Math 170B HW #2 Partial Solutions
=================================

1a. p_3 = 2.828427125.
  
1b. |p_3 - sqrt(8)| = 2.5 x 10^(-10).

1c. p_4 = 2.828431373. (2.82843137254902 using MATLAB).

1d. |p_4 - sqrt(8)| = 4.248 x 10^(-6).


2a. Using p_0 = 2 and p_1 = 3 and MATLAB, 
    p_2 = 2.80000000000000,
    p_3 = 2.82758620689655,
    p_4 =  2.82840236686391.

2b. The absolute error is -2.475788228029074e-005.

2c. No. Although we know the root is bracketed, we are never guaranteed 
    any specific reduction in size of |p_n - p|. Consider f(x) = -1 + 1/x.


3a. d(x) = (x-3)^2 + (x^3 + 1)^2.

3b. p_2 = 1.24780486488897 (due to MATLAB).


4a. Since f'(x) = a, just compute p_0 - f(p_0)/f'(p_0) using f(p_0) = a*p_0 - b 
    and f'(p_0) = a. The answer should be -b/a which is exactly what you get in 
    finding the root of a linear function.

4b. For any p_0 and p_1 just plug in to the Secant method formula and you will 
    get p_2 = -b/a.


5. Again, just plug into the Secant method formula. You answers should look similar 
   to those of 4a and 4b. Why?


6a. p_1 = -p_0,    p_2 = p_0,    p_3 = -p_0.

6b. Examine what would happen if we chose p_0 < 0 (rather than > 0 in part a). All in all, 
    Newton's method will not converge for any nonzero initial guess.

    This does not violate the theorem on Newton's method convergence because our function 
    f does not even satisfy the hypotheses of the theorem; hence, it can not violate the 
    theorem. Which hypothesis or hypotheses are not satisfied? (Hint: Take a derivative of 
    f and note the first hypothesis given in Thm 2.5 on Pg. 66.)


7. For all of these, g(x) = x - f(x)/f'(x).


8. We convert to a fixed point problem. (WHY???).
   Show that g(x) := x - f(x)/f'(x) satisfies the hypotheses of the fixed point theorem. 
   This means you must show three things:
 
           (i) g is continuous on [3/4, 5/4].
          (ii) g(x) is in [3/4, 5/4] for all x in [3/4, 5/4].
         (iii) g' exists on (3/4, 5/4) and |g'(x)| <= k < 1 for all x in (3/4, 5/4).
  
   Showing these three items is just calculus so I leave it to you.