Suppose that the sequence {x} is generated by the fixed point iteration method în = g(xn-1), n ≥ 1. Further, suppose that the sequence convergence linearly to the fixed point p. Show that (a) g'(p) ~ (b) en Xn-In-1 xn-1-Xn-2' g'(p) g'(p)-1 |xn - xn-11.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Suppose that the sequence {n} is generated by the fixed point iteration method xn = g(xn-1),
n ≥ 1. Further, suppose that the sequence convergence linearly to the fixed point p. Show that
(a) g'(p) ~ _In-n-1
Xn-1 Xn-2
9
g' (p)
(b) |en| ~₁ |In — Xn-1|-
g' (p)-1
Transcribed Image Text:Suppose that the sequence {n} is generated by the fixed point iteration method xn = g(xn-1), n ≥ 1. Further, suppose that the sequence convergence linearly to the fixed point p. Show that (a) g'(p) ~ _In-n-1 Xn-1 Xn-2 9 g' (p) (b) |en| ~₁ |In — Xn-1|- g' (p)-1
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