Theor 721 If A is strictly diagonally dominant, then for any choice of x", both the Jacobi and Gauss-Scidel methods give sequences (x , that converge to the unique solution of Ax -b.
Theor 721 If A is strictly diagonally dominant, then for any choice of x", both the Jacobi and Gauss-Scidel methods give sequences (x , that converge to the unique solution of Ax -b.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please send proof of theorem.... Should short and Authentic....
![(1) x-x| S ITI|x - x|l:
di) |k-x®I< –
We have seen that the Jacobi and Gauss-Seidel iterative techniques can be written
x = Tx-+e and x = T,x"-+e.
using the matrices
T, = D(L+ U) and T,- (D- L)-U.
If p(T,) or p(T,) is less than 1, then the corresponding sequence (x will converge to
the solution x of Ax = b. For example, the Jacobi scheme has
` =D'(L + U)xª-D + D-'b,
and, if (x1 converges to x, then
X=DL+U)x +Db.
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.
We can now give easily verified sufficiency conditions for convergence of the Jacobi
and Gauss-Seidel methods. (To prove convergence for the Jacobi scheme see Exercise 14,
and for the Gauss-Seidel scheme see (Or2), p. 120.)
erle ARRoeet Mete p det t pet c ep d
iy i d Cip L em ia
d ri
7.3 The Jacobi and Gauss-Siedel Iterative Techniques
459
Theor 7.21 If A is strictly diagonally dominant, then for any choice of x, both the Jacobi and
Gauss-Scidel methods give sequences (x , that converge to the unique solution of
Ax-b.
The relations ne rapidity of converge
the spectral radius of the iteration
be seen from Corollary 7.20. The inequale
norm, so it follows from the statement after Theorem 7.15 on page 446 that
Ix" - || = p(Ty'Ix® – x|l.
(7.12)
Thus we would like to select the iterative technique with minimal p(T) < for a particular
system Ax = b. 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. The proof of this result can
be found in [Y], pp. 120-127.
Theorem 7.22 (Stein-Rosenberg)
If a s0, for each i +j and a, > 0, for each i= 1,2, ...,n, then one and only one of the
following statements holds:
(i) Os P(T,) < p(T) < I:
(i) p(T) = p(T,) = 0;
(ii) I< p(T) < p(T):
(iv) p(T) = p(T,) = 1.
For the special case described in Theorem 7.22, we see from part (i) that when one
method gives convergence, then both give convergence, and the Gauss-Seidel method con-
verges faster than the Jacobi method. Part (ii) indicates that when one method diverges then
both diverge, and the divergence is more pronounced for the Gauss-Seidel method.
ERCISE SET 73
3
1. Find the first two iterations of the Jacobi method for the following linear systems, using x"
=0
a. 3 - + -1,
3 + 6r + 2ky =0.
3 + 3x +7x) = 4.
10 -
-A+ 10r - 2,7
- 2 + 10r, = 6.
= 9,
. I0, + S
Sx, + 10 - 4xy
-6,
d.
4 + + +
X 6.
- 25.
- - 3k + + 4
-6,
4x + 8 - xa= -11.
2 + + Sa - 4- AS = 6,
-A- Ay- + 4
- + S -1.
-6,
2 - + a+ 4xs = 6.
Find the first two iterations of the Jacobi method for the following linear systems, using x e
2.
h -2+ + a-4,
X-2 - a = -4,
a.
4x + A - = 5,
- + 3 + = -4,
2 + 2x + Sx = 1.
N+ 2-0.
C.
XI + 4 - - =-I,
4x + - + -2.
d.
-4
-0,
-N +4x -
- 5.
- - + S+ 0,
- A + 4
- 0,
+ 4-
- 4+ 4r, - -2,
- A+ 4- 6.
X-+ + 3=1.
+ L
-6,
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa29dd1fd-ff33-4092-aedd-03e5d354d4ee%2Fbcb7116f-4016-4fcf-b0ac-3ed5b6f5fe4a%2Fn15cbcg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(1) x-x| S ITI|x - x|l:
di) |k-x®I< –
We have seen that the Jacobi and Gauss-Seidel iterative techniques can be written
x = Tx-+e and x = T,x"-+e.
using the matrices
T, = D(L+ U) and T,- (D- L)-U.
If p(T,) or p(T,) is less than 1, then the corresponding sequence (x will converge to
the solution x of Ax = b. For example, the Jacobi scheme has
` =D'(L + U)xª-D + D-'b,
and, if (x1 converges to x, then
X=DL+U)x +Db.
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.
We can now give easily verified sufficiency conditions for convergence of the Jacobi
and Gauss-Seidel methods. (To prove convergence for the Jacobi scheme see Exercise 14,
and for the Gauss-Seidel scheme see (Or2), p. 120.)
erle ARRoeet Mete p det t pet c ep d
iy i d Cip L em ia
d ri
7.3 The Jacobi and Gauss-Siedel Iterative Techniques
459
Theor 7.21 If A is strictly diagonally dominant, then for any choice of x, both the Jacobi and
Gauss-Scidel methods give sequences (x , that converge to the unique solution of
Ax-b.
The relations ne rapidity of converge
the spectral radius of the iteration
be seen from Corollary 7.20. The inequale
norm, so it follows from the statement after Theorem 7.15 on page 446 that
Ix" - || = p(Ty'Ix® – x|l.
(7.12)
Thus we would like to select the iterative technique with minimal p(T) < for a particular
system Ax = b. 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. The proof of this result can
be found in [Y], pp. 120-127.
Theorem 7.22 (Stein-Rosenberg)
If a s0, for each i +j and a, > 0, for each i= 1,2, ...,n, then one and only one of the
following statements holds:
(i) Os P(T,) < p(T) < I:
(i) p(T) = p(T,) = 0;
(ii) I< p(T) < p(T):
(iv) p(T) = p(T,) = 1.
For the special case described in Theorem 7.22, we see from part (i) that when one
method gives convergence, then both give convergence, and the Gauss-Seidel method con-
verges faster than the Jacobi method. Part (ii) indicates that when one method diverges then
both diverge, and the divergence is more pronounced for the Gauss-Seidel method.
ERCISE SET 73
3
1. Find the first two iterations of the Jacobi method for the following linear systems, using x"
=0
a. 3 - + -1,
3 + 6r + 2ky =0.
3 + 3x +7x) = 4.
10 -
-A+ 10r - 2,7
- 2 + 10r, = 6.
= 9,
. I0, + S
Sx, + 10 - 4xy
-6,
d.
4 + + +
X 6.
- 25.
- - 3k + + 4
-6,
4x + 8 - xa= -11.
2 + + Sa - 4- AS = 6,
-A- Ay- + 4
- + S -1.
-6,
2 - + a+ 4xs = 6.
Find the first two iterations of the Jacobi method for the following linear systems, using x e
2.
h -2+ + a-4,
X-2 - a = -4,
a.
4x + A - = 5,
- + 3 + = -4,
2 + 2x + Sx = 1.
N+ 2-0.
C.
XI + 4 - - =-I,
4x + - + -2.
d.
-4
-0,
-N +4x -
- 5.
- - + S+ 0,
- A + 4
- 0,
+ 4-
- 4+ 4r, - -2,
- A+ 4- 6.
X-+ + 3=1.
+ L
-6,
-
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