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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. linagr

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. linagr

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.5: Iterative Methods For Solving Linear Systems
Problem 5EQ
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Please answe number 17
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 X, + 4x, = 5 3. 2x, - x2 2. -4x, + 2x, = -6 Зх, — 5х, 3 1 4. 4x1 + x2 + X3 = 7 X1 - 3x, + xạ = -2 -x, + x2 - 3x3 = -6 X1 - 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.x3 = 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 x, + 2x, = 3 22. 4x, + 2.x, - 2x, = 0 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 (x1, 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 2.x, + x2 = 11. 2х, - Зх, 10. — х, + 4х, 3D1 3x, – 2x, = 2 12. х, + 3х, — х, —5 23. 4x, + 12- 3 X1 + 6x2 - 2rz + X2 + 5x3 2x2 X4 - Xg -6 = -7 - X + x6 -5 X, + 3x, - 10x, = 3x, 9. 3x, - x2 = 5 + 5x, - Xg - x, - Xgs = Xz = 13 X2 + 2x3 = 1 -X3 - X4 + 6x, - x6 - X + 5x6 - Xg = 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. -1 -x, + 4x - X4 - 4 15. -3 2 16. 1 -4 1 -x3 + 4x, – Xs - X6 13 -3 -X4 + 4xs X7 = 26 -Xs + 4x6 X7 - Xg = 16 -x, + 4x, - Xg = 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. -x7 + 4.xg = 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.
Transcribed Image Text: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 X, + 4x, = 5 3. 2x, - x2 2. -4x, + 2x, = -6 Зх, — 5х, 3 1 4. 4x1 + x2 + X3 = 7 X1 - 3x, + xạ = -2 -x, + x2 - 3x3 = -6 X1 - 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.x3 = 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 x, + 2x, = 3 22. 4x, + 2.x, - 2x, = 0 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 (x1, 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 2.x, + x2 = 11. 2х, - Зх, 10. — х, + 4х, 3D1 3x, – 2x, = 2 12. х, + 3х, — х, —5 23. 4x, + 12- 3 X1 + 6x2 - 2rz + X2 + 5x3 2x2 X4 - Xg -6 = -7 - X + x6 -5 X, + 3x, - 10x, = 3x, 9. 3x, - x2 = 5 + 5x, - Xg - x, - Xgs = Xz = 13 X2 + 2x3 = 1 -X3 - X4 + 6x, - x6 - X + 5x6 - Xg = 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. -1 -x, + 4x - X4 - 4 15. -3 2 16. 1 -4 1 -x3 + 4x, – Xs - X6 13 -3 -X4 + 4xs X7 = 26 -Xs + 4x6 X7 - Xg = 16 -x, + 4x, - Xg = 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. -x7 + 4.xg = 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.
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