Chapter 2.5, Problem 11ES

Use Steffensen’s method to approximate the solutions of the following equations to within 10⁻⁵.

1. a. x = (2 − e^x + x2)/3, where g is the function in Exercise 13(a) of Section 2.2.
2. b. x = 0.5(sin x + cos x), where g is the function in Exercise 13(f) of Section 2.2.
3. c. x = (e^x/3)^1/2, where g is the function in Exercise 13(c) of Section 2.2.
4. d. x = 5^−x, where g is the function in Exercise 13(d) of Section 2.2.

13. For each of the following equations, determine an interval [a, b] on which fixed-point iteration will converge. Estimate the number of iterations necessary to obtain approximations accurate to within 10⁻⁵ and perform the calculations.

1. a. $x=\frac{2-e^x+x^2}{3}$
2. b. $x=\frac{5}{x^2}+2$
3. c. x = (e^x/3)^1/2
4. d. x = 5^−x
5. e. x = 6^−x
6. f. x = 0.5(sin x + cos x)