1.
Iterative Techniques in Matrix Algebra
Jacobi & Gauss-Seidel Iterative Techniques II
Numerical Analysis (9th Edition)
R L Burden & J D Faires
Beamer Presentation Slides
prepared by
John Carroll
Dublin City University
c
2011 Brooks/Cole, Cengage Learning

2.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38

3.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38

4.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38

5.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi & Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38

6.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi & Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 3 / 38

7.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Looking at the Jacobi Method
A possible improvement to the Jacobi Algorithm can be seen by
re-considering
x(k)
i =
1
aii
2
Xn
664
j=1
j6=i
−aijx(k−1)
j
+ bi
3
775
, for i = 1, 2, . . . , n
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 4 / 38

8.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Looking at the Jacobi Method
A possible improvement to the Jacobi Algorithm can be seen by
re-considering
x(k)
i =
1
aii
2
Xn
664
j=1
j6=i
−aijx(k−1)
j
+ bi
3
775
, for i = 1, 2, . . . , n
The components of x(k−1) are used to compute all the
components x(k)
i of x(k).
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 4 / 38

9.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Looking at the Jacobi Method
A possible improvement to the Jacobi Algorithm can be seen by
re-considering
x(k)
i =
1
aii
2
Xn
664
j=1
j6=i
−aijx(k−1)
j
+ bi
3
775
, for i = 1, 2, . . . , n
The components of x(k−1) are used to compute all the
components x(k)
i of x(k).
But, for i 1, the components x(k)
1 , . . . , x(k)
i−1 of x(k) have already
been computed and are expected to be better approximations to
the actual solutions x1, . . . , xi−1 than are x(k−1)
1 , . . . , x(k−1)
i−1 .
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 4 / 38

10.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Instead of using
x(k)
i =
1
aii
2
Xn
664
j=1
j6=i
−aijx(k−1)
j
+ bi
3
775
, for i = 1, 2, . . . , n
it seems reasonable, then, to compute x(k)
i using these most recently
calculated values.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 5 / 38

11.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Instead of using
x(k)
i =
1
aii
2
Xn
664
j=1
j6=i
−aijx(k−1)
j
+ bi
3
775
, for i = 1, 2, . . . , n
it seems reasonable, then, to compute x(k)
i using these most recently
calculated values.
The Gauss-Seidel Iterative Technique
x(k)
i =
1
aii
2
4−
Xi−1
j=1
(aijx(k)
j ) −
Xn
j=i+1
(aijx(k−1)
j ) + bi
3
5
for each i = 1, 2, . . . , n.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 5 / 38

12.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutions
to
10x1 − x2 + 2x3 = 6
−x1 + 11x2 − x3 + 3x4 = 25
2x1 − x2 + 10x3 − x4 = −11
3x2 − x3 + 8x4 = 15
,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 6 / 38

13.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutions
to
10x1 − x2 + 2x3 = 6
−x1 + 11x2 − x3 + 3x4 = 25
2x1 − x2 + 10x3 − x4 = −11
3x2 − x3 + 8x4 = 15
,
starting with x = (0, 0, 0, 0)t
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 6 / 38

14.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutions
to
10x1 − x2 + 2x3 = 6
−x1 + 11x2 − x3 + 3x4 = 25
2x1 − x2 + 10x3 − x4 = −11
3x2 − x3 + 8x4 = 15
,
starting with x = (0, 0, 0, 0)t and iterating until
kx(k) − x(k−1)k1
kx(k)k1
10−3
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 6 / 38

15.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutions
to
10x1 − x2 + 2x3 = 6
−x1 + 11x2 − x3 + 3x4 = 25
2x1 − x2 + 10x3 − x4 = −11
3x2 − x3 + 8x4 = 15
,
starting with x = (0, 0, 0, 0)t and iterating until
kx(k) − x(k−1)k1
kx(k)k1
10−3
Note: The solution x = (1, 2, −1, 1)t was approximated by Jacobi’s
method in an earlier example.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 6 / 38

16.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Solution (1/3)
For the Gauss-Seidel method we write the system, for each
k = 1, 2, . . . as
x(k)
1 =
1
10
x(k−1)
2 −
1
5
x(k−1)
3 +
3
5
x(k)
2 =
1
11
x(k)
1 +
1
11
x(k−1)
3 −
3
11
x(k−1)
4 +
25
11
x(k)
3 = −
1
5
x(k)
1 +
1
10
x(k)
2 +
1
10
x(k−1)
4 −
11
10
x(k)
4 = −
3
8
x(k)
2 +
1
8
x(k)
3 +
15
8
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 7 / 38

17.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Solution (2/3)
When x(0) = (0, 0, 0, 0)t , we have
x(1) = (0.6000, 2.3272, −0.9873, 0.8789)t .
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 8 / 38

18.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Solution (2/3)
When x(0) = (0, 0, 0, 0)t , we have
x(1) = (0.6000, 2.3272, −0.9873, 0.8789)t . Subsequent iterations give
the values in the following table:
k 0 1 2 3 4 5
x(k)
1 0.0000 0.6000 1.030 1.0065 1.0009 1.0001
x(k)
2 0.0000 2.3272 2.037 2.0036 2.0003 2.0000
x(k)
3 0.0000 −0.9873 −1.014 −1.0025 −1.0003 −1.0000
x(k)
4 0.0000 0.8789 0.984 0.9983 0.9999 1.0000
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 8 / 38

19.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Solution (3/3)
Because
kx(5) − x(4)k1
kx(5)k1
=
0.0008
2.000 = 4 × 10−4
x(5) is accepted as a reasonable approximation to the solution.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 9 / 38

20.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Solution (3/3)
Because
kx(5) − x(4)k1
kx(5)k1
=
0.0008
2.000 = 4 × 10−4
x(5) is accepted as a reasonable approximation to the solution.
Note that, in an earlier example, Jacobi’s method required twice as
many iterations for the same accuracy.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 9 / 38

21.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations
To write the Gauss-Seidel method in matrix form,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 10 / 38

22.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations
To write the Gauss-Seidel method in matrix form, multiply both sides of
x(k)
i =
1
aii
2
4−
Xi−1
j=1
(aijx(k)
j ) −
Xn
j=i+1
(aijx(k−1)
j ) + bi
3
5
by aii and collect all kth iterate terms,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 10 / 38

23.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations
To write the Gauss-Seidel method in matrix form, multiply both sides of
x(k)
i =
1
aii
2
4−
Xi−1
j=1
(aijx(k)
j ) −
Xn
j=i+1
(aijx(k−1)
j ) + bi
3
5
by aii and collect all kth iterate terms, to give
ai1x(k)
1 + ai2x(k)
2 + · · · + aiix(k)
i = −ai,i+1x(k−1)
i+1 − · · · − ainx(k−1)
n + bi
for each i = 1, 2, . . . , n.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 10 / 38

24.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations (Cont’d)
Writing all n equations gives
(k)
1 = −a12x
a11x
(k−1)
2 − a13x
(k−1)
3 − · · · − a1nx
(k−1)
n + b1
(k)
1 + a22x
a21x
(k)
2 = −a23x
(k−1)
3 − · · · − a2nx
(k−1)
n + b2
...
an1x(k)
1 + an2x(k)
2 + · · · + annx(k)
n = bn
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 11 / 38

25.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations (Cont’d)
Writing all n equations gives
(k)
1 = −a12x
a11x
(k−1)
2 − a13x
(k−1)
3 − · · · − a1nx
(k−1)
n + b1
(k)
1 + a22x
a21x
(k)
2 = −a23x
(k−1)
3 − · · · − a2nx
(k−1)
n + b2
...
an1x(k)
1 + an2x(k)
2 + · · · + annx(k)
n = bn
With the definitions of D, L, and U given previously, we have the
Gauss-Seidel method represented by
(D − L)x(k) = Ux(k−1) + b
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 11 / 38

26.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
(D − L)x(k) = Ux(k−1) + b
Re-Writing the Equations (Cont’d)
Solving for x(k) finally gives
x(k) = (D − L)−1Ux(k−1) + (D − L)−1b, for each k = 1, 2, . . .
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 12 / 38

27.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
(D − L)x(k) = Ux(k−1) + b
Re-Writing the Equations (Cont’d)
Solving for x(k) finally gives
x(k) = (D − L)−1Ux(k−1) + (D − L)−1b, for each k = 1, 2, . . .
Letting Tg = (D − L)−1U and cg = (D − L)−1b, gives the Gauss-Seidel
technique the form
x(k) = Tgx(k−1) + cg
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 12 / 38

28.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
(D − L)x(k) = Ux(k−1) + b
Re-Writing the Equations (Cont’d)
Solving for x(k) finally gives
x(k) = (D − L)−1Ux(k−1) + (D − L)−1b, for each k = 1, 2, . . .
Letting Tg = (D − L)−1U and cg = (D − L)−1b, gives the Gauss-Seidel
technique the form
x(k) = Tgx(k−1) + cg
For the lower-triangular matrix D − L to be nonsingular, it is necessary
and sufficient that aii6= 0, for each i = 1, 2, . . . , n.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 12 / 38

29.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 13 / 38

30.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (1/2)
To solve Ax = b given an initial approximation x(0):
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 14 / 38

31.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (1/2)
To solve Ax = b given an initial approximation x(0):
INPUT the number of equations and unknowns n;
the entries aij , 1 i, j n of the matrix A;
the entries bi , 1 i n of b;
the entries XOi , 1 i n of XO = x(0);
tolerance TOL;
maximum number of iterations N.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 14 / 38

32.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (1/2)
To solve Ax = b given an initial approximation x(0):
INPUT the number of equations and unknowns n;
the entries aij , 1 i, j n of the matrix A;
the entries bi , 1 i n of b;
the entries XOi , 1 i n of XO = x(0);
tolerance TOL;
maximum number of iterations N.
OUTPUT the approximate solution x1, . . . , xn or a message
that the number of iterations was exceeded.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 14 / 38

33.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38

34.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Step 3 For i = 1, . . . , n
set xi =
1
aii
2
4−
Xi−1
j=1
aijxj −
Xn
j=i+1
aijXOj + bi
3
5
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38

35.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Step 3 For i = 1, . . . , n
set xi =
1
aii
2
4−
Xi−1
j=1
aijxj −
Xn
j=i+1
aijXOj + bi
3
5
Step 4 If ||x − XO|| TOL then OUTPUT (x1, . . . , xn)
(The procedure was successful)
STOP
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38

36.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Step 3 For i = 1, . . . , n
set xi =
1
aii
2
4−
Xi−1
j=1
aijxj −
Xn
j=i+1
aijXOj + bi
3
5
Step 4 If ||x − XO|| TOL then OUTPUT (x1, . . . , xn)
(The procedure was successful)
STOP
Step 5 Set k = k + 1
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38

37.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Step 3 For i = 1, . . . , n
set xi =
1
aii
2
4−
Xi−1
j=1
aijxj −
Xn
j=i+1
aijXOj + bi
3
5
Step 4 If ||x − XO|| TOL then OUTPUT (x1, . . . , xn)
(The procedure was successful)
STOP
Step 5 Set k = k + 1
Step 6 For i = 1, . . . , n set XOi = xi
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38

38.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Step 3 For i = 1, . . . , n
set xi =
1
aii
2
4−
Xi−1
j=1
aijxj −
Xn
j=i+1
aijXOj + bi
3
5
Step 4 If ||x − XO|| TOL then OUTPUT (x1, . . . , xn)
(The procedure was successful)
STOP
Step 5 Set k = k + 1
Step 6 For i = 1, . . . , n set XOi = xi
Step 7 OUTPUT (‘Maximum number of iterations exceeded’)
STOP (The procedure was unsuccessful)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38

39.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that aii6= 0, for each
i = 1, 2, . . . , n.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 16 / 38

40.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that aii6= 0, for each
i = 1, 2, . . . , n. If one of the aii entries is 0 and the system is
nonsingular, a reordering of the equations can be performed so
that no aii = 0.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 16 / 38

41.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that aii6= 0, for each
i = 1, 2, . . . , n. If one of the aii entries is 0 and the system is
nonsingular, a reordering of the equations can be performed so
that no aii = 0.
To speed convergence, the equations should be arranged so that
aii is as large as possible.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 16 / 38

42.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that aii6= 0, for each
i = 1, 2, . . . , n. If one of the aii entries is 0 and the system is
nonsingular, a reordering of the equations can be performed so
that no aii = 0.
To speed convergence, the equations should be arranged so that
aii is as large as possible.
Another possible stopping criterion in Step 4 is to iterate until
kx(k) − x(k−1)k
kx(k)k
is smaller than some prescribed tolerance.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 16 / 38

43.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that aii6= 0, for each
i = 1, 2, . . . , n. If one of the aii entries is 0 and the system is
nonsingular, a reordering of the equations can be performed so
that no aii = 0.
To speed convergence, the equations should be arranged so that
aii is as large as possible.
Another possible stopping criterion in Step 4 is to iterate until
kx(k) − x(k−1)k
kx(k)k
is smaller than some prescribed tolerance.
For this purpose, any convenient norm can be used, the usual
being the l1 norm.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 16 / 38

44.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 17 / 38

45.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Introduction
To study the convergence of general iteration techniques, we need
to analyze the formula
x(k) = Tx(k−1) + c, for each k = 1, 2, . . .
where x(0) is arbitrary.
The following lemma and the earlier Theorem on convergent
matrices provide the key for this study.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 18 / 38

46.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies (T) 1, then (I − T)−1 exists, and
(I − T)−1 = I + T + T2 + · · · =
1X
j=0
Tj
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 19 / 38

47.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies (T) 1, then (I − T)−1 exists, and
(I − T)−1 = I + T + T2 + · · · =
1X
j=0
Tj
Proof (1/2)
Because Tx = x is true precisely when (I − T)x = (1 − )x, we
have as an eigenvalue of T precisely when 1 − is an
eigenvalue of I − T.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 19 / 38

48.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies (T) 1, then (I − T)−1 exists, and
(I − T)−1 = I + T + T2 + · · · =
1X
j=0
Tj
Proof (1/2)
Because Tx = x is true precisely when (I − T)x = (1 − )x, we
have as an eigenvalue of T precisely when 1 − is an
eigenvalue of I − T.
But || (T) 1, so = 1 is not an eigenvalue of T, and 0
cannot be an eigenvalue of I − T.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 19 / 38

49.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies (T) 1, then (I − T)−1 exists, and
(I − T)−1 = I + T + T2 + · · · =
1X
j=0
Tj
Proof (1/2)
Because Tx = x is true precisely when (I − T)x = (1 − )x, we
have as an eigenvalue of T precisely when 1 − is an
eigenvalue of I − T.
But || (T) 1, so = 1 is not an eigenvalue of T, and 0
cannot be an eigenvalue of I − T.
Hence, (I − T)−1 exists.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 19 / 38

50.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/2)
Let
Sm = I + T + T2 + · · · + Tm
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 20 / 38

51.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/2)
Let
Sm = I + T + T2 + · · · + Tm
Then
(I −T)Sm = (1+T +T2+· · ·+Tm)−(T +T2+· · ·+Tm+1) = I −Tm+1
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 20 / 38

52.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/2)
Let
Sm = I + T + T2 + · · · + Tm
Then
(I −T)Sm = (1+T +T2+· · ·+Tm)−(T +T2+· · ·+Tm+1) = I −Tm+1
and, since T is convergent, the Theorem on convergent matrices
implies that
lim
m!1
(I − T)Sm = lim
(I − Tm+1) = I
m!1
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 20 / 38

53.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/2)
Let
Sm = I + T + T2 + · · · + Tm
Then
(I −T)Sm = (1+T +T2+· · ·+Tm)−(T +T2+· · ·+Tm+1) = I −Tm+1
and, since T is convergent, the Theorem on convergent matrices
implies that
lim
m!1
(I − T)Sm = lim
(I − Tm+1) = I
m!1
Thus, (I − T)−1 = limm!1 Sm = I + T + T2 + · · · =
P1j
=0 Tj
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 20 / 38

54.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Theorem
For any x(0) 2 IRn, the sequence {x(k)}1k
=0 defined by
x(k) = Tx(k−1) + c, for each k 1
converges to the unique solution of
x = Tx + c
if and only if (T) 1.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 21 / 38

55.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38

56.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1. Then,
x(k) = Tx(k−1) + c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38

57.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1. Then,
x(k) = Tx(k−1) + c
= T(Tx(k−2) + c) + c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38

58.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1. Then,
x(k) = Tx(k−1) + c
= T(Tx(k−2) + c) + c
= T2x(k−2) + (T + I)c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38

59.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1. Then,
x(k) = Tx(k−1) + c
= T(Tx(k−2) + c) + c
= T2x(k−2) + (T + I)c
...
= Tkx(0) + (Tk−1 + · · · + T + I)c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38

60.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1. Then,
x(k) = Tx(k−1) + c
= T(Tx(k−2) + c) + c
= T2x(k−2) + (T + I)c
...
= Tkx(0) + (Tk−1 + · · · + T + I)c
Because (T) 1, the Theorem on convergent matrices implies that T
is convergent, and
lim
k!1
Tkx(0) = 0
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38

61.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/5)
The previous lemma implies that
lim
k!1
x(k) = lim
k!1
Tkx(0) +
0
@
1X
j=0
Tj
1
Ac
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 23 / 38

62.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/5)
The previous lemma implies that
lim
k!1
x(k) = lim
k!1
Tkx(0) +
0
@
1X
j=0
Tj
1
Ac
= 0 + (I − T)−1c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 23 / 38

63.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/5)
The previous lemma implies that
lim
k!1
x(k) = lim
k!1
Tkx(0) +
0
@
1X
j=0
Tj
1
Ac
= 0 + (I − T)−1c
= (I − T)−1c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 23 / 38

64.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/5)
The previous lemma implies that
lim
k!1
x(k) = lim
k!1
Tkx(0) +
0
@
1X
j=0
Tj
1
Ac
= 0 + (I − T)−1c
= (I − T)−1c
Hence, the sequence {x(k)} converges to the vector x (I − T)−1c
and x = Tx + c.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 23 / 38

65.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (3/5)
To prove the converse, we will show that for any z 2 IRn, we have
limk!1 Tkz = 0.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 24 / 38

66.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (3/5)
To prove the converse, we will show that for any z 2 IRn, we have
limk!1 Tkz = 0.
Again, by the theorem on convergent matrices, this is equivalent
to (T) 1.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 24 / 38

67.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (3/5)
To prove the converse, we will show that for any z 2 IRn, we have
limk!1 Tkz = 0.
Again, by the theorem on convergent matrices, this is equivalent
to (T) 1.
Let z be an arbitrary vector, and x be the unique solution to
x = Tx + c.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 24 / 38

68.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (3/5)
To prove the converse, we will show that for any z 2 IRn, we have
limk!1 Tkz = 0.
Again, by the theorem on convergent matrices, this is equivalent
to (T) 1.
Let z be an arbitrary vector, and x be the unique solution to
x = Tx + c.
Define x(0) = x − z, and, for k 1, x(k) = Tx(k−1) + c.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 24 / 38

69.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (3/5)
To prove the converse, we will show that for any z 2 IRn, we have
limk!1 Tkz = 0.
Again, by the theorem on convergent matrices, this is equivalent
to (T) 1.
Let z be an arbitrary vector, and x be the unique solution to
x = Tx + c.
Define x(0) = x − z, and, for k 1, x(k) = Tx(k−1) + c.
Then {x(k)} converges to x.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 24 / 38

70.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38

71.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
so
x − x(k) = T
x − x(k−1)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38

72.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
so
x − x(k) = T
x − x(k−1)
= T2
x − x(k−2)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38

73.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
so
x − x(k) = T
x − x(k−1)
= T2
x − x(k−2)
=
...
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38

74.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
so
x − x(k) = T
x − x(k−1)
= T2
x − x(k−2)
=
...
= Tk
x − x(0)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38

75.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
so
x − x(k) = T
x − x(k−1)
= T2
x − x(k−2)
=
...
= Tk
x − x(0)
= Tkz
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38

76.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (5/5)
Hence
lim
k!1
Tkz = lim
k!1
Tk
x − x(0)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 26 / 38

77.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (5/5)
Hence
lim
k!1
Tkz = lim
k!1
Tk
x − x(0)
= lim
k!1
x − x(k)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 26 / 38

78.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (5/5)
Hence
lim
k!1
Tkz = lim
k!1
Tk
x − x(0)
= lim
k!1
x − x(k)
= 0
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 26 / 38

79.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (5/5)
Hence
lim
k!1
Tkz = lim
k!1
Tk
x − x(0)
= lim
k!1
x − x(k)
= 0
But z 2 IRn was arbitrary, so by the theorem on convergent
matrices, T is convergent and (T) 1.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 26 / 38

80.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Corollary
kTk 1 for any natural matrix norm and c is a given vector, then the
sequence {x(k)}1k
=0 defined by
x(k) = Tx(k−1) + c
converges, for any x(0) 2 IRn, to a vector x 2 IRn, with x = Tx + c, and
the following error bounds hold:
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 27 / 38

81.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Corollary
kTk 1 for any natural matrix norm and c is a given vector, then the
sequence {x(k)}1k
=0 defined by
x(k) = Tx(k−1) + c
converges, for any x(0) 2 IRn, to a vector x 2 IRn, with x = Tx + c, and
the following error bounds hold:
(i) kx − x(k)k kTkkkx(0) − xk
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 27 / 38

82.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Corollary
kTk 1 for any natural matrix norm and c is a given vector, then the
sequence {x(k)}1k
=0 defined by
x(k) = Tx(k−1) + c
converges, for any x(0) 2 IRn, to a vector x 2 IRn, with x = Tx + c, and
the following error bounds hold:
(i) kx − x(k)k kTkkkx(0) − xk
(ii) kx − x(k)k kTkk
1−kTkkx(1) − x(0)k
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 27 / 38

83.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Corollary
kTk 1 for any natural matrix norm and c is a given vector, then the
sequence {x(k)}1k
=0 defined by
x(k) = Tx(k−1) + c
converges, for any x(0) 2 IRn, to a vector x 2 IRn, with x = Tx + c, and
the following error bounds hold:
(i) kx − x(k)k kTkkkx(0) − xk
(ii) kx − x(k)k kTkk
1−kTkkx(1) − x(0)k
The proof of the following corollary is similar to that for the Corollary to
the Fixed-Point Theorem for a single nonlinear equation.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 27 / 38

84.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 28 / 38

85.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Using the Matrix Formulations
We have seen that the Jacobi and Gauss-Seidel iterative techniques
can be written
x(k) = Tjx(k−1) + cj and
x(k) = Tgx(k−1) + cg
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 29 / 38

86.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Using the Matrix Formulations
We have seen that the Jacobi and Gauss-Seidel iterative techniques
can be written
x(k) = Tjx(k−1) + cj and
x(k) = Tgx(k−1) + cg
using the matrices
Tj = D−1(L + U) and Tg = (D − L)−1U
respectively.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 29 / 38

87.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Using the Matrix Formulations
We have seen that the Jacobi and Gauss-Seidel iterative techniques
can be written
x(k) = Tjx(k−1) + cj and
x(k) = Tgx(k−1) + cg
using the matrices
Tj = D−1(L + U) and Tg = (D − L)−1U
respectively. If (Tj ) or (Tg) is less than 1, then the corresponding
sequence {x(k)}1k
=0 will converge to the solution x of Ax = b.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 29 / 38

88.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Example
For example, the Jacobi method has
x(k) = D−1(L + U)x(k−1) + D−1b,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 30 / 38

89.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Example
For example, the Jacobi method has
x(k) = D−1(L + U)x(k−1) + D−1b,
and, if {x(k)}1k
=0 converges to x,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 30 / 38

90.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Example
For example, the Jacobi method has
x(k) = D−1(L + U)x(k−1) + D−1b,
and, if {x(k)}1k
=0 converges to x, then
x = D−1(L + U)x + D−1b
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 30 / 38

91.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Example
For example, the Jacobi method has
x(k) = D−1(L + U)x(k−1) + D−1b,
and, if {x(k)}1k
=0 converges to x, then
x = D−1(L + U)x + D−1b
This implies that
Dx = (L + U)x + b and (D − L − U)x = b
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 30 / 38

92.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Example
For example, the Jacobi method has
x(k) = D−1(L + U)x(k−1) + D−1b,
and, if {x(k)}1k
=0 converges to x, then
x = D−1(L + U)x + D−1b
This implies that
Dx = (L + U)x + b and (D − L − U)x = b
Since D − L − U = A, the solution x satisfies Ax = b.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 30 / 38

93.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
The following are easily verified sufficiency conditions for convergence
of the Jacobi and Gauss-Seidel methods.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 31 / 38

94.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
The following are easily verified sufficiency conditions for convergence
of the Jacobi and Gauss-Seidel methods.
Theorem
If A is strictly diagonally dominant, then for any choice of x(0), both the
Jacobi and Gauss-Seidel methods give sequences {x(k)}1k
=0 that
converge to the unique solution of Ax = b.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 31 / 38

95.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Is Gauss-Seidel better than Jacobi?
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 32 / 38

96.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Is Gauss-Seidel better than Jacobi?
No general results exist to tell which of the two techniques, Jacobi
or Gauss-Seidel, will be most successful for an arbitrary linear
system.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 32 / 38

97.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Is Gauss-Seidel better than Jacobi?
No general results exist to tell which of the two techniques, Jacobi
or Gauss-Seidel, will be most successful for an arbitrary linear
system.
In special cases, however, the answer is known, as is
demonstrated in the following theorem.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 32 / 38

98.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
(Stein-Rosenberg) Theorem
If aij 0, for each i6= j and aii 0, for each i = 1, 2, . . . , n, then one
and only one of the following statements holds:
(i) 0 (Tg) (Tj ) 1
(ii) 1 (Tj ) (Tg)
(iii) (Tj ) = (Tg) = 0
(iv) (Tj ) = (Tg) = 1
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 33 / 38

99.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
(Stein-Rosenberg) Theorem
If aij 0, for each i6= j and aii 0, for each i = 1, 2, . . . , n, then one
and only one of the following statements holds:
(i) 0 (Tg) (Tj ) 1
(ii) 1 (Tj ) (Tg)
(iii) (Tj ) = (Tg) = 0
(iv) (Tj ) = (Tg) = 1
For the proof of this result, see pp. 120–127. of
Young, D. M., Iterative solution of large linear systems, Academic
Press, New York, 1971, 570 pp.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 33 / 38

100.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 (Tg) (Tj ) 1
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 34 / 38

101.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 (Tg) (Tj ) 1
that when one method gives convergence, then both give
convergence,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 34 / 38

102.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 (Tg) (Tj ) 1
that when one method gives convergence, then both give
convergence, and the Gauss-Seidel method converges faster than
the Jacobi method.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 34 / 38

103.
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 (Tg) (Tj ) 1
that when one method gives convergence, then both give
convergence, and the Gauss-Seidel method converges faster than
the Jacobi method.
Part (ii), namely
1 (Tj ) (Tg)
indicates that when one method diverges then both diverge, and
the divergence is more pronounced for the Gauss-Seidel method.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 34 / 38

104.
Questions?

105.
Eigenvalues Eigenvectors: Convergent Matrices
Theorem
The following statements are equivalent.
(i) A is a convergent matrix.
(ii) limn!1 kAnk = 0, for some natural norm.
(iii) limn!1 kAnk = 0, for all natural norms.
(iv) (A) 1.
(v) limn!1 Anx = 0, for every x.
The proof of this theorem can be found on p. 14 of Issacson, E. and H.
B. Keller, Analysis of Numerical Methods, John Wiley Sons, New
York, 1966, 541 pp.
Return to General Iteration Methods — Introduction
Return to General Iteration Methods — Lemma
Return to General Iteration Methods — Theorem

106.
Fixed-Point Theorem
Let g 2 C[a, b] be such that g(x) 2 [a, b], for all x in [a, b]. Suppose, in
addition, that g0 exists on (a, b) and that a constant 0 k 1 exists
with
|g0(x)| k, for all x 2 (a, b).
Then for any number p0 in [a, b], the sequence defined by
pn = g(pn−1), n 1
converges to the unique fixed point p in [a, b].
Return to the Corrollary to the Fixed-Point Theorem

107.
Functional (Fixed-Point) Iteration
Corrollary to the Fixed-Point Convergence Result
If g satisfies the hypothesis of the Fixed-Point Theorem then
|pn − p|
kn
1 − k
|p1 − p0|
Return to the Corollary to the Convergence Theorem for General Iterative Methods