Ingeniero

Numerical Methods Dr Dana Mackey (Dana.Mackey@dit.ie)

1 October 2012

Problem Sheet 1: The bisection method. The Newton-Raphson method

1. Let f (x) = 3x4 − 8x2 + 1. (a) Using signchanges, show that f (x) = 0 has four roots between -2 and 2. (b) Use the bisection method to evaluate one root of your choice. (c) Use Newton’s method to evaluate the same root as in (b). (d) How dothese two methods compare? Use an error tolerance of = 0.01. (Answer: −1.592226039, −0.3626057200, 0.3626057200, 1.592226039) √ 2. Approximate 3 13 to three decimal places by applying the bisectionmethod to the equation x3 − 13 = 0. (Answer: 2.351334688...) 3. It can be shown that the equation (The value is 4.261483697...)
3 2

x−6−

1 2

sin(2x) = 0 has a unique real root.

(a)Find an interval on which the root is guaranteed to exist. (b) Using the bisection method, approximate this root to within a tolerance of 10−4 . 4. Let f (x) = x3 − x2 + 3x − 1. (a) Show that f (x) =0 has at least one root between -1 and 1. (b) Use five iterations of Newton’s method to find an approximation for this root. (c) How many iterations of the bisection method would be needed in orderto produce the same accuracy as in part (ii)? Answer: 0.3611030805...

5. Consider the function f (x) = tan(πx) − x − 6. (a) Show that f (x) = 0 has a root between 0 and 1. (b) UseNewton-Raphson method to evaluate this root. Try the following initial approximations: 0.48, 0.4 and 0. (Use 6 exact digits and do not exceed 10 iterations each time.) Comment briefly on the results. (Theroot is 0.4510472588.) 6. Consider the nonlinear equation x4 − 18x2 + 45 = 0. (a) Show that the equation has a root in the interval (1, 2). (b) Use five iterations of Newton’s method to find anapproximation for this root. (c) How many iterations of the bisection method would be needed in order to produce the same accuracy as in part (ii)? (Hint: The exact root is √ 3 = 1.73205080.)