UVU CS-3320 – Numerical Software Development -Assignment 2

Description

SECTION 1.1
Exercise 5) Consider the equation x ⁴ = x ³ + 10.

a) Find an interval [a, b] of length one inside which the equation has a solution

b) Starting with [a, b], how many steps of the Bisection Method are required to calculate the solution
within 10 ^-10? Answer with an integer.

Computer Problem 2) Use the Bisection Method to find the root to eight correct decimal places.
● MATLAB Code for Bisection Method (code from textbook):

function xc = bisect(f, a, b, tol)
if sign(f(a)) * sign(f(b)) >= 0
error(‘f(a)f(b) < 0 not satisfied!’) % Ceases execution
end

a) x ⁵ + x = 1

b) sin(x) = 6x + 5

c) ln(x) + x ² = 3

SECTION 1.2
Exercise 1) Find all fixed points of the following g(x).
a) x/3

b) x ² – 2x + 2

c) x ² – 4x + 2

Exercise 7) Use Theorem 1.6 to determine whether Fixed-Point Iteration of g(x) is locally convergent to
the given fixed point r.
Theorem 1.6 : Assume that g is continuously differentiable, that g(r) = r, and that s = | g’(r) | < 1.
Then Fixed-Point Iteration converges linearly with rate S to the fixed point r for initial guess
sufficiently close to r .
a)g(x) = (2x – 1) ^1/3, r = 1

b) g(x) =x³ +1 /2 , r=1

c) g(x) = sin(x) + x, r = 0

Computer Problem 1) Apply Fixed-Point Iteration to find the solution of each equation to eight correct
decimal places.

a) x ³ = 2x + 2

b) e ^x + x = 7

c) e ^x + sin(x) = 4