1. CS 284a Lecture Wednesday, 26 November, 1997
CS 284a, 26 November 1997
Copyright (c) , John Thornley

2. Final Homework: Skyline Matrix Problem
LU factorization of special kind of sparse matrix.
Without pivoting.
Part of the “Salishan Problems”.
Need to avoid storage/computation of zero elements.
CS 284a, 26 November 1997
Copyright (c) , John Thornley

3. Copyright (c) 1997-98, John Thornley
A Skyline Matrix 1 2 3 4 5 6 7 7 8 4
0  lower[i]  i
for k in 0 .. lower[i] - 1 : A[i][k] == 0
for k in lower[i] .. i : A[i][k] != 0
0  upper[j]  j
for k in 0 .. upper[j] - 1 : A[k][j] == 0
for k in upper[j] .. j : A[k][j] != 0
for i in 0 .. n - 1 :
for j in 0 .. n - 1 : 1 3 5 3 2 8 1 2 2 7 3 2 4 7 4 1 4 8 5 8 4 5 7 6 3 7 5 1 6 9 1 8 9 4 7 4 6 2 8 A
lower[0 .. 7] = {0, 1, 1, 1, 3, 1, 2, 4}
upper[0 .. 7] = {0, 1, 2, 0, 4, 2, 0, 7}
CS 284a, 26 November 1997
Copyright (c) , John Thornley

4. Skyline Matrix Representation7 1 1 3 2 8 2 1 3 2 4 3 8 5 2 7 4 8 4 5 5 7 6 3 7 5 2 4 8 5 6 9 1 8 9 6 4 3 7 1 4 1 9 7 4 6 2 7 8 L U
CS 284a, 26 November 1997
Copyright (c) , John Thornley

5. Skyline Matrix Type Definition
typedef struct {
int n;
int *lower, *upper;
float **L, **U;
} skyline_matrix;
CS 284a, 26 November 1997
Copyright (c) , John Thornley

6. Skyline Matrix Problem Declaration (Sequential)
void solve(skyline_matrix A, float b[], float x[]);
/* Given A and B, compute x such that Ax = b. */
CS 284a, 26 November 1997
Copyright (c) , John Thornley

7. Solving Ax = b Using LU Factorization
Goal: Solve Ax = b for x.
Step 1: Solve A = LU for L and U.
Step 2: Solve Ly = b for y (by forward substitution).
Step 3: Solve Ux = y for x (by back substitution).
CS 284a, 26 November 1997
Copyright (c) , John Thornley

8. Copyright (c) 1997-98, John Thornley
LU Factorization
Factorize A into unit lower-triangular and general upper-triangular factors:
for 0  i < n, 0  j < i : LU[i][j] = (A[i][j] - LU[i][0..j-1]*LU[0..j-1][j])/LU[j][j]
for 0  i < n, i  j < n : LU[i][j] = A[i][j] - LU[i][0..i-1]*LU[0..i-1][j] 6 3 5 9 4 7 8 3 4 1 6 3 5 9 4 7 8 3 4 18 13 18 23 9 22 28 11 9 3 1 4 3 -4 -3 1 4 2 -3 42 37 45 52 23 50 71 37 20 7 4 1 -2 5 7 -3 -1 8 4 24 4 2 81 68 15 23 58 54 4 -2 6 1 7 4 7 5 2 8 18 37 32 65 43 82 91 58 67 3 7 2 8 1 6 4 1 3 4 12 26 31 11 18 81 70 24 40 2 5 -3 4 5 1 5 6 9 7 36 30 25 56 94 11 25 62 18 6 3 7 -3 7 -4 1 4 3 -1 18 29 38 29 5 78 80 2 24 3 5 -4 6 2 -2 3 1 6 4 42 37 37 86 41 50 94 64 39 7 4 5 2 -3 2 2 -4 1 3 A L U
CS 284a, 26 November 1997
Copyright (c) , John Thornley

9. Multithreaded LU Factorization of Skyline Matrix
LU is same shape as A.
Therefore, write LU over A as usual.
Could use barriers, but this limits concurrency.
Blocking and counters should perform better.
CS 284a, 26 November 1997
Copyright (c) , John Thornley