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(a) Explain why the Jacobi and Gauss-Seidel methods may sometimes be preferred over direct methods (such as elimination and factorization) when solving a 3 x 3 linear system Ax = b.

(a) Explain why the Jacobi and Gauss-Seidel methods may sometimes be preferred over direct methods (such as elimination and factorization) when solving a 3 x 3 linear system Ax = b.

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
Section2.1: Introduction To Systems Of Linear Equations
Problem 26EQ: The systems in Exercises 25 and 26 exhibit a lower triangular pattern that makes them easy to solve...
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Instruction: Apply three decimal place rounding to calculations where applicable. (a) Explain why the Jacobi and Gauss-Seidel methods may sometimes be preferred over direct methods (such as elimination and factorization) when solving a 3 x 3 linear system Ax = b. (b) The 3 x 3 system below has solution ₁=1, x₂ = -2 and x3 = 0. 4x1 + 2x₂ - 2x3 0 2x16x22x3 = 14 3x₁x₂ + 4x3 = 5 - (i) Explain why this system is not strictly diagonally dominant (hereafter SDD). (0) (0) (ii) Using the initial approximation z(0) = 0.75, x= -1.75 and r = -0.25, apply three iterations of the Jacobi Method to the above system. (iii) Taking the initial approximation (0) = 0.75, x) = -1.75 and 20) = -0.25, apply three iterations of the Gauss-Seidel Method to the system. (iv) Will the Jacobi and Gauss-Seidel iterates for all 3 x 3 linear systems that are not SDD always diverge? If not, give an example of a system that is not SDD, but for which the Jacobi and Gauss-Seidel iterates converge to the solution of the system.
Transcribed Image Text:Instruction: Apply three decimal place rounding to calculations where applicable. (a) Explain why the Jacobi and Gauss-Seidel methods may sometimes be preferred over direct methods (such as elimination and factorization) when solving a 3 x 3 linear system Ax = b. (b) The 3 x 3 system below has solution ₁=1, x₂ = -2 and x3 = 0. 4x1 + 2x₂ - 2x3 0 2x16x22x3 = 14 3x₁x₂ + 4x3 = 5 - (i) Explain why this system is not strictly diagonally dominant (hereafter SDD). (0) (0) (ii) Using the initial approximation z(0) = 0.75, x= -1.75 and r = -0.25, apply three iterations of the Jacobi Method to the above system. (iii) Taking the initial approximation (0) = 0.75, x) = -1.75 and 20) = -0.25, apply three iterations of the Gauss-Seidel Method to the system. (iv) Will the Jacobi and Gauss-Seidel iterates for all 3 x 3 linear systems that are not SDD always diverge? If not, give an example of a system that is not SDD, but for which the Jacobi and Gauss-Seidel iterates converge to the solution of the system.
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(b) The 3 x 3 system below has solution ₁ = 1, 2 = -2 and x3 = 0. 0 4x1 + 2x2 - 2x3 2x16x22x3 = 14 3x1x₂ + 4x3 = 5 (i) Explain why this system is not strictly diagonally dominant (hereafter SDD). (0) (0) (0) (ii) Using the initial approximation z = 0.75, x=-1.75 and three iterations of the Jacobi Method to the above system. = = -0.25, apply (iii) Taking the initial approximation (0) = 0.75, x = -1.75 and r=-0.25, apply three iterations of the Gauss-Seidel Method to the system. (iv) Will the Jacobi and Gauss-Seidel iterates for all 3 x 3 linear systems that are not SDD always diverge? If not, give an example of a system that is not SDD, but for which the Jacobi and Gauss-Seidel iterates converge to the solution of the system.
Transcribed Image Text:(b) The 3 x 3 system below has solution ₁ = 1, 2 = -2 and x3 = 0. 0 4x1 + 2x2 - 2x3 2x16x22x3 = 14 3x1x₂ + 4x3 = 5 (i) Explain why this system is not strictly diagonally dominant (hereafter SDD). (0) (0) (0) (ii) Using the initial approximation z = 0.75, x=-1.75 and three iterations of the Jacobi Method to the above system. = = -0.25, apply (iii) Taking the initial approximation (0) = 0.75, x = -1.75 and r=-0.25, apply three iterations of the Gauss-Seidel Method to the system. (iv) Will the Jacobi and Gauss-Seidel iterates for all 3 x 3 linear systems that are not SDD always diverge? If not, give an example of a system that is not SDD, but for which the Jacobi and Gauss-Seidel iterates converge to the solution of the system.
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