Gokaraju Rangaraju Institute of Engineering & Technology
(Autonomous Institute under JNTU Hyderabad)
First Year B.Tech
Mathematics III: Numerical Methods: Problem Sheet 2 - Unit I/ II
(Solution of linear system of equations)
Solution of linear systems: Gauss elimination, Gauss elimination with partial pivoting, Gauss-Jordan method,
Jacobi and Gauss-Seidel iterative methods. Convergence of iterative methods (without proof).
1. Find the solution of the following system of equations by Gauss elimination method.
1 2 3  x1   8 
2 1 1  x1    5






(a) 2 3 4 x2  11 .
(b) 1 3 6  x2    1 .

   

   
1 5 1  x3   4 
 3 1 7  x3   6
2
1
(c) 
1

4
4 1 1  x   4 
1 1 1  y  0.5
     .
2 4 2  z   1 
   
2 3 5  w  0 
1
4 1
1 5
2
(d) 
2  3 3

3 1  1
1  x1  2.4
1  x2  0.7
     .
2  x3  3.5
   
5  x4  2.7
2. Find the solution of the system of equations given in Q.1(a), (b), (c) by Gauss elimination method
with partial pivoting.
3. Find the solution of the following system of equations using the Gauss-Jordan method.
2 1 1  x1    5
4  1 2  x1   1 






(a) 1 3 6 x2  1 .
(b) 1  5 1  x2    1 .

   

   
 3 1 7  x3   6
2 3 6  x3   9
 3 2 1  x   0 
(c) 1 4 2  y     5.

   
2 3 5  z   6
4. Find the inverse of the coefficient matrices using the Gauss-Jordan method and hence solve the
system of equations in Q.3.(a), (c).
5. Find the solution of the following system of equations correct to two decimal places using (i)
Jacobi iteration method, (ii) Gauss-Seidel iteration method.
 4  1 1   x1  4
(a)  1 4  1  x2   2.

   
 1  1 4   x3  4
15 3 0   x1   6 
(c)  1 10 1   x2    4.

   
 0 2 12  x3   5 
6 2 1  x1  5
(b) 1 6 1  x2   3.

   
2 4 7  x3  5
4 0 2   x1   4 
(d) 0 5 2   x2    3.

   
5 4 10  x3   2 
1
6. In Q. 5 (c), (d) set up the Jacobi method in matrix form, find the iteration matrices and the rate of
convergence in the problems, if the method converges.
7. In Q. 5 (a), (c), (d) set up the Gauss-Seidel method in matrix form, find the iteration matrices and
the rate of convergence in the problems, if the method converges.
UNIT II (Interpolation for uniform data)
Interpolation – I: Finite differences – Forward, backward and central differences. Relationships
between operators. Differences of a polynomial. Newton’s forward and backward difference
formulas. Gauss central difference formulas.
1. Prove the following using differences.
2 x 2 x (1  x)
. Interval is unity. (JNTU-08)

x!
( x  1)!
(b) [ x( x  1)( x  2)( x  3)]  4( x  1)( x  2)( x  3) . Interval is unity. (JNTU-08)
(a) 
(c) [1 / f ( x )] 
 f ( x )
.
f ( x ) f ( x  1)
(JNTU-09)
(d) 2 f i2  ( f i  f i 1 )f i .
(JNTU-06)
(e) ( f i g i )  f i g i  g i 1f i .
(f) ( f i / g i )  ( g i f i  f i g i ) /( g i g i 1 ).
2. If y x is the value of y at x for which the fifth differences are constant and y1  y 7  784 ,
y 2  y 6  686, y 3  y 5  1088 ,
find y 4 .
(JNTU-07)
3. Find the second difference of the polynomial x 4 12 x 3  42 x 2  30 x  9 with interval of
differencing h = 2.
4. Derive Newton-Gregory forward interpolation formula.
(JNTU-09S)
5. Find y(2.5) using Newton’s forward formula from
x 0 1 2
3
4
5
6
(JNTU-06)
y 0 1 16 81 256 625 1296
6. Given that sin( 45 0 )  0.7077 , sin( 50 0 )  0.766, sin( 55 0 )  0.8192 , sin( 60 0 )  0.866, find sin( 52 0 )
using Newton’s forward difference formula.
(JNTU-06)
7. Use Newton’s forward difference formula to find the polynomial satisfied by (0, 5), (1, 12), (2, 37),
(3, 86).
(JNTU-10)
8. Find y(1.6) using Newton’s forward formula from
x
1
1.4
1.8
2.2
(JNTU-06)
y 3.49 4.82
5.96 6.5
9. Find the polynomial that fits the data
x 0
1
2
3
4
y 5 12 37 86 165
Use Newton’s forward formula.
(JNTU-09S)
10. Find f(9) by Newton’s backward formula given that f (2)  94.8, f (5)  87.9, f (8)  81.3,
(JNTU-07S)
f (11)  75.1.
11. Find y(22) using Newton’s backward formula from
x
20
25
30
35
40
45
(JNTU-07S)
y
354 332 291 260 231 204
12. From the following table of half yearly premiums for policies at quinquennial ages, estimate the
premium for policies at the age 63.
Age x
45
50
55
60
65
(JNTU-09S)
Premium y 114.84 96.16 83.32 74.48 68.48
2
13. Find y(25) given that y 20  24, y 24  32, y 28  35, y 32  40, using Gauss forward difference
formula.
(JNTU-06S)
14. Find f(1.3) given x = 2, 1, 0, 1, 2, 3; f(x) = 6, 5, 2, 2, 3, 6, using Gauss forward difference
formula.
(JNTU-09S)
15. Find f(22) using Gauss forward formula from
x
20
25
30
35
40
45
f(x)
354 332 291 260 231 204
16. Using Gauss backward difference formula find y(8)
x
0
5
10
15
20
25
y
7
11
14
18
24 32
17. Using Gauss backward difference formula find y(5)
x
y
0
5
2
23
4
89
6
251
8
557
(JNTU-07)
(JNTU-07)
10
1055
3