... 12.
Similarly we have obtained
∆
3
u
0
= 6 and
∆
4
u
0,
∆
5
u
0
, are all zero as
u
r
=
r
3
is a polynomial
of third degree.
130
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
PROBLEM ... ax
−
−−
=
0
!
n
n
nhax
=
!
n
n
nha
132
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Hence, we have nth difference of the polynomial is constant and so a...
... the Range
Lower limit a = 0
Upper limit b = 6
Enter the number of subintervals = 6
Value of the integral is: 1.3571
598
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Enter the value of y2 ... ALGORITHM FOR TRAPEZOIDAL RULE
Step 1. Start of the program for numerical integration
Step 2. Input the upper and lower limits a and b
Step 3. Obtain the number of sub...
... 1.796307
62
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
2.6.2 Rate of Convergence of Iteration Method
Let f(x) = 0 be the equation which is being expressed as x = g(x). The iterative formula for ... (0.6072) + 1] = 0.6071
Now, x
5
and x
6
being almost same. Hence the required root is given by 0.607.
56
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Fir...
... 66
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Putting i = 2 in (1) x
3
= 0.02439
Therefore reciprocal of 41 is 0.0244.
Example 8. Find the square root of 20 correct to 3 decimal places ... 0.6071
68
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
From which we have, h = –
()
()
0
0
fx
fx
′
, where [f ′(x
0
)¹ 0].
Hence, if x
0
be the initial approximati...
...
∇
and
4
log 50 = 0.0508
∇−
Example 10. Given that:
12345678
1 8 27 64 125 216 343 512
x
y
Construct backward difference table and obtain
4
()
f8∇
.
108
COMPUTER BASED NUMERICAL AND STATISTICAL ...
2
2
()
d
fx
dx
and so on.
The operator ∆ is an analogous to the operator D of differential calculus. In finite differences,
we deal with ratio of simultaneous increments o...
... = 0.047875
190
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Example 4. Use Gauss’s forward formula to find a polynomial of degree four which takes the
following values of the function ... No. of Persons earning wages between Rs. 60 to 70 is 423.59375 – 370 = 53.59375 or
54000. (Approx.)
186
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Take the mean o...
... DIFFERENTIATION
The method of obtaining the derivatives of a function using a numerical technique is known as
numerical differentiation. There are essentially two situations where numerical differentiation ... Newton
forward formula, and if the same is required at a point near the end of the set of given tabular
294
INTERPOLATION WITH UNEQUAL INTERVAL
289
Example 1. Obta...
... Ans.
324
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Example 6. Evaluate the integral
6
3
0
1
dx
x+
∫
by using Weddle’s rule.
Sol. Divide the interval [0,6] into 6 equal parts each of ... NUMERICAL AND STATISTICAL TECHNIQUES
6.5 TRAPEZOIDAL RULE
Putting n =1 in equation (2) and taking the curve y = f(x) through (x
0
, y
0
) and (x
0
, y
0
) as a polynomial
of...
... 218295
13
afa f
or x ≤ .988.
340
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Example 10. Obtain y when x = 0.1, x = 0.2 Given that
;
dy
xy
dx
=+
y(0) = 1, Check the result
with exact value.
Sol. ... to 4 decimal places,
then using the first neglected term, namely
13
218295
15
x
as an approximation of the error, we have.
13
218295
15
x
≤ .00005.
Taking logarithm,...
...
()
()
3
1;03
dy
xxyy
dx
=+ =
Compute the value of
()
0.1y
and
()
0.2y
.[Ans. 3.005, 3.020]
348
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Now, using Euler’s modified formula, we obtain
()
()()()( )
221
2
0.1
.828 ... simply replace
()
000
,,
xyz
by
()
111
,,
xyz
in the above formulae.
350
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Since,
() ()...