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MATH 5006 Numerical Methods

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MATH 5006 Numerical Methods

 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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