6. 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+- In+1 = where ro is the largest integer for which a? < N. Show that if by change a? the sequence is constant, i.e., r, = simply Newton's method applied to N, VN for all n = 1,2, ... Show that the sequence is f(x) = x² – N How many iterations does it take to calculate V411 to three decimal places using To = 20?

6. 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+- In+1 = where ro is the largest integer for which a? < N. Show that if by change a? the sequence is constant, i.e., r, = simply Newton's method applied to N, VN for all n = 1,2, ... Show that the sequence is f(x) = x² – N How many iterations does it take to calculate V411 to three decimal places using To = 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

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6. 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+-
In+1 =
where ro is the largest integer for which a? < N. Show that if by change a?
the sequence is constant, i.e., r, =
simply Newton's method applied to
N,
VN for all n = 1,2, ... Show that the sequence is
f(x) = x² – N
How many iterations does it take to calculate V411 to three decimal places using
To = 20?

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