Unit – 3
Numerical Methods
Example 1:
Newton Forward differences
Y(+ rh) = + + ….
Newton Backward difference
Y(x) = y () = + + + …….
The population of a city in the decimal census is given in the following table
Year X | 1891 | 1901 | 1911 | 1921 | 1931 |
Population Y (in thousand) | 46 | 66 | 81 | 93 | 101 |
Approximate the population of the city in the years 1895 and 1925
Ans Difference Table
X | Y | y | y | y | |
1891 | 46 | 20 |
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1901 | 66 | 15 | -5 | -2 |
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1911 | 81 | 12 | -3 | -1 | -3 |
1921 | 93 | 8 | -4 |
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1931 | 101 |
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Compute the population in 1895
Using Newton forward differences formula
Y(x) = + + + …
Let = 1891 h = 10 X = = 1895
r =
y(1895) = 46 + 0.4(20)+ (-5)+ (2) +
= 54.85 thousand (Approx)
Compute the population in 1925 using backward difference formula
Y(x) = + r+ ……..
Let = 1931 r =
Y(1925) = 101 + (0.6) 8 + +
= 96.84 thousand only.
Interpolation with unequal intervals
Newton divided difference formula
f() =
f() =
(x) = f() + (x - )f() + (x -)(x - )f() + ……
e.g using divided difference interpolation Find f(x)
x | 0 | 1 | 2 | 4 | 5 | 6 |
Y = f(x) | 1 | 14 | 15 | 5 | 6 | 19 |
The divided difference table
x | 4 | d.d of order1 | d.d of order2 | d.d of order3 | d.d of order4 | d.d of order5 |
0 | 1 |
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1 | 14 | 13 | -6 | 1 |
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2 | 15 | 1 | -2 | 1 | 0 |
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3 | 5 | -5 | 2 | 1 | 0 | 0 |
4 | 6 | 1 | 6 |
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5 |
| 13 |
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6 | 19 |
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Using formula
Y = f(x)= + (x - ) + (x - ) (x- )f + (x - )(x - )(x - ) f +…….
= 1 + (x - 0)(13) + (x - 0)(x-1)(-6)+ (x - 0)(x-1)(x -2)(1) + (x -0)(x-1)(x - 2)(x - 4)(0) + (x -0)(x - 1)(x -1) (x - 2)(x - 4)(x -5)0+ 0
= 1 + 13x + ( - x)(-6)+ (+ 2x)(1) + 0
= 1 + 21x - 9
f(x) =
Using Lagranye’s interpolation Solve the following data
x | 300 | 304 | 305 | 307 |
2.477 | 2.482 | 2.484 | 2.487 |
Calculate the approximate value of
Ans. Using formula
y = = + +
= (2.477) + (2.482) + (2.484) + (2.4871)
= (2.477) + (2.4829) + (2.4843) + (2.4871)
= 1.2739 + 4.9658 – 4.4717 + 0.7106
= 2.4786
Newton forward difference formula
Y = + P+ + + ….
=
=
( =
Where h =
=
Backward difference
=
( =
(1) Find from the following table
x | 3 | 5 | 11 | 27 | 34 |
f(x) | -13 | 23 | 899 | 17315 | 35606 |
x | 1 | 1.2 | 1.4 | 1.6 | 1.8 | 2 | 2.2 |
4 | 2.71 | 3.32 | 4.05 | 4.95 | 6.04 | 7.3 | 9.05 |
(2)
Find and at x = 1.2 , x = 1.6 and x = 2.2
Numerical integration f(x) by interoperating Polynomial (x) and obtain
Which Will be taken as a approximate value of . This is also called quadrature.
Trapezoidal rule
= h where h =
Simpson’s rule
=
Simpson’s rule
=
e.g Solve by (i) Trapezoidal rule , (ii) Simpson’s rule ,(iii) Simpson’s rule
Ans Let h = 1 or Let n = 6
x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
f(x) = | 1 | 0.500 | 0.200 | 0.100 | 0.05 | 0.038 | 0.027 |
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By Trapezoidal rule
I = = h
= 1
= 1.41
By Simpson’s rule
I =
=
= 1.366
Simpson’s rule
I =
=
= 1.357
Truncation of error:
The errors that result from using an approximation in place of an exact mathematical procedure
Ex:
approximation truncation errors
example 1:
maclurian series ln(1+x)
S = (-1 < x 1)
Solution:
Consider the given maclurian series is ln(1+x)
And assume n=5
S = (-1 < x 1)
s = 1-
In 2 = 0.693
=
Example 2:
Estimate the truncation error if we calculate e as
Solution:
Given
This is the maclurian series of f(x) = ex with x=1 and n=7.
Thus the bound of the truncation error is
Therefore the actual truncation error is about 0.2786
Example 3:
How many terms are required in calculation of using a maclurian series expansion,in which the result is correct to atleast 3 significant figure.
Solution:
Considering the maclurian series
Finding of error for 3 significant values is,
Let equation be
Then,
→ Using Runge kutta method of fourth order determine y (0.1) and y(0.2) correct to four decimal place given that where y(0)=2 and h=0.1.
Ans.
To find y(0.1) we have
For
Reference Books:
1. Erwin Kreyszig, “Advanced Engineering Mathematics”, Wiley India,10th Edition.
2. M.D. Greenberg, “Advanced Engineering Mathematics”, Pearson Education, 2 nd Edition.
3. Peter. V and O‟Neil, “Advanced Engineering Mathematics”, Cengage Learning,7th Edition.
4. S.L. Ross, “Differential Equations”, Wiley India, 3rd Edition.
5. S. C. Chapra and R. P. Canale, “Numerical Methods for Engineers”, McGraw-Hill, 7th Edition.
6. J. W. Brown and R. V. Churchill, “Complex Variables and Applications”, McGraw-Hill Inc, 8th Edition.