Exercise Problems
Q1
Solve the following set of equations by Gauss-Sidel iteration, start with
(x, y, z) = (-1, 0, 1). Perform only two iterations.
4x –y – 2z = 0
-2x – y + 5z =1
-x + 4y – z = 1
Q2
The Redlich_Kuong equation is
P =
RT
A (T )
−
V−b
V(V + b )
Measurements of P = 83.7, T = 486.9, and V = 12.0 have been made. It is known that A (T) = 0.0837, and
R = 1.98. Find the value of b that satisfies the equation using Newton’s method with an initial guess of 2.0.
Carry out single iteration.
Q3
The non-ideal gas law is given by
a 

 p + 2 (v − b ) = RT
v 

where constants a =3.6, b = 0.04 and universal gas constant R = 0.082 atm/mol.K. A chemical engineering project
requires you to accurately estimate the molar volume (v) when temperature T=300K and pressure p=10 atm.
a) Find molar volume (v) by Newton’s method. Start with initial guess of v =2.5. Stop after two iterations.
b)
Find v by x = g(x) method. Start iterations with v =2.5 and stop after 2 iterations.
c) Compare the results.
Q4
Solve the set of equations below by Guass—Sidel iteration method. Start with an initial guess of 0.2, 0.4, and
0.6 for x, y, and z, respectively and stop after the 2nd iteration.
20x - 2y – z = 10
x + y + 10z = 20
2x - 30y + z =6
Q5 For matrix B to be an inverse of matrix A, find x and y.

1



A = 1
2

1
 3


x 
2


1
1 
3
4

1
1 
4 5

1


30 
9 − 36




B = − 36 y
− 180




30 − 180 180 




Q6
Determine the roots of the following simultaneous non-linear equations, starting with initial guesses of x =
2, y =3. Carry out only single iteration.
x2 sin(y) + yx = 4
y cos(x) + x2 y = -4
Q7
Concentration of a chemical in a completely mixed reactor is given by
C(t) = Cinf (1-exp(-0.04t)) + Co exp(-0.02t)
where Cinf =10 (inflow concentration); Co = 4 (initial concentration). Compute the time required for C to
be 9.3 by using
(a) Secant Method, starting with t1 = 53 and t2 = 56. Carry out only single iteration.
(b) Newton’s method, starting with t1 = 54. Carry out only single iteration.
Q8
Solve the following differential equation
Ux = xU + x + U ,
U(0) = 1,
0<x<1
by Euler method, considering h = 0.5.
Q9
You are asked to find the root of f(x) = ex-3x2 = 0 by x = g(x) method, starting with an initial value of
x = 0.0. Before starting the iterations, do you think that you would have a convergence? Justify. If it is
convergent, find the root until satisfying the tolerance limit of 0.02.
Q10
Determine the roots of simultaneous non-linear equations. Start with initial guesses of x = 2 and y=3.4.
Stop iterations when the tolerance limit is equal to 0.1.
(x - 4)2 + (y - 4)2 = 4
x2 + y2 = 16
Q11
Concentration of chemical in a completely mixed reactor is given by
C = Cinf (1-exp(-0.04t)) + Co exp(-0.04t)
where Cinf =10 (inflow concentration); Co = 4 (initial concentration). Compute the time required for C to be
9.3 by using Bisection Method. Start with xL = 53 and xR=56. Stop calculations when the tolerance limit is
equal to 0.2.
Q12
Find U at x=0.3 by Modified Euler method. Consider h=0.25.
Uxx = UUx
U(0) = 1.0
Ux(0) = -1.0
Q13
Find the root of f(x) = e x- 3x2 = 0 by x = g(x) method. Start with an initial value of x = 0.0. Do the
iteration until satisfying the tolerance limit of 0.01.