Differences

This shows you the differences between two versions of the page.

--- gibson:teaching:fall-2016:math753:hw2 [2016/09/19 12:05]
gibson
+++ gibson:teaching:fall-2016:math753:hw2 [2016/09/20 09:34] (current)
gibson
@@ Line 1: / Line 1: @@
 ====== Math 753/853 HW2 ======
+Here is the {{:gibson:teaching:fall-2016:math753:math753-hw2.ipynb|Julia Notebook for HW2}}. Please download the notebook, fill it in, save your completed work as a notebook file with name ''math753-hw2-lastname.ipynb'', and submit it either through Canvas or via email.
+For your convenience reading online, the questions for HW2 are listed below.
 -------
 **Problem 1.** Write a Julia function ''bisectsearch'' that take a function $f(x)$ and an interval $a,b$ such that $f(a) f(b) < 0$, returns a root $r$ of $f$ within that interval. Find the root to as many digits accuracy as possible.
@@ Line 55: / Line 57: @@
 ------
-**Problem 5.** Modify your bisection search and Newton search codes so that, along with the root $r$ they return a vector of errors $e_n = | x_n - r|$ for $n=1,2,...$. Then solve $\cos^2 x + 6 = x$ using both bisection and Newton method, and make a plot comparing $e_n$ versus $n$ for the two methods. Put both the bisection errors and the Newton method errors on the same plot. Use blue circles for bisection errors and red squares for Newton method. Make the vertical $e_n$ axis logarithmic.
+**Problem 5.** Modify your bisection-search and Newton-method functions so that, along with the root $r$ they return a vector of errors $e_n = | x_n - r|$ for $n=1,2,...$. Then solve $\cos^2 x + 6 = x$ using both bisection and Newton method, and make a plot comparing $e_n$ versus $n$ for the two methods. Put both the bisection errors and the Newton method errors on the same plot. Use blue circles for bisection errors and red squares for Newton method. Make the vertical $e_n$ axis logarithmic.
 We have theoretical estimates of the convergence rates of bisection and Newton method. Do your error plots fit this theory, or not? Explain your answer in reference to the error plots.
+------
+** Bonus problem 6.** Use your Newton-method function to find the root of $f(x) = x^2$ starting with initial guess $x_0=1$, and plot the error $e_n$ versus $n$ as in problem. Does the Newton method converegence toward the true solution at the expected rate? Why or why not?
+------
+** Bonus problem 7.** Consider $f(x) = (1-3/(4x))^{1/3}$. What is the root $r$ of this function? What happens when you apply your Newton-method algorithm to this function with a starting guess $x_0 = 1$? Why?

channelflow.org

User Tools

Site Tools

Differences

Page Tools