2016 Numerical Analysis Midterm Exam
When is the best time to have the midterm exam?
 Tuesday Oct 25th 16:30 -- 18:30 [see note]   0 Thursday Oct 27th 16:30 -- 18:30 [see note]   6 Tuesday Nov 1st 16:30 -- 18:30 [see note]   4 Thursday Nov 3rd 16:30 -- 18:30   12 Friday Nov 4th 14:00 -- 16:00   1 Friday Nov 4th 16:30 -- 18:30   0 Friday Nov 4th 18:00 -- 20:00   1
Poll ended at 2:03 am on Tuesday October 18th 2016. Total votes: 24. Total voters: 17.
Thursday November 3rd 2016
16:30 — 18:30

NO NOTES OR BOOKS; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
 10.25.16
 Question #1
Consider the function $f=\sin(x)$ with $x$ in radians. Find the root $f=0$ for the initial interval $\frac{1}{2}\pi \le x \le \frac{3}{2} \pi$ using the bisection method. Do so in two different ways:
 (a) By hand, with enough iterations to yield a root accurate to at least 4 significant digits. How many iterations are needed to find a root accurate to at least 4 significant digits? (b) With a C code that starts as follows: #include #include #include #include double f(double x){  double ret;  ret=sin(x);  return(ret);}int main(void){
 Question #2
Consider the system of equations $AX=B$ with $A$ equal to: $$A=\left[ \begin{array}{ccc} -2 & 2 & -1 \\ 6 & -6 & 7 \\ 3 & -8 & 4 \end{array} \right]$$ and $B$ equal to: $$B=\left[ \begin{array}{c} -1 \\ -7 \\ -6 \end{array} \right]$$ Find $X$ using partial pivoting (by hand).
 Question #3
Consider a real number stored with 5 bytes. Bit #1 is reserved for the sign, while bits #2 to #10 are reserved for the biased exponent, and bits #11 to #40 are related to the significand. Do the following:
 (a) Find the minimum and maximum possible exponent $p$ (b) Find the smallest possible positive number (c) Find the largest possible number (d) Find the smallest possible positive subnormal number
 Question #4
Consider the following non-linear system of equations: $$x_2 x_1 x_3 = 5$$ $$\frac{1}{2}x_1^2 + \frac{1}{2}x_2^2 = 100$$ $$x_2 + x_3 =0$$ Do the following:
 (a) Find $x_1$, $x_2$, and $x_3$ using Newton's method using the initial conditions $x_1=0$, $x_2=1$, and $x_3=1$. Do so by hand and solve the first 2 iterations only. (b) Using the results obtained in (a) estimate the order of convergence of the method.
 Answers
 1. 13. 2. 2, $\frac{1}{4}$, $-\frac{5}{2}$. 3. 255, -254, $3.454 \times 10^{-77}$, $1.15792089 \times 10^{77}$, $3.217\times 10^{-86}$. 4. 4.91244, 51.125, -51.125, 0.1, -1.1, -1.18.
 12.25.16
 $\pi$