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Computing Skills Assignment Help

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Question 1

Consider the following nonlinear equation:

f (x) = x2 − 10 cos(x) = 0,

where x is in radians. The equation has two solutions ±1.3793646.

(a)   Write a complete Fortran program (documentation is not necessary) that:

(i)    finds a solution using Secant method with x0 = 100 and x1 = 50;

(ii)    stops when |xN − xN−1| ≤ 10−8 or the number of iterations exceeds 1000;

(iii)    prints the result using the following format:

T h e  s o l u t i o n i s  * * . * * * * * * *

(7 marks)

(b)  Run your program and show the numerical solution to the considered nonlinear

equation.                                                                                                                                                  (2 marks)

Question 2

Let f (x) = x − 2X  and g(x) = 2X.

(a)   Show that function g has a unique fixed point on [ 13 , 1].

(b)    Write a program (documentation is not necessary) that:

(i)    uses fixed point iteration to find, to an accuracy of 10−4 f (x) = 0 that lies in [ 13 , 1];

(ii)    prints the result using the following format:

T h e  s o l u t i o n i s  * . * * * *

(6 marks)

(c)  Run your program and show the numerical solution to the considered nonlinear

equation.                                                                                                                                                  (2 marks)

(d)  Estimate the number of iterations required to achieve 10−4 accuracy.

(6 marka)

Question 3                                                                                                 14 Marks

2 MARKS

Consider the following nonlinear function

f (x) = 110Φ(d(x)) − 120 exp(−0.02)Φ(d(x) − x),               x > 0

where

d(x) =

ln

110

+

0.02 +

X2

S1

120

2

and

Φ(s) =

exp(

T2

)dt

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