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. 2x, - X2 2. -4x, + 2x, = -6 3x, - 5x, = 1 4. 4x, + x2 + x3 = x - 7x, + 2x, = -2 3x1 = 2 7 X - 3x2 + x, = -2 -X, + x2 - 3x3 = -6 5. Apply the Gauss-Seidel method to Exercise 1. 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,, xz, . . . , x,) = (0, 0, . . . , 0). + 4x3 = 11 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, + 2x, - 2x, = 0 X, + 2x, = 3 X, - 3x, - x3 = 7 3x, - x2 + 4.x; = 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). * 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 = 11. 2х, - Зх, 10. -x, + 4x, = 1 Зх, — 2х, 32 12. x, + 3x, - X3 = 5 23. 41, + 12* 3 3 X + 6x2 X2 + 5x3 2x2 2x3 + X4 - Xg -6 = -7 - Xg + X6 + 5x4 - Xg X4 + 6x, - x6 - x + 5x6 + 4x7 -5 X, + 3x, - 10x, = 3x1 9 3x, - x2 = 5 X = X3 = 13 X2 + 2x3 = 1 X = 12 - Er- = - 12 In Exercises 13-16, determine whether the matrix is strictly diago- nally dominant. -X3 X = -2 - x, + 5x, = -X4 -x4 - Xg 13. 14. 24. 4x, - X2 - = 18 -x, + 4x, - X3 - -х, + 4x, — X4 = 18 [12 - 3 5 -1 1 2 -3 6 X - Xs 4 15. 2 16. 1 -4 -X, + 4x, - X6 4 6 13 = 26 -x4 + 4x5 -Xg + 4x, - X6 - X7 Xg - Xg = 16 + 4x, - x, = 10 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, = 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. 2x, - X2 2. -4x, + 2x, = -6 3x, - 5x, = 1 4. 4x, + x2 + x3 = x - 7x, + 2x, = -2 3x1 = 2 7 X - 3x2 + x, = -2 -X, + x2 - 3x3 = -6 5. Apply the Gauss-Seidel method to Exercise 1. 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,, xz, . . . , x,) = (0, 0, . . . , 0). + 4x3 = 11 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, + 2x, - 2x, = 0 X, + 2x, = 3 X, - 3x, - x3 = 7 3x, - x2 + 4.x; = 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). * 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 = 11. 2х, - Зх, 10. -x, + 4x, = 1 Зх, — 2х, 32 12. x, + 3x, - X3 = 5 23. 41, + 12* 3 3 X + 6x2 X2 + 5x3 2x2 2x3 + X4 - Xg -6 = -7 - Xg + X6 + 5x4 - Xg X4 + 6x, - x6 - x + 5x6 + 4x7 -5 X, + 3x, - 10x, = 3x1 9 3x, - x2 = 5 X = X3 = 13 X2 + 2x3 = 1 X = 12 - Er- = - 12 In Exercises 13-16, determine whether the matrix is strictly diago- nally dominant. -X3 X = -2 - x, + 5x, = -X4 -x4 - Xg 13. 14. 24. 4x, - X2 - = 18 -x, + 4x, - X3 - -х, + 4x, — X4 = 18 [12 - 3 5 -1 1 2 -3 6 X - Xs 4 15. 2 16. 1 -4 -X, + 4x, - X6 4 6 13 = 26 -x4 + 4x5 -Xg + 4x, - X6 - X7 Xg - Xg = 16 + 4x, - x, = 10 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, = 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
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