School of Mathematics, Thapar University, Patiala
UMA007 : Numerical Analysis
Assignment 2
Roots of Non-linear Equations-A
1. Use the bisection method to find solutions accurate to within 103 for the following problems.
a. x 2x = 0 for 0 x 1 b. ex x2 + 3x 2 = 0 for 0 x 1
c. x + 1 2 sin(x) = 0 for 0 x 0.5 and 0.5 x 1.
2. Find an approximation to 3
25 correct to within 103 using the bisection algorithm.
3. Find a bound for the number of iterations needed to achieve an approximation by bisection method with
accuracy 102 to the solution of x3 x 1 = 0 lying in the interval [1, 2]. Find an approximation to the
root with this degree of accuracy.
4. Sketch the graphs of y = x and y = tan x. Use the bisection method to find an approximation to within
103 to the first positive value of x with x = tan x.
5. For each of the following equations, use the given interval or determine an interval [a, b] on which fixed-point
iteration will converge. Estimate the number of iterations necessary to obtain approximations accurate to
within 102 , and perform the calculations.
a. 2 + sin x x = 0 use [2, 3] b. x3 2x 5 = 0 use [2, 3] c. 3x2 ex = 0 d. x cos x = 0.
6. Use the fixed-point iteration method to find smallest and second smallest positive roots of the equation
tan x = 4x, correct to 4 decimal places.
7. Show that g(x) = + 0.5 sin(x/2) has a unique fixed point on [0, 2]. Use fixed-point iteration to find an
approximation to the fixed point that is accurate to within 102 . Also estimate the number of iterations
required to achieve 102 accuracy, and compare this theoretical estimate to the number actually needed.
8. Find all the zeros of f (x) = x2 + 10 cos x by using the fixed-point iteration method for an appropriate
iteration function g. Find the zeros accurate to within 102 .
9. The iterates xn+1 = 2 (1 + c)xn + cx3n will converge to = 1 for some values of constant c (provided that
x0 is sufficiently close to ). Find the values of c for which convergence occurs? For what values of c, if any,
convergence is quadratic.
10. What is the order of convergence of the iteration
xn (x2n + 3a)
xn+1 =
3x2n + a
as it converges to the fixed point = a?
11. Let A be a given positive constant and g(x) = 2x Ax2 .
a. Show that if fixed-point iteration converges to a nonzero limit, then the limit is = 1/A, so the inverse
of a number can be found using only multiplications and subtractions.
b. Find an interval about 1/A for which fixed-point iteration converges, provided x0 is in that interval.
12. Consider the root-finding problem f (x) = 0 with root , with f 0 (x) 6= 0. Convert it to the fixed-point
problem
x = x + cf (x) = g(x)
with c a nonzero constant. How should c be chosen to ensure rapid convergence of
xn+1 = xn + cf (xn )
to (provided that x0 is chosen sufficiently close to )? Apply your way of choosing c to the root-finding
problem x3 5 = 0.
CONTINUED
2
13. A particle starts at rest on a smooth inclined plane whose angle is changing at a constant rate
d
= < 0.
dt
At the end of t seconds, the position of the object is given by
e et
t
g
x(t) = 2 sin t .
2 2
Suppose the particle has moved 1.7 ft in 1 s. Find, to within 105 , the rate at which changes. Assume
that g = 32.17 ft/s2 .
