Solution of non linear & transcendental equations(Error in Number Representation)

2. Class- M.sc (C.S.) 2nd semester
Session-2018
Submitted by-
Ayushi Dubey
Submitted to-
Prof. Mr. Yugal Sharma

4. An error is defined as the difference between the
actual value and the approximate value obtained from
the experimental observation or form numerical
computation .
The error in computed results may be due to error in
input data or computational algorithm.
TRUE VALUE = APPROXIMATION + ERROR
(or)
ERROR (E) = TRUE VALUE ‐ APPROXIMATION

6. Inherent Errors-
Inherent errors are those that are present in the data
supplied to the model.
Inherent errors (also known as input errors)
Contain two components, namely data errors &
conversion errors.
These are errors which occur due to inaccurate
measurements or observations which may be due to
limitation of the measuring device.
(a)Data Errors-

7. (b)Conversion Errors-
C. E. (also known as representation error) arise due to
the limitations of computer to store the data exactly.
As we have already seen, many numbers can not be
represented exactly in a given numbers of decimal
digits. In some cases a decimal number can not be
represented exactly in binary form.

8. Numerical errors are introduced during the process at
implementation of a numerical method.
These are of two types, round off errors and truncation
errors.
Numerical Errors-
Round off error occur when a fixed number of digits
are used to represent numbers. Since the numbers are
stored at every stage of computation. Round off error is
introduced at the end at every arithmetic operation.
Round off Errors-

9. Rounding a number an be done in two ways, one is
chopping & other is symmetric rounding.
Chopping:-
•Chopping means dropping the extra digits.
•Suppose we are using a computer with a fixed word
length of four digits, then a number like 64.8234 will be
stored as 64.82 and the digits 34 will be dropped.
Symmetric Round off:-
•In this method, the last retained significant digit is
rounded up by 1 if the first discarded digit is larger or
equal to 5, otherwise the last retained digit is
unchanged.
•For example 34.7694 would become 34.77 & the
number 34.7623 would become 34.76.

10. Truncation Errors-
•Truncation errors arise from using an approximation in
place of an exact mathematical procedure.
•It is the error result in from the truncation of the
numerical process.
•We often use some finite number of terms to estimate
the sum of an infinite series. For example,
is replaced by the finite sum,
the series has been truncated.
xias i
i




0
xias i
n
i


0

11. Some fundamental definition of
errors
(measures of accuracy) -

12. Absolute error-
ERROR (E) = TRUE VALUE ‐ APPROXIMATION
Relative Error-
Relative error is the ratio of the absolute error to the
actual value of a variable. if x is the (true)actual value
and xa is the approximate value of a variable, than the
relative error is given by-
relative error = =
Percentage Error-
If x is the actual value at xa is the approximate value of a
variable, then the percentage error is –
percentage error = =
x
xx a
100
x
xx a

13. Example-
Number round off to four significant figures = 37.46235
Solution-
Absolute error =
Relative error =
Percentage error = Relative error*100
=0.0000627*100
=0.00627
00235.046000.3746235.37  axx
0000627.0
46235.37
00235.0


x
xx a