8) Use Newton's method to derive the ancient and long standing divide and average method for finding square roots by hand. It is also known as the Babylonian method. √ can be approximated by iterating *;+ 1 = 12 ( x ₁ + 1 = 1 ) using an initial guess xo. Because √ is the positive solution to the equation x² - r = 0, derive the iterating formula above by simplifying the Newton's method formula using ƒ(x) = x² −r. Note that since r is a constant, d/dx(r) = 0.
8) Use Newton's method to derive the ancient and long standing divide and average method for finding square roots by hand. It is also known as the Babylonian method. √ can be approximated by iterating *;+ 1 = 12 ( x ₁ + 1 = 1 ) using an initial guess xo. Because √ is the positive solution to the equation x² - r = 0, derive the iterating formula above by simplifying the Newton's method formula using ƒ(x) = x² −r. Note that since r is a constant, d/dx(r) = 0.
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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Transcribed Image Text:8) Use Newton's method to derive the ancient and long standing divide and average method for finding
square roots by hand. It is also known as the Babylonian method.
√r can be approximated by iterating X¡ +1²
(x₁+1=1)
using an initial guess x₁.
Because √ is the positive solution to the equation x² - r = 0, derive the iterating formula above by
simplifying the Newton's method formula using f(x)=x²-r. Note that since r is a constant, d/dx(r) = 0.
9) Use the divide and average method derived in Exercise 8 and modify your code from Exercise 5 (three
times) to calculate the square roots of the following numbers to within 8 decimal places
(a) r = 2
(b) r = 3
(c) r = 5
You do not need to include all of the code for these problems in the HW you turn in, as most of it will
just be the same as number 5. However, write down the first 5 to 8 lines (depending on the language
used) that define the function used, the initial guess, and the tolerance, along with the roots found in the
end.
Expert Solution
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Newton's method uses the formula: to perform the iterations and to approximate the root of the equation: using the initial approximation of the root: .
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