... Approximate value (x′) = 0 .33 3
Therefore, Absolute Error,
1
0 .33 3 0 .33 333 3 0 .33 3 0.00 033 3
3
a
Exx
′
=− =− = − =
Relative Error, E
r
=
0.00 033 3
0.000999
0 .33 333 3
a
E
x
==
and
Percentage Error, E
p
= ... second
decimal places.
Example 4. If 0 .33 3 is the approximate value of
1
3
, then find its absolute, relative and percentage
errors.
Sol. Given that True val...
... problems. A major
advantage for numerical technique is that a numerical answer can be obtained even when a
problem has no analytical solution. However, result from numerical analysis is an approximation,
in ... significant figures at each step of
computation. At each step of computations, retain at least one more significant figure
than that given in the data, perform the last...
... obtained by
x
1
=
2
22
44
and
22
bb ac bb ac
aa
−+ − −− −
=x
These are called closed form solution. Similar formulae are also available for cubic and
biquadratic polynomial equations but we rarely ... 26
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
The basic operations specified by IEEE arithmetic are first and foremost addition, subtraction,
multiplication, and divi...
... 122
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Example 22. Evaluate: ∆
n
[sin (ax + b)]
Sol. We know
f∆
(x) = f(x + h) – f (x) therefore
∆
sin (ax + b) = sin [a (x + h)+b] – sin)(ax ... hD hD
hD
116
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
Example 14. Evaluate the following:
I. ∆
2
(cos 2x)
II. ∆
2
(3e
x
)
III. ∆ tan
–1
x
IV. ∆(x + cos x)
the inte...
... 22
1
2
n
nn
an bn c n a n b n a n
−
−++−∆+∆+∆
142
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
9. Evaluate the production of wool in the year 1 935 from the given data:
( ) 1 931 1 932 1 933 ... given a set of equidistant values of arguments and its corresponding value of f(x). Suppose
for n + 1 equidistant argument values x = a, a + h, a + 2h,...
... R.H.S.
3. 9 FACTORIAL NOTATIONS
The product of n consecutive factors each at a constant difference and the first factor being x is
called a factorial function or a factorial polynomial of degree n and ...
n
C
3
x
3
E
3
+ ]u
0
= u
0
+
n
C
1
u
1
x +
n
C
2
u
2
x
2
+
n
C
3
u
3
x
3
+ = L.H.S.
150
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
CorCor...
... life. [Ans. 15.67]
13. Find the value of tan 33 ° by Lagrange’s formula if tan 30 ° = 0.5774, tan 32 ° = 0.6249, tan
35 ° = 0.7002, tan 38 ° = 0.78 13. [Ans. 0.64942084]
14. Apply Lagrange’s formula to ...
n
ax ax ax ax
−− −−
11. Certain corresponding values of x and log
10
x are given as
10
: 30 0 30 4 30 5 30 7
log : 2.4771 2.4829 2.48 43 2.4871
x
x
Find the log
10
30 1...
... co-efficients a
1
, a
2
a
n
will produce little
error for small x near zero. But probably substantial error near the ends of the interval
268
COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES
7. Apply ... approximation to a given function.
For evaluating a function f(x) on a computer it is generally more efficient of space and time
to have an analytic approxi...
... 0.5 230 cm. and a standard
deviation of 0.0 032 cm. Calculate the 2-sigma and 3- sigma upper and lower control
limits for mean of sample of 4.
Ans. 2–sigma
UCL 0.5262 cm
LCL 0.5198 cm
CL 0.5 230 ... concern has an average sale of Rs. 10000/- daily estimated over a long
period. A salesman claims that he will increase the average sales by Rs. 700/- a day. The concern
is...