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7. Let A be an invertible n xn matrix over R. Consider the system of linear equation Ax = b or Eaijt; = bi, i = 1, ..., n. j=1 Let A = C - R. This is called a splitting of the matrix A and R is the defect matrix of the splitting. Consider the iteration Cx(k+1) = Rx(*) + b, k = 0, 1,2,.... Let (3) -() 4 00 C = |0 4 0 0 0 4 () (€) 4 3 A = -1 4 b = 2 x(0) 4 2 The iteration converges if p(C-'R) < 1, where p(C-1R) denotes the spectral radius of C-1R. Show that p(C-1R) < 1. Perform the iteration.

7. Let A be an invertible n xn matrix over R. Consider the system of linear equation Ax = b or Eaijt; = bi, i = 1, ..., n. j=1 Let A = C - R. This is called a splitting of the matrix A and R is the defect matrix of the splitting. Consider the iteration Cx(k+1) = Rx(*) + b, k = 0, 1,2,.... Let (3) -() 4 00 C = |0 4 0 0 0 4 () (€) 4 3 A = -1 4 b = 2 x(0) 4 2 The iteration converges if p(C-'R) < 1, where p(C-1R) denotes the spectral radius of C-1R. Show that p(C-1R) < 1. Perform the iteration.

Advanced Engineering Mathematics
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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7. Let A be an invertible n x n matrix over R. Consider the system of linear equation Ax = b or 2aija; = bị, AijX'j i = 1,..., n. Let A = C - R. This is called a splitting of the matrix A and R is the defect matrix of the splitting. Consider the iteration Cx(k+1) = Rx(k) +b, k = 0, 1,2,.. . Let (3) :). 4 0 0 4 0 0 0 4 (3) 4 -1 -1 4 -2 A = -1 C = b = 2 *(0) 4 2 The iteration converges if p(C-!R) < 1, where p(C-1R) denotes the spectral radius of C-1R. Show that p(C-!R)< 1. Perform the iteration.
Transcribed Image Text:7. Let A be an invertible n x n matrix over R. Consider the system of linear equation Ax = b or 2aija; = bị, AijX'j i = 1,..., n. Let A = C - R. This is called a splitting of the matrix A and R is the defect matrix of the splitting. Consider the iteration Cx(k+1) = Rx(k) +b, k = 0, 1,2,.. . Let (3) :). 4 0 0 4 0 0 0 4 (3) 4 -1 -1 4 -2 A = -1 C = b = 2 *(0) 4 2 The iteration converges if p(C-!R) < 1, where p(C-1R) denotes the spectral radius of C-1R. Show that p(C-!R)< 1. Perform the iteration.
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