Question 2 Consider the following nonlinear equation: f (x) = x3 — x2 — 10 cos(x) = 0. (a) Let x0 = 100 and x1 = 50. Calculate x3 using the following Secant scheme: Xn Xn-1 Xn+1 = Xn f (x.) f(xn) — f...

Question 2 Consider the following nonlinear equation: f (x) = x3 — x2 — 10 cos(x) = 0. (a) Let x0 = 100 and x1 = 50. Calculate x3 using the following Secant scheme:
Xn Xn-1 Xn+1 = Xn f (x.) f(xn) — f (xn_i)'
n= 1, 2, .
(4 marks)
(b) Write a complete Fortran program that: (i) finds a solution using the iteration scheme given in (a) with xo = 100 and xi = 50; (ii) stops when Ix, — xn-ii

(6 marks) (c) Run your program and show the numerical solution to the considered nonlinear equation. (2 marks)
Question 3 Consider the following linear system of equations:
{xi + 2x2 + 3x3 = 14 2x1 + 5x2 + 2x3 = 18 3x1 + x2 + 5x3 = 20
Use LU (Doolittle) factorisation method to solve the above system.
Question 4 Suppose we are given the following values of a function:
sill 0.32 sin 0.34 sin 0.36
0.314567 0.333487 0.352274
(a) Construct a Lagrange polynomial of degree one to approximate sin 0.3367, and show the approximation error. (6 marks) (b) Construct a Lagrange polynomial of degree two to approximate sin 0.3367, and show the approximation error. (6 marks)