NUMERICAL COMPUTATIONQUESTION ONE
(15 MARKS)
a) The Newton-Raphson method is generally given as:
f ( xi )
xi +1 = xi −
f ( xi )
Give the Algorithm of finding the root of a function using this method.
(2 Marks)
b) Use Newton-Rahpson method to find the root of the function f ( x) = x 3 − 467 to the third iteration
to 5 decimal places, where f ( x) = 3 x 2 . Take initial guess to be xo = 6 .
(3 Marks)
c) Find a root of x4-x-10 = 0 using the fixed point iteration technique. Perform 5 iterations for the
start points x0=1.0, 2.0 and 4.0, all to 5 decimal places.
(5 Marks)
d) Consider the set of equations:
x1 + x2 = 4
x1 − 2×2 = 1
Use eight rounds of the Gauss-Seidel Iteration to solve for x1 and x2
QUESTION TWO
(15 MARKS)
a) Use Gaussian elimination technique to solve the following system of equations
i. x − 3y + z = 4
ii. 2x − 8y + 8z = −2
iii. −6x + 3y − 15z = 9
b) i) Given
(5 Marks)
x =
(4 Marks)
b−a
b
, show the formula for approximating the integral f (x )dx for n partitions
n
a
using the trapezoidal rule.
(2Marks)
ii) Approximate the integral
sin (x)dx using trapezoidal rule for n = 8 .
0
a) i) Use the equation below to define what is meant by Linearization of Differential Equations
(3 Marks)
a) ii) Linearize the following differential equation with an input value of u=16.
(6 Marks)
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