1.  (Roots of unity revisited:  adapted from LM 3.1–24)  In Problem 2 of Homework 1, we calculated all five roots of x5 ` 1, but it required a number of lines of code. In this problem, we will do this more compactly for a more general xn ` 1. Recall that

´1 “ eπi  “ e3πi  “ e5πi  “ ¨ ¨ ¨ ,

but, if we take the nth root, we obtain n distinct values

eπi{n, e3πi{n, e5πi{n , . . . ,

due to the periodicity of the complex exponentials; think Euler.

(a) Write a script which, given a positive integer n, finds all n roots of xn ` 1 at once, using ONE statement. It must also print out all n of these roots neatly using either disp or fprintf in tabular form. A loop may be used for printing results, but is not allowed

in the calculation of the roots.

(b) Run the script with n “ 3, 5, 7, and 11.

Note.  Print out the contents of your script m-file using type.

2.  (Catastrophic cancellation; LM 9.3– 10) Consider the function

f pxq “ $’&    if x ‰ 0

Here we explore the catastrophic cancellation which occurs as x Ñ 0 since ex Ñ 1 as x Ñ 0.

(a)  Use the Taylor series expansion of ex  to prove that f is continuous at 0.

(b)  Now calculate f pxq numerically for x  “ 10´k  where k P Nr1, 20s in three slightly different ways:

i. Calculate f pxq as written.

f1 pxq “  ,    for x ‰ 0.

(You and I know that analytically f1 pxq ” f pxq for all nonzero x – but MATLAB doesn’t.)

iii. MATLAB has a function which analytically subtracts 1 from the exponential to avoid catastrophic cancellation before the result is calculated numerically.  So define the function f2 pxq to be the same as f pxq except that ex ´ 1 is replaced by expm1(x).

Tabulate the results using disp or fprintf. The table should have four columns with the first being x, the second using f pxq, the third using f1 pxq, and the fourth using f2 pxq, with all shown to full accuracy. Do it as efficiently as you can, without using a loop.

Note. Write your code for this part in a single code block. No external script needs to be written.

(c)  Comment on the results obtained in the previous part. Explain why certain methods work well while others do not.

3.  (Improved triangular substitutions; adapted from FNC 2.3.5)  If B P Rnˆp  has columns b1 , . . . , bp , then we can pose p linear systems at once by writing AX “ B , where X P Rnˆp whose jth column xj  solves Axj  “ bj  for j “ 1, . . . , p:

»                          fi      »                          fi

A —x1    x2     ¨ ¨ ¨   xp  ffi “  —b1    b2     ¨ ¨ ¨   bp  ffi .

“X                                              “B

(a) Modify backsub .m and forelim .m from lecture so that they solve the case where the second input is an n ˆ p matrix, for p ě 1. Include the programs at the end of your live script.

(b) If AX  “ I , then X  “ A´1 .  Use this fact to write a MATLAB function ltinverse that uses your modified forelim to compute the inverse of a lower triangular matrix. Include the program at the end of your live script.  Then test your function using the following matrices, that is, compare your numerical solutions against the given exact solutions.

4.  (Vectorizing mylu .m; FNC 2.4.7) Below is an instructional version of LU factorization code presented in lecture.