Use the same notations as on Page 6 of the lecture notes: A is the coefficient matrix for each linear system, D is the diagonal matrix with diagonal value ai, and D-L is the lower triangular matrix of A. For the following linear systems 3x₁ + x₂ + x3 = 0 x₁ + 3x₂x3 = 4 (3x₁ + x₂5x3 = -6 (1) - (x₁ + 3x₂ − 3x3 = 2 (3) x₁ + x₂ + x3 = 2 (3x₁ + 3x₂ + x3 = 4 (x₁ + 2x₂ - 2x3 = 1 (2) x₁ + x₂ + x3 = 2 (2x₁ + 2x₂ + x3 = 3 2x₁ = x₂ + x3 = -3 (4) 2x₁ + 2x₂ + 2x3 = 4 (-x₁ - x₂ + 2x3 = 1 a) For system (3), find Tj, TĠ, p(Tj), and p(TG). Does the Jacobi iterative method converge for method converge for system (4)? Does Gauss-Seidel iterative method converge for system (4)?
Use the same notations as on Page 6 of the lecture notes: A is the coefficient matrix for each linear system, D is the diagonal matrix with diagonal value ai, and D-L is the lower triangular matrix of A. For the following linear systems 3x₁ + x₂ + x3 = 0 x₁ + 3x₂x3 = 4 (3x₁ + x₂5x3 = -6 (1) - (x₁ + 3x₂ − 3x3 = 2 (3) x₁ + x₂ + x3 = 2 (3x₁ + 3x₂ + x3 = 4 (x₁ + 2x₂ - 2x3 = 1 (2) x₁ + x₂ + x3 = 2 (2x₁ + 2x₂ + x3 = 3 2x₁ = x₂ + x3 = -3 (4) 2x₁ + 2x₂ + 2x3 = 4 (-x₁ - x₂ + 2x3 = 1 a) For system (3), find Tj, TĠ, p(Tj), and p(TG). Does the Jacobi iterative method converge for method converge for system (4)? Does Gauss-Seidel iterative method converge for system (4)?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 22EQ: Consider the matrix A=[2314]. Show that any of the three types of elementary row operations can be...
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Transcribed Image Text:Use the same notations as on Page 6 of the lecture notes: A is the coefficient matrix for each linear
system, D is the diagonal matrix with diagonal value ai, and D-L is the lower triangular matrix of
A. For the following linear systems
3x₁ + x₂ + x3 = 0
x₁ + 3x₂x3 = 4
(3x₁ + x₂ = 5x3 = -6
(1)
(x₁ + 3x₂ 3x3 = 2
(3) x₁ + x₂ + x3 = 2
(3x₁ + 3x₂ + x3 = 4
(x₁ + 2x₂ - 2x3 = 1
(2) x₁ + x₂ + x3 = 2
(2x₁ + 2x₂ + x3 = 3
(2x₁ - x₂ + x3 = -3
(4) 2x₁ + 2x₂ + 2x3 = 4
-x₁x₂ + 2x3 = 1
a)
For system (3), find T₁, TĠ, p(Tj), and p(TG). Does the Jacobi iterative method converge for
method converge for system (4)? Does Gauss-Seidel iterative method converge for system (4)?
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