An ancient technique for extracting the square root of an integer N > 1 going back to the Babylonions involves using the iteration process N 1 Xn +- 2 In+1 = In where ro is the largest integer for which x2 < N. Show that if by change x² the sequence is constant, i.e., x„ = /N for all n = simply Newton's method applied to N, 1, 2, ... Show that the sequence is %3D f(x) = x² – N 1 How many iterations does it take to calculate v411 to three decimal places using xo = 20?

An ancient technique for extracting the square root of an integer N > 1 going back to the Babylonions involves using the iteration process N 1 Xn +- 2 In+1 = In where ro is the largest integer for which x2 < N. Show that if by change x² the sequence is constant, i.e., x„ = /N for all n = simply Newton's method applied to N, 1, 2, ... Show that the sequence is %3D f(x) = x² – N 1 How many iterations does it take to calculate v411 to three decimal places using xo = 20?

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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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

An ancient technique for extracting the square root of an integer N > 1 going back to
the Babylonions involves using the iteration
process
N
1
Xn +-
2
In+1 =
In
where ro is the largest integer for which x2 < N. Show that if by change x²
the sequence is constant, i.e., x„ = /N for all n =
simply Newton's method applied to
N,
1, 2, ... Show that the sequence is
%3D
f(x) = x² – N
1
How many iterations does it take to calculate v411 to three decimal places using
xo = 20?

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