b) and (11a) involve row operations All elements are within the pro le lji in (10b) is computed as uij and re ected { Backward substitution involves a column operation Must test if an operation is inside the pro le Rewrite the operation using (row) saxpys 20 Pro le Schemes Example 6. Column Identi cation 2 6 6 6 6 6 L=6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 The pro le { The local (lower and upper) bandwidth of row i is pi = i ; mi { The bandwidth of A is p = max1 i n pi { The number of multiplications and divisions to factor A is = N ;1 X i=1 pi(pi + 3) (12a) { The number of elements within the pro le is P= 21 N X i=1 pi (12b) Pro le Reduction Zero elements within the pro le become nonzero during the factorization { Fill-in depends on the ordering of equations and unknowns { Methods for reducing bandwidth will reduce pro le Methods of reducing bandwidth are simpler Methods for reducing pro le are more e ective { Example 7. Consider the 5 5 matrix 2 3 1 6 72 6 7 6 73 6 7 A= 6 74 4 5 5 12345 A is symmetric but only the lower triangular portion is shown All elements within the pro le will ll in ( ) Reversing the order of equations and unknowns produces a system with no ll in 2 3 5 6 74 6 7 6 73 6 7 A= 6 72 4 5 1 54321 22 Combat Graph Theory Reading: Saad (1996) A graph G (V E ) is de ned by sets of vertices and edges vN g E = fe1 e2 with ek consisting of pairs (vi vj ), vi vj 2 V V = fv1 v2 eN g v e The adjacency graph representation of a symmetric matrix A: i. Vertices of the graph correspond to rows of A (unknowns) ii. Edges of the graph correspond to nonzero elements of A { Graph vertex i and j are connected if and only if aij 6= 0, i 6= j Example 8. The graph of a symmetric matrix A arising from a scalar PDE is often the mesh 2 6 6 6 6 6 A= 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 1234567 23 1 2 3 4 5 6 7 5 11 00 11 00 11 00 2 1 1 0 1 0 1 07 1 0 1 0 1 11 0 004 1 11 0 00 11 00 1 0 1 0 3 1 06 1 0 Combat Graph Theory Example 9. Draw graphs of the matrices of Example 7 { A symmetric permutation is equivalent to reordering the vertices of a graph { A symmetric permutation changes the rows and columns of a matrix so that the eigenvalues do not change The degree of a vertex of a graph is the number of edges incident on it { Example 10. The degree of Node 1 of Example 8 is 2, that of Node 4 is 4 A level set consists of all unmarked neighbors of all nodes visited in a previous level set 24 Cuthill-McKee Algorithm Cuthill-McKee (CM) pro le minimization technique: { Traverse a graph in level sets { Traverse the level set in order of increasing degree { Select the initial level set to consist of a single vertex v1 Select v1 on the periphery of G function Q = cm(V ) % cm: Cuthill-McKee procedure to number a set of vertices % V to reduce the pro le of a sparse matrix. The procedure % begins with a starting vertex v1 2 V . On return, Q % stores the renumbered vertices. L = fv1g Q=L while ~L 2 ( ) For each vertex in L (in the order in which they are numbered) Append their unnumbered neighbors to Q in order of increasing degree L = f vertices just numbered g end end This algorithm is slightly di erent than S...
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