Numerical Methods Contents - SAM

Two arrays:
double * val size P ,
unsigned int * dptr size n
P := n +
n∑
m i (A) .
i=1
(2.6.6)
⎛
⎞
∗ 0 ∗ 0 0 ∗ 0 ∗ 0
A = ⎜ ∗ ∗ 0 0 ⎝
0 0 ∗
⎟
⎠
∗ ∗ 0 0 ∗ ∗ 0
0 0 ∗ ∗ 0 0 ∗
Indexing rule:
dptr[j] = k
⇕
val[k] = a jj
Another example:Reflection at cross diagonal ➤ reduction of # env(A)
⎛
⎞
⎛
⎞
∗ 0 ∗ ∗ ∗
∗ 0 ⎜ 0 ∗ 0 0 0
⎟
⎝
⎠ −→ ⎜ ∗ ∗ ∗ 0 ∗
⎟
⎝
∗ 0 ⎠
0 0 0 ∗ 0
∗ 0 0 ∗ ∗ ∗
∗ ∗ ∗ 0 0 ∗
i ← N + 1 − i
# env(A) = 30 # env(A) = 22
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
val a 11 a 22 a 31 a 32 a 33 a 44 a 52 a 53 a 54 a 55 a 65 a 66 a 73 a 74 a 75 a 76 a 77
dptr 0 1 2 5 6 10 12 17
✸
Example 2.6.30 (Reducing fill-in by reordering).
✸
Minimizing bandwidth:
Goal: Minimize m i (A),A = (a ij ) ∈ R N,N , by permuting rows/columns of A
Example 2.6.29 (Reducing bandwidth by row/column permutations).
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Recall: cyclic permutation of rows/columns of arrow matrix → Ex. 2.6.14
envelope arrow matrix
envelope arrow matrix
0
0
2
2
4
4
M: 114×114 symmetric matrix (from computational
PDEs)
Code 2.6.31: preordering in MATLAB
1 spy (M) ;
2 [ L ,U] = lu (M) ; spy (U) ;
3 r = symrcm(M) ;
4 [ L ,U] = lu (M( r , r ) ) ; spy (U) ;
5 m = symamd(M) ;
6 [ L ,U] = lu (M(m,m) ) ; spy (U) ;
0
20
40
60
80
6
6
Pattern of M
➥
100
8
8
(Here: no row swaps from pivoting !) 0 20 40 60 80 100
nz = 728
10
10
Examine patterns of LU-factors (→ Sect. 2.2) after reordering:
12
0 2 4 6 8 10 12
nz = 31
12
0 2 4 6 8 10 12
nz = 31
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