Show that if A is a real number such that A > 0, then thesequence (Tk) defined by1AXk+1k >022xkconverges to A for all xo > VA/2.Let us consider the simple iteration (xk) defined byf(Ek)Ik+1 = *kf(Tk + f(xk)) – f (Xk – f(Xk))2f(xk)where y(xk) :=for k > 0 for a given xo E R.Assuming• f :R → R is three times differentiable, and• (xk) converges to a root x, of f s.t. f(x.) = 0, f'(x.) + 0, f"(x.) # 0,deduce the order of convergence for the sequence (x1) to x..

Given, A∈ℝ, A>0, xk is defined by, xk+1=12xk+A2xk ,k≥0....1x0≥A2 and xk≥0 ∀k≥0 Now,…

Answered: Show that if A is a real number such…

Show that if A is a real number such that A > 0, then the sequence (xk) defined by A 2xk 1 Xk+1 k >0 converges to vA for all xo > VA/2.

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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Question

Show that if A is a real number such that A > 0, then the
sequence (Tk) defined by
1
A
Xk+1
k >0
2
2xk
converges to A for all xo > VA/2.
Let us consider the simple iteration (xk) defined by
f(Ek)
Ik+1 = *k
f(Tk + f(xk)) – f (Xk – f(Xk))
2f(xk)
where y(xk) :=
for k > 0 for a given xo E R.
Assuming
• f :R → R is three times differentiable, and
• (xk) converges to a root x, of f s.t. f(x.) = 0, f'(x.) + 0, f"(x.) # 0,
deduce the order of convergence for the sequence (x1) to x..

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