3x, - 2x, = 2 12. X, + 3x, - X3 = 5 3x, = 5 X2 + 2x3 = 1 ether the matrix is strictly diago-
3x, - 2x, = 2 12. X, + 3x, - X3 = 5 3x, = 5 X2 + 2x3 = 1 ether the matrix is strictly diago-
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please answer number 20 with step by step solution
![SECTION 10.2
EXERCISES
In Exercises 1-4, apply the Jacobi method to the given system of
linear equations, using the initial approximation (x, x2, ..., x,) =
(0, 0, ..., 0). Continue performing iterations until two successive
approximations are identical when rounded to three significant digits.
19. Interchange the rows of the system of linear equations in
Exercise 11 to obtain a system with a strictly diagonally dom-
inant coefficient matrix. Then apply the Gauss-Seidel method
to approximate the solution to two significant digits.
20. Interchange the rows of the system of linear equations in
Exercise 12 to obtain a system with a strictly diagonally dom-
inant coefficient matrix. Then apply the Gauss-Seidel method
to approximate the solution to two significant digits.
1. Зх, — х, 3D 2
2. - 4x, + 2x, = - 6
3x, - 5x, = 1
4. 4x, + x2 + X3 = 7
X2 =
X, + 4x, = 5
3. 2х, -
X2
2
%3D
X - 3x, + x, = -2
-x, + x2 - 3xz = -6
5. Apply the Gauss-Seidel method to Exercise 1.
X - 7x, + 2x, = -2
In Exercises 21 and 22, the coefficient matrix of the system of linear
equations is not strictly diagonally dominant. Show that the Jacobi
and Gauss-Seidel methods converge using an initial approximation
of (x, X, . .. ,x,) = (0, 0, . , 0).
%3D
3x1
+ 4x3 = 11
%3D
6. Apply the Gauss-Seidel method to Exercise 2.
7. Apply the Gauss-Seidel method to Exercise 3.
8. Apply the Gauss-Seidel method to Exercise 4.
21. -4x, + 5x, = 1
x, + 2x, = 3
22. 4х, + 2х, -2х, 3 0
х, — Зх, — х, 3D 7
3x, - x2 + 4.xz = 5
%3D
In Exercises 9-12, show that the Gauss-Seidel method diverges for
the given system using the initial approximation (x, x2, . . . , x,„) =
(0, 0, ..., 0).
%3D
Y In Exercises 23 and 24, write a computer program that applies the
Gauss-Siedel method to solve the system of linear equations.
9. x, - 2x, = -1
2x, + x2 = 3
11. 2х, - Зх,
10. — х, + 4х, 3 1
3x, – 2x, = 2
12. х, + 3x, — X, — 5
23. 4л, +
3
x, + 6x2 - 2x3 + X4 - Xg
-6
= -7
%3D
X2 + 5x
2x2
- Xe + X6
-5
%3D
х, + 3х, — 10х, —
3x, - x2
= 5
+ 5x, - X
- X7
X7 - Xg =
3x,
X3 = 13
Xz + 2x3 = 1
Xz = 12
= - 12
-X3 -
X + 6x, - X6
Xg + 5x6
In Exercises 13–16, determine whether the matrix is strictly diago-
nally dominant.
-x3
+ 4x, - x = -2
- x, + 5x, =
-X4
-X4 - X.
2
2
13.
1]
- 1
14.
24. 4x, –
X2 - X3
= 18
-x, + 4x2 - X3 -
-x, + 4x -
X4
= 18
12
6
[7
X - xs
= 4
15. 2 -3
16. 1
-4
1
-x + 4x, -
I's -
= 4
6
13
2
-3
-X4 + 4xs -
-Xs + 4x, –
X6 -
X7
= 26
X, - xg = 16
%3D
17. Interchange the rows of the system of linear equations in
Exercise 9 to obtain a system with a strictly diagonally domi-
nant coefficient matrix. Then apply the Gauss-Seidel method
to approximate the solution to two significant digits.
-x, + 4x, - Xg = 10
-x, + 4xg = 32
%3D
18. Interchange the rows of the system of linear equations in
Exercise 10 to obtain a system with a strictly diagonally dom-
inant coefficient matrix. Then apply the Gauss-Seidel method
to approximate the solution to two significant digits.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2938bd5a-763a-4bd0-9f18-fdc610b49a1b%2F6d841cec-e4e0-4a21-8f09-73daac410e17%2Fiqlzv4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:SECTION 10.2
EXERCISES
In Exercises 1-4, apply the Jacobi method to the given system of
linear equations, using the initial approximation (x, x2, ..., x,) =
(0, 0, ..., 0). Continue performing iterations until two successive
approximations are identical when rounded to three significant digits.
19. Interchange the rows of the system of linear equations in
Exercise 11 to obtain a system with a strictly diagonally dom-
inant coefficient matrix. Then apply the Gauss-Seidel method
to approximate the solution to two significant digits.
20. Interchange the rows of the system of linear equations in
Exercise 12 to obtain a system with a strictly diagonally dom-
inant coefficient matrix. Then apply the Gauss-Seidel method
to approximate the solution to two significant digits.
1. Зх, — х, 3D 2
2. - 4x, + 2x, = - 6
3x, - 5x, = 1
4. 4x, + x2 + X3 = 7
X2 =
X, + 4x, = 5
3. 2х, -
X2
2
%3D
X - 3x, + x, = -2
-x, + x2 - 3xz = -6
5. Apply the Gauss-Seidel method to Exercise 1.
X - 7x, + 2x, = -2
In Exercises 21 and 22, the coefficient matrix of the system of linear
equations is not strictly diagonally dominant. Show that the Jacobi
and Gauss-Seidel methods converge using an initial approximation
of (x, X, . .. ,x,) = (0, 0, . , 0).
%3D
3x1
+ 4x3 = 11
%3D
6. Apply the Gauss-Seidel method to Exercise 2.
7. Apply the Gauss-Seidel method to Exercise 3.
8. Apply the Gauss-Seidel method to Exercise 4.
21. -4x, + 5x, = 1
x, + 2x, = 3
22. 4х, + 2х, -2х, 3 0
х, — Зх, — х, 3D 7
3x, - x2 + 4.xz = 5
%3D
In Exercises 9-12, show that the Gauss-Seidel method diverges for
the given system using the initial approximation (x, x2, . . . , x,„) =
(0, 0, ..., 0).
%3D
Y In Exercises 23 and 24, write a computer program that applies the
Gauss-Siedel method to solve the system of linear equations.
9. x, - 2x, = -1
2x, + x2 = 3
11. 2х, - Зх,
10. — х, + 4х, 3 1
3x, – 2x, = 2
12. х, + 3x, — X, — 5
23. 4л, +
3
x, + 6x2 - 2x3 + X4 - Xg
-6
= -7
%3D
X2 + 5x
2x2
- Xe + X6
-5
%3D
х, + 3х, — 10х, —
3x, - x2
= 5
+ 5x, - X
- X7
X7 - Xg =
3x,
X3 = 13
Xz + 2x3 = 1
Xz = 12
= - 12
-X3 -
X + 6x, - X6
Xg + 5x6
In Exercises 13–16, determine whether the matrix is strictly diago-
nally dominant.
-x3
+ 4x, - x = -2
- x, + 5x, =
-X4
-X4 - X.
2
2
13.
1]
- 1
14.
24. 4x, –
X2 - X3
= 18
-x, + 4x2 - X3 -
-x, + 4x -
X4
= 18
12
6
[7
X - xs
= 4
15. 2 -3
16. 1
-4
1
-x + 4x, -
I's -
= 4
6
13
2
-3
-X4 + 4xs -
-Xs + 4x, –
X6 -
X7
= 26
X, - xg = 16
%3D
17. Interchange the rows of the system of linear equations in
Exercise 9 to obtain a system with a strictly diagonally domi-
nant coefficient matrix. Then apply the Gauss-Seidel method
to approximate the solution to two significant digits.
-x, + 4x, - Xg = 10
-x, + 4xg = 32
%3D
18. Interchange the rows of the system of linear equations in
Exercise 10 to obtain a system with a strictly diagonally dom-
inant coefficient matrix. Then apply the Gauss-Seidel method
to approximate the solution to two significant digits.
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