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
{ Fillin 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 CuthillMcKee Algorithm
CuthillMcKee (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: CuthillMcKee 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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 Fall '98
 JosephE.Flaherty
 Vertex, WI, Diagonal matrix, pro le

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