Lab 6
Finding roots - Newton-Raphson Method

A. Newton-Raphson root-finder
Write a code Newton.m that implements the Newton-Raphson method for finding a root of F(x) = 0.
It requires F(x) to be differentiable with computable derivative DF(x).
The user should provide: x0: initial guess, maxIT: max number of iterations, and TOL: tolerance for convergence.
Newton converges fast typically, so maxIT=20 usually suffices.
For convenience, you may set maxIT=20 and TOL = 1.e-14 in the code so only x0 needs to be provided as argument.
• The code should:   accept input   x0 ,
print out the iterates:   n   xn   Fn   (neatly, in columns, use format: %d %f %e ),
and, upon convergence, print out the value of the root, the residual (%e format), and how many iterations it took.
• The (formula for) F(x) and DF(x) should be coded in a subprogram:   function [Fn , DFn] = FCN( xn )

B. To do:
1. Debug the code on a simple problem with known solution, e.g. F(x) = x2 − 2.
2. Use your Newton code to find all three roots of the cubic: F(x) = x3 + x2 − 3x − 3
Try various x0, like: 2, 1, 0.5, 0.1, 0.0, −0.5, −1, etc.
Do you notice anything unexpected/interesting? what?
3. Also try some big x0:  100, 1000, −100, etc
Do you notice anything unexpected/interesting? what?
4. Try some other maxIT and TOL.
Q1: What are the three roots you found ? How many iterations did each one take ? with what x0 ?
Q2: Is Newton more efficient than Bisection? how did you decide?
Q3: What is the effect of changing the initial guess x0 ?
Q4: What is the effect of changing maxIT ?
Q5: What is the effect of changing the tolerance ?
6. Now find the real root of "Newton's cubic": x3 − 2x − 5 = 0
(comment out previous formulas in FCN and insert new one).
This is the only equation Newton ever bothered to solve with his method, in 1671 !

C. Prepare Lab6 for submission. In a plain text file Lab6.txt , insert:
1. Name, Date, Lab6
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2. Answers to the questions in B.5  (each labeled and separated by %------------------------%)
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3. What are the roots of "Newton's cubic" ? in how many iterations, with what x0 for each ?
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