CS 662 Theory of Parallel Algorithms
Numerical Methods part 3
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San Diego State University  This page last updated May 2, 1996
Jacobi
Let Ax = b then we have:

So



But need x's to compute x's!
Iterate
Let
= a guess for
= t'th iteration for
by the equation:

Example
and
then



What to use for guess?
Does it work?
Two Guesses
t  x1  x2  x1  x2 
0  0  0  1000  1000 
1  0.20  0.3  599.80  500 
2  0.35  0.4  300.35  300 
3  0.41  0.4  179.59  150 
4  0.46  0.5  90.46  89.5 
5  0.47  0.5  53.53  45 
6  0.49  0.5  27.49  26.5 
7  0.49  0.5  15.71  14 
8  0.50  0.5  8.60  7.6 
9  0.50  0.5  4.36  4.5 
10  0.50  0.5  2.93  1.93 
11  0.50  0.5  0.96  1.7 
12  0.50  0.5  1.23  0.23 
13  0.50  0.5  0.06  0.9 
14  0.50  0.5  0.72  0.3 
15  0.50  0.5  0.37  0.6 
16  0.50  0.5  0.57  0.4 
17  0.50  0.5  0.46  0.5 
18  0.50  0.5  0.52  0.5 
19  0.50  0.5  0.49  0.5 
20  0.50  0.5  0.51  0.5 
Example 2
Let
and
then
and
t  x1  x2  x1  x2  x1  x2 
0  0  0  1  1  5  5 
1  0.20  0.5  0.40  2.5  3.20  11 
2  0.50  0.9  1.70  0.3  6.50  6.9 
3  0.74  1.5  0.02  3.9  4.34  14 
4  1.10  2  2.54  0.5  8.30  9.2 
5  1.39  2.7  0.52  5.6  5.71  17 
6  1.82  3.3  3.55  1.5  10.46  12 
7  2.17  4.1  1.13  7.6  7.35  21 
8  2.68  4.8  4.76  2.8  13.05  15 
9  3.10  5.9  1.85  10  9.32  27 
10  3.72  6.7  6.21  4.2  16.16  19 
11  4.22  7.9  2.73  13  11.68  33 
12  4.96  8.9  7.95  6  19.89  24 
13  5.56  10  3.77  16  14.52  40 
14  6.46  12  10.04  8  24.37  30 
15  7.17  13  5.02  21  17.92  49 
16  8.25  15  12.55  11  29.75  36 
17  9.11  17  6.53  26  22.01  60 
18  10.40  19  15.56  14  36.20  45 
19  11.43  21  8.34  32  26.91  73 
20  12.98  23  19.17  17  43.94  54 
Convergence
When does this work?
Let D be the diagonal matrix whose diagonal entries are equal to diagonal
entries of A
Let B = A D
Then
can be written as

converges to x for all initial guesses if and only if

This happens if and only if the eigenvalues of
are all less than 1
Def. A square matrix A with entries
is row diagonally dominant if

for all j
Theorem.
 If A is row diagonally dominant than the Jacobi method converges
Example 1
Example 2
GaussSeidel Method
Let Ax = b and let

if
converges it converges to x
Theorem
 If the Jacobi method converges for a given A then the GaussSeidel method
also converges
Where is the Parallelism?
GaussSeidel Method

Jacobi Method
