Consider the function f(x) = r* – 4r' + 6z – 4z + 1. (6.1) Determine the order of the root z = 1. (6.2) If the Newton-Raphson algorithm is used to approximate the root z = 1, what will the order of convergence be? Motivate your answer, referring to (6.1). (6.3) Assume that I, → 1 as n → 0, when z, is determined by the Newton-Raphson method. Confirm your answer in (6.2) by determining lim n e for the relevant value of r, without calculating iterations. Hints: e, = 1- I, and we can factorize f and f' as follows: f(z) = (x – 1)' and f'(x) = 4(x – 1)*.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Consider the function f(x) = a* – 4 + 6x² – 4x + 1.
(6.1) Determine the order of the root z = 1.
(6.2) If the Newton-Raphson algorithm is used to approximate the root z = 1, what will
the order of convergence be? Motivate your answer, referring to (6.1).
(6.3) Assume that a, → 1 as n → oc, when r, is determined by the Newton-Raphson
method. Confirm your answer in (6.2) by determining
En+1
lim
for the relevant value of r, without calculating iterations. Hints: e, = 1-In and we
can factorize f and f' as follows: f(x) = (x – 1)' and f'(x) = 4(x – 1)°.
Transcribed Image Text:Consider the function f(x) = a* – 4 + 6x² – 4x + 1. (6.1) Determine the order of the root z = 1. (6.2) If the Newton-Raphson algorithm is used to approximate the root z = 1, what will the order of convergence be? Motivate your answer, referring to (6.1). (6.3) Assume that a, → 1 as n → oc, when r, is determined by the Newton-Raphson method. Confirm your answer in (6.2) by determining En+1 lim for the relevant value of r, without calculating iterations. Hints: e, = 1-In and we can factorize f and f' as follows: f(x) = (x – 1)' and f'(x) = 4(x – 1)°.
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