Numerical Analysis Questions and Answers – Bessel’s Formula

This set of Numerical Analysis Multiple Choice Questions & Answers (MCQs) focuses on “Bessel’s Formula”.

1. What will be the solution of the table using Bessel’s formula?

x f(x)
22 2854
24 3162
26 3544
28 3992

x = 25

a) 3344.25
b) 3344
c) 3444.50
d) 3374.75
View Answer

Answer: a
Explanation:

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Bessel's method to find solution

h=24-22=2
Taking x0=24 then p=(x-x0)/h=(x-24)/2

The difference table is

x p=(x-24)/2 y Δy Δ2y Δ3y
22 -1 2854
308
24 0 3162 74
382 -8
26 1 3544 66
448
28 2 3992

x=25
p=(x-x0)/h=(25-24)/2=0.5
y0=3162, Δy0=382, Δ2y-1 = 74, Δ3y-1 = -8

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Applying Bessel’s formula,

y0.5 =(3162+3544)/2+(0.5-(1/2))⋅(382)+0.5(0.5-1)/2⋅(74+66)/2+(0.5-1/2)0.5(0.5-1)/6⋅(-8)
y0.5 =3353+0-8.75+0
y0.5 =3344.25

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Solution of Bessel's interpolation is y(25)=3344.25.

2. What will be the solution of the table using Bessel’s formula?

x f(x)
20 24
24 32
26 35
22 40

x = 25

a) 0.619
b) 0.611
c) 0.614
d) Solution not possible
View Answer

Answer: d
Explanation: Given,

The value of table for x and y

x 20 24 26 22
y 24 32 35 40

Here difference of x is not same. So, solution using Bessel's method is not possible.

3. What will be the solution of the equation 2x3 +1 using Bessel's formula where x1 = 2 and x2 = 4 and x = 2.1 and step value (h) = 0.25?
a) 11.522
b) 19.522
c) 14.522
d) 29.522
View Answer

Answer: b
Explanation: Given,
Equation is f(x)=2x3 + 1

Bessel's method to find solution

h=2.25-2=0.25
Taking x0=3 then p=(x-x0)/h=(x-3)/0.25
The difference table is

x p=(x-3)/0.25 y Δy Δ2y Δ3y Δ4y
2 -4 17
6.7812
2.25 -3 23.7812 1.6875
8.4688 0.1875
2.5 -2 32.25 1.875 0
10.3438 0.1875
2.75 -1 42.5938 2.0625 0
12.4062 0.1875
3 0 55 2.25 0
14.6562 0.1875
3.25 1 69.6562 2.4375 0
17.0938 0.1875
3.5 2 86.75 2.625 0
19.7188 0.1875
3.75 3 106.4688 2.8125
22.5312
4 4 129

x=2.1
p=(x-x0)/h=(2.1-3)/0.25=-3.6
y0=55, Δy0=14.6562, Δ2y-1 = 2.25, Δ3y-1 = 0.1875, Δ4y-2 = 0
Applying Bessel’s formula,
y-3.6 = (55+69.6562)/2+(-3.6-(1/2))⋅(14.6562)+-3.6(-3.6-1)/2⋅(2.25+2.4375)/2+(-3.6-(1/2))(-3.6)(-3.6-
1)/6⋅(0.1875)+(-3.6+1)(-3.6)(-3.6-1)(-3.6-2)/24⋅(0)/2
y-3.6 = 62.3281-60.090625+19.40625-2.12175+0
y-3.6 = 19.522

Solution of Bessel's interpolation is y(2.1)=19.522.

4. What will be the solution of the equation 3x+1 using Bessel's formula where x1 = 2 and x2 = 4 and x = 2.1 and step value (h) = 0.25?
a) 4.6
b) 2.5
c) 7.3
d) 1.9
View Answer

Answer: c
Explanation: Given,
Equation is f(x)=3x+1

Bessel's method to find solution

h=2.25-2=0.25
Taking x0=3 then p=(x-x0)/h=(x-3)/0.25
The difference table is

x p=(x-3)/0.25 y Δy Δ2y
2 -4 7
0.75
2.25 -3 7.75 0
0.75
2.5 -2 8.5 0
0.75
2.75 -1 9.25 0
0.75
3 0 10 0
0.75
3.25 1 10.75 0
0.75
3.5 2 11.5 0
0.75
3.75 3 12.25 0
0.75
4 4 13

 x=2.1
p=(x-x0)/h=(2.1-3)/0.25=-3.6
y0=10, Δy0=0.75, Δ2y-1 = 0

Bessel's formula is

y-3.6 =(10+10.75)/2+(-3.6-(1/2))⋅(0.75)+(-3.6)(-3.6-1)/2⋅(0)/2
y-3.6 =10.375-3.075+0
y-3.6 =7.3

Solution of Bessel's interpolation is y(2.1)=7.3.

5. What will be the solution of the equation x using Bessel's formula where x1 = 2 and x2 = 4 and
x = 2.1 and step value (h) = 0.25?
a) 1
b) 2.1
c) 2
d) 1.2
View Answer

Answer: b

Explanation: Given,
Equation is f(x)=x

Bessel's method to find solution

h=2.25-2=0.25
Taking x0=3 then p=(x-x0)/h=(x-3)/0.25
The difference table is

x p=(x-3)/0.25 y Δy Δ2y
2 -4 2
0.25
2.25 -3 2.25 0
0.25
2.5 -2 2.5 0
0.25
2.75 -1 2.75 0
0.25
3 0 3 0
0.25
3.25 1 3.25 0
0.25
3.5 2 3.5 0
0.25
3.75 3 3.75 0
0.25
4 4 4

x=2.1
p=(x-x0)/h=(2.1-3)/0.25=-3.6
y0=3, Δy0=0.25, Δ2y-1 =0

Bessel's formula is

y-3.6 = (3+3.25)/2+(-3.6-(1/2))⋅(0.25)+(-3.6)(-3.6-1)/2⋅(0)/2
y-3.6 = 3.125-1.025+0
y-3.6 = 2.1
Solution of Bessel's interpolation is y(2.1)=2.1.

6. What will be the solution of the equation log(x) using Bessel's formula where x1 = 2 and x2 = 4 and x = 2.1 and step value (h) = 0.25?
a) 1.3799
b) 1.3791
c) 1.3794
d) -0.5045
View Answer

Answer: d
Explanation: Given,
Equation is f(x)=log(x)

Bessel's method to find solution

h=2.25-2=0.25
Taking x0=3 then p=(x-x0)/h=(x-3)/0.25
The difference table is

x p=(x-3)/0.25 y Δy Δ2y Δ3y Δ4y Δ5y
2 -4 0.301
0.0512
2.25 -3 0.3522 -0.0054
0.0458 0.001
2.5 -2 0.3979 -0.0044 -0.0003
0.0414 0.0008 0.0001
2.75 -1 0.4393 -0.0036 -0.0002
0.0378 0.0006 0.0001
3 0 0.4771 -0.003 -0.0001
0.0348 0.0004 0
3.25 1 0.5119 -0.0026 0
0.0322 0.0004 0
3.5 2 0.5441 -0.0022 0
0.03 0.0003
3.75 3 0.574 -0.0019
0.028
4 4 0.6021

p=(x-x0)/h=-3/0.25=-12
y0=0.4771, Δy0=0.0348, Δ2y-1 = -0.003, Δ3y-1 = 0.0004, Δ4y-2 = -0.0001, Δ5y-2 = 0

Bessel's formula is

y-12 = (0.4771+0.5119)/2+(-12-(1/2))⋅(0.0348)+(-12(-12-1))/2⋅(-0.003-0.0026)/2+(-12-(1/2))-12(-12-
1)/6⋅(0.0004)+(-12+1)-12(-12-1)(-12-2)/24⋅(-0.0001)/2+(-12-(1/2))(-12+1)-12(-12-1)(-12-2)/120⋅(0)
y-12 = 0.4945-0.4345263282-0.2185512232-0.1459353163-0.1109642258-0.089011542
y-12 = -0.5045

Solution of Bessel's interpolation is y()=-0.5045.

7. What will be the root of the equation cos(x) using Bessel's formula where x1 = 2 and x2 = 4 and
x = 2.1 and step value (h) = 0.25?
a) 1.2045
b) 2.2354
c) 3.8547
d) 4.6235
View Answer

Answer: a
Explanation: Given,
Equation is f(x)=cos(x)

Bessel's method to find solution

h=2.25-2=0.25
Taking x0=3 then p=(x-x0)/h=(x-3)/0.25
The difference table is

x p=(x-3)/0.25 y Δy Δ2y Δ3y Δ4y Δ5y Δ6y Δ7y
2 -4 -0.4161
-0.212
2.25 -3 -0.6282 0.0391
-0.173 0.0108
2.5 -2 -0.8011 0.0498 -0.0031
-0.1232 0.0077 -0.0005
2.75 -1 -0.9243 0.0575 -0.0036 0.0002
-0.0657 0.0041 -0.0003 0
3 0 -0.99 0.0616 -0.0038 0.0002
-0.0041 0.0003 0 0
3.25 1 -0.9941 0.0618 -0.0038 0.0002
0.0577 -0.0036 0.0002
3.5 2 -0.9365 0.0582 -0.0036
0.1159 -0.0072
3.75 3 -0.8206 0.051
0.1669
4 4 -0.6536

p=(x-x0)/h=-3/0.25=-12
y0=-0.99, Δy0=-0.0041, Δ2y-1 = 0.0616, Δ3y-1 = 0.0003, Δ4y-2 = -0.0038, Δ5y-2 = 0, Δ6y-3 = 0.0002, Δ7y-3 = 0

Bessel's formula is

y-12 = (-0.99±0.9941)/2+(-12-(1/2))⋅(-0.0041)+(-12(-12-1))/2⋅(0.0616+0.0618)/2+(-12-(1/2))-12(-12-1)/6⋅(0.0003)+(-12+1)(-12)(-12-1)(-12-2)/24⋅(-0.0038-0.0038)/2+(-12-(1/2))(-12+1)(-12)(-12-1)(-12-2)/120⋅(0)+(-12+1)(-12+1)(-12)(-12-1)(-12-2)(-12-2)/720⋅(0.0002)/2+(-12-(1/2))(-12+1)(-12+1)(-12)(-12-1)(-12-2)(-12-2)/5040⋅(0)
y-12 = -0.9921+0.0517147435+4.8111611636-0.0835996791-3.8388952996+0.0400232382+1.2252441039-0.0091243207
y-12 = 1.2045

Solution of Bessel's interpolation is y()=1.2045.

8. What will be the solution of the table using Bessel’s formula?

x f(x)
10 0.1
20 2.1
30 3.4
40 4.5

x = 25

a) 2.8062
b) 4.2563
c) 1.2354
d) -0.2396
View Answer

Answer: a
Explanation:

Bessel's method to find solution

h = 20-10 = 10

Taking x0 = 20 then p = (x-x0)/h = (x-20)/10
The difference table is

x p=(x-20)/10 y Δy Δ2y Δ3y
10 -1 0.1
2
20 0 2.1 -0.7
1.3 0.5
30 1 3.4 -0.2
1.1
40 2 4.5

x=25
p = (x-x0)/h = (25-20)/10 = 0.5
y0 = 2.1 ,Δy0 = 1.3, Δ2y-1 = -0.7, Δ3y-1 = 0.5

Bessel's formula is

y0.5 = (2.1+3.4)/2+(0.5-(1/2))⋅(1.3)+0.5(0.5-1)/2⋅(-0.7-0.2)/2+(0.5-(1/2))0.5(0.5-1)/6⋅(0.5)
y0.5 = 2.75+0+0.05625+0
y0.5 = 2.8062

Solution of Bessel's interpolation is y(25)=2.8062.

9. What will be the solution of the table using Bessel’s formula?

x f(x)
10 0.1
20 2.1
30 3.4
40 4.5
50 5.6

x = 15

a) 1.9874
b) 1.3254
c) 1.2589
d) 1.2715
View Answer

Answer: d
Explanation:

Bessel's method to find solution

h = 20-10 = 10
Taking x0=30 then p=(x-x0)/h=(x-30)/10
The difference table is

x p=(x-30)/10 y Δy Δ2y Δ3y Δ4y
10 -2 0.1
2
20 -1 2.1 -0.7
1.3 0.5
30 0 3.4 -0.2 -0.3
1.1 0.2
40 1 4.5 0
1.1
50 2 5.6

x = 15
p = (x-x0)/h = (15-30)/10 = -1.5
y0 = 3.4, Δy0 = 1.1, Δ2y-1 = -0.2, Δ3y-1 = 0.2, Δ4y-2 = -0.3

Bessel's formula is
y-1.5 = (3.4+4.5)/2+(-1.5-(1/2))⋅(1.1)+(-1.5(-1.5-1))/2⋅(-0.2)/2+(-1.5-(1/2))(-1.5)(-1.5-1)/6⋅(0.2)+(-1.5+1)(-1.5)(-1.5-1)(-1.5-2)/24⋅(-0.3)/2
y-1.5 = 3.95-2.2-0.1875-0.25-0.041015625
y-1.5 = 1.2715

Solution of Bessel's interpolation is y(15) = 1.2715.

10. What will be the root of the equation x+1 using Bessel's formula where x1 = 4 and x2 = 6 and
x = 2.1 and step value (h) = 0.50?
a) 3.1
b) 1.62
c) 2
d) 2.56
View Answer

Answer: a
Explanation: Given,
Equation is f(x)=x+1
The value of table for x and y

x 4 4.5 5 5.5 6
y 5 5.5 6 6.5 7

Bessel's method to find solution

h = 4.5-4 = 0.5
Taking x0 = 5 then p = (x-x0)/h = (x-5)/0.5
The difference table is

x p=(x-5)/0.5 y Δy Δ2y
4 -2 5
0.5
4.5 -1 5.5 0
0.5
5 0 6 0
0.5
5.5 1 6.5 0
0.5
6 2 7

x = 2.1
p = (x-x0)/h = (2.1-5)/0.5 = -5.8
y0 = 6, Δy0 = 0.5, Δ2y-1 = 0

Bessel's formula is

y-5.8 = (6+6.5)/2+(-5.8-(1/2))⋅(0.5)+(-5.8(-5.8-1))/2⋅(0)/2
y-5.8 = 6.25-3.15+0
y-5.8 = 3.1

Solution of Bessel's interpolation is y(2.1)=3.1.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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