2/14/2020 Homework 4
Homework 4 Problem 1
(Analytical) In Homework 3 Problem 4 you were asked to use Newton’s Method to get an iterative method for finding the square root of a number . Apply the tools we have from the theory of fixed point iterations directly to this formula to show that the method is exactly quadratically convergent. Present your work in a Markdown cell.
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Problem 2
(Julia) Use Newton’s Method to find the two distinct roots of , the double root at and the simple root at , by choosing appropriate initial values (other than 0 or -1). Print out your iterates to observe the convergence. For the double root, modify the problem so that you get quadratic convergence. You may use a for-loop with 12 iterations to observe the convergence.
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Problem 3
(Julia) Use the
1. 2. 3.
method in the
package in Julia to find the roots of
Plot each curve to make sure your results are reasonable. If the option. Give some thought to the accuracy.
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Problem 4
(Analytical) Find by hand the quadratic interpolating polynomial to results in a Markdown cell.
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through
method fails, try another
. Typeset your
https://notebooks.hpc.uiowa.edu/11837075-648397/user/pudchen/nbconvert/html/Homework 4.ipynb?download=false
1/1
0 = 𝑥
3,2,1 = 𝑥 )𝑥(nl
𝚗𝚘𝚒𝚝𝚒𝚜𝚘𝙿 𝚎𝚜𝚕𝚊𝙵
0𝑥 2𝑥 + 3𝑥 = )𝑥(𝑓
1− = 𝑥
𝚕𝚓.𝚜𝚝𝚘𝚘𝚁
𝚗𝚘𝚒𝚝𝚒𝚜𝚘𝙿 𝚎𝚜𝚕𝚊𝙵
𝑎
2)1 − 𝑥( = )𝑥(𝑓 )𝑥(nl 2𝑥 = )𝑥(𝑓 )𝑥(soc 𝑥 = )𝑥(𝑓