Numerical Calculus Assignment #4

Instruction: Write your answers clearly and show all relevant work including details.
You may use a calculator and 5- digit chopping unless specified otherwise.
Note the typo in Question 3(c) corrected on 2/16/2009.
Section 2.3 :
1. (3 pts.) Let f(x) = e^2x – x² + cos x. Use Newton’s method to approximate a root to the
equation f (x) = 0 after 3 iterations, by using the initial approximation = –1.0.
2. (4 pts.) Define the same function f(x) = e^2x – x² + cos x as in Question 1. Find an
approximation to a root for equation f(x) = 0 with = –1.0 and = 0.0 after 3
iterations, based on (a) the Secant method; and (b) the method of false position.
Section 2.4:
3. (7 pts.) Define f(x) = x² ln (1+x) for x > −1.
(a) Show that x = 0 is a zero of multiplicity 3 for the function f(x). (Hint: use
Theorem 2.11 and prove that f(0) = f’(0) = f’’(0) = 0.)
(b) Use Algorithm 2.3 (Newton’s method)
to compute a root for the
equation f(x) = x² ln (1+x) = 0 using an initial approximation = 0.5 and an
accuracy to within 10^-5.
(c) Redo Part (b) by computing an approximation to the root for the equation f(x) =
x² ln (1+x) = 0 using an initial approximation = 0.5 based on the modified
Newton-Raphson method described in Equation 2.11 of the text. Compare this
algorithm ’s speed of convergence to the speed of Newton’s method of Part (a).
4. (4 pts.) Suppose the sequence converges to p as n → ∞ and for all
n ≥ 1. (a) Show that ; and (b) in general, find a
relationship between for any n ≥ 2.
Section 2.5:
5. (4 pts.) Let f(x) = x² ln (1+x) = 0 as in Question 3. Apply Steffensen’s technique to
Newton’s method and compute approximation as done in Example 2 of this
section (p. 85) of the text.
Section 2.6:
6. (8 pts.) Let f(x) = x³ – 9x² – 9x – 10.
(a) Apply Newton’s method and Horner’s method using synthetic division to
approximate the real root of f(x) = 0 after 3 iterations, using initial approximation
= 8.0. Show your work in each of the steps .
(b) Use Algorithm 2.8 (Muller’s method) 
to approximate the two complex
roots for f (x) = 0 within 10^-5, using the initial approximations.0, 1, and 2. In
addition , find the exact values of the two complex roots. (using the quadratic
formula ).