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--- gibson:teaching:fall-2016:math753:hw2 [2016/09/19 09:18]
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 ====== 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.
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 **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.
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 //Challenges:// (you don't have to do these, but you should do some if you want an A)
-Add checks to your ''bisectsearch'' function that verify that the starting conditions for bisection search are met. The checks should print a helpful error message and exit the function while returning the most reasonable value for the root from available information.
+  * Add checks to your ''bisectsearch'' function that verify that the starting conditions for bisection search are met. The checks should print a helpful error message and exit the function while returning the most reasonable value for the root from available information.
-Think carefully about the stopping conditions for bisection search. What kinds of problems are possible for this algorithm when it's implemented in floating-point arthmetic? Add checks to your function that stop iteration and return the most reasonable value for the root. In some cases you might want to print an error or warning message.
+  * Think carefully about the stopping conditions for bisection search. What kinds of problems are possible for this algorithm when it's implemented in floating-point arthmetic? Add checks to your function that stop iteration and return the most reasonable value for the root. In some cases you might want to print an error or warning message.
-Add some diagnostic printing to your function that prints out $a,b$, and $f(c)$ at each iteration. Add another argument ''diagnostics'' to the function that turns the printing on and off. Make ''diagnostics'' default to ''false''.
+  * Add some diagnostic printing to your function that prints out $a,b$, and $f(c)$ at each iteration. Add another argument ''diagnostics'' to the function that turns the printing on and off. Make ''diagnostics'' default to ''false''.
-Write your ''bisectsearch'' function so that it operates equally well on any floating-point type: Float16, Float32, Float64, or BigFloat.
+  * Write your ''bisectsearch'' function so that it operates equally well on any floating-point type: Float16, Float32, Float64, or BigFloat.
-Instead of using ''if a; b; end'' statements for your checks, use the more Juliaesque ''a && b'' short-circuit conditional evaluation. Likewise, instead of using ''if a; b ; else c ; end'' statements, use the ''a ? b : c'' ternary operator. See [[http://docs.julialang.org/en/release-0.4/manual/control-flow/#man-conditional-evaluation]].
+  * Instead of using ''if a; b; end'' statements for your checks, use the more Juliaesque ''a && b'' short-circuit conditional evaluation. Likewise, instead of using ''if a; b ; else c ; end'' statements, use the ''a ? b : c'' ternary operator. See [[http://docs.julialang.org/en/release-0.4/manual/control-flow/#man-conditional-evaluation]].
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-**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?

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