IE4004 Errors.pdf - IE4004 NUMERICAL ANALYSIS FOR...

IE4004
NUMERICAL ANALYSIS
FOR INDUSTRIAL ENGINEERS
Assist. Prof. ilayda ÜLKÜ
2/4/2020
2019/2020 Spring

REVIEW OF CALCULUS
AND
MATHEMATICAL PRELIMINARIES

Introduction
•
List of certain analytical results to which reference will be made
•
we deal with real and continuous functions in the range considered
and also they possess as many continuous derivatives as required
•
Definition of limit : let f be a function defined on a set X of real numbers, f
is said to have the limit L at x
0
and written
If, given any real number  > 0, there exists a real number  > 0 such that
| f(x) – L | <  and 0 < | x – x
0
| < 
L x f
x x
=
→
) (
lim
0

•
Definition of Continuity : function f is said to be continuous at x
0
if
•
Definition of convergence : Let be an infinite sequence of real
or complex numbers.
The sequence is said to converge to a number x , if for any  > 0, there
exists a positive integer N(  ) such that n > N(  ) implies | x
0
– x | < 
The notation
•
means that the sequence converges to x
) ( ) (
lim
0
0
x f x f
x x
=
→

= 1
} {
n n
x
x x
n
x
=
 →
lim

•
Definition of differentiability : if f is a function defined in an open
interval containing x
0
, f is said to be differentiable at x
0
, if
exists. When this limit exists it is denoted f ´ (x
0
) and is called the
derivative of f at x
0
A function that has a derivative at each
number in a set is said to be differentiable on X .
0
0
0
) ( ) (
lim
x x
x f x f
x x
−
−
→

ERROR DEFINITIONS

Introduction
Final results of computations are generally approximations and involve
errors.
If ã is an approximate value of a quantity a (true value), the error is defined
as;
error = true value - approximate value
 = a – ã (sometimes refered as absolute error )
Relative error, 
r
of ã is defined by
relative error = absolute error/true value

r
=  / a
For example, ã = 10.5 is an approximation of a = 10.2,
its error is  = -0.3

In practice an error bound is defined by
|  |  
Which tells us how far away the computed value ã the
unknown a can at most lie.
Similarly for the relative error, an error bound is a
number defined by
| 
r
|  
r

Error Analysis
Possible sources of errors in numerical analysis
•
Truncation error
•
Round-off error
•
Error in original data
•
Programming bugs
•
Propagated error

Truncation Error
•
Truncation error refers to the error involved in using a truncated or