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. 3x, - x, = 2 X, + 4x, = 5 3. 2х, - х, X - 3x, + x3 = -2 -x, + x2 - 3x3 = -6 2. - 4x, + 2x, = -6 Зх, — 5х, 3 1 4. 4x + x2 + x3 = 7 X - 7x, + 2x3 = -2 3x1 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 (x1, X2, . .. , x,) = (0, 0, . . . , 0). + 4.x = 11 5. Apply the Gauss-Seidel method to Exercise 1. 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 22. 4x, + 2.x, - 2x, = 0 X, + 2x, = 3 x, - 3x, - x, = 7 3x, - x2 + 4.x3 = 5 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). O 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 2.x, + x2 = 11. 2x, - 3x2 10. — х, + 4х, 3D1 3x, - 2x, = 2 12. х, + 3x, — х, —5 = 5 23. 4x, + 12- 3 3 X + 6x2 - 2xz + X2 + 5x3 2x2 X4 - Xg = -6 = -7 - X + X6 -5 X, + 3x, - 10x, = 3x, 9. 3x, - x2 + 5x, - Xg - x, - Xg = X3 = 13 X2 + 2x3 = 1 -X3 - X4 + 6x, - x6 X + 5x, - X = 12 = -12 In Exercises 13-16, determine whether the matrix is strictly diago- nally dominant. -X3 + 4x, - Xg = - x, + 5x = -X4 -2 -X4 - x 13. 14. 24. 4x, - x2 - X3 = 18 -x, + 4x2 - X3 - X4 = 18 12 7. 5 -1 1 -x, + 4x, - x4 - 4 15. -3 2 16. 1 -4 -x3 + 4x, - Xs - X6 %3D 6. 13 -3 -X4 + 4xs -Xs + 4x6 X7 = 26 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. X7 - Xg = 16 -x, + 4x, - xg = 10 -x7 + 4xg = 32 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.
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. 3x, - x, = 2 X, + 4x, = 5 3. 2х, - х, X - 3x, + x3 = -2 -x, + x2 - 3x3 = -6 2. - 4x, + 2x, = -6 Зх, — 5х, 3 1 4. 4x + x2 + x3 = 7 X - 7x, + 2x3 = -2 3x1 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 (x1, X2, . .. , x,) = (0, 0, . . . , 0). + 4.x = 11 5. Apply the Gauss-Seidel method to Exercise 1. 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 22. 4x, + 2.x, - 2x, = 0 X, + 2x, = 3 x, - 3x, - x, = 7 3x, - x2 + 4.x3 = 5 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). O 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 2.x, + x2 = 11. 2x, - 3x2 10. — х, + 4х, 3D1 3x, - 2x, = 2 12. х, + 3x, — х, —5 = 5 23. 4x, + 12- 3 3 X + 6x2 - 2xz + X2 + 5x3 2x2 X4 - Xg = -6 = -7 - X + X6 -5 X, + 3x, - 10x, = 3x, 9. 3x, - x2 + 5x, - Xg - x, - Xg = X3 = 13 X2 + 2x3 = 1 -X3 - X4 + 6x, - x6 X + 5x, - X = 12 = -12 In Exercises 13-16, determine whether the matrix is strictly diago- nally dominant. -X3 + 4x, - Xg = - x, + 5x = -X4 -2 -X4 - x 13. 14. 24. 4x, - x2 - X3 = 18 -x, + 4x2 - X3 - X4 = 18 12 7. 5 -1 1 -x, + 4x, - x4 - 4 15. -3 2 16. 1 -4 -x3 + 4x, - Xs - X6 %3D 6. 13 -3 -X4 + 4xs -Xs + 4x6 X7 = 26 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. X7 - Xg = 16 -x, + 4x, - xg = 10 -x7 + 4xg = 32 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.
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
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,