Introduction and Analysis of Error Pertemuan 1 Matakuliah: S0262-Analisis Numerik Tahun: 2010.
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Transcript of Introduction and Analysis of Error Pertemuan 1 Matakuliah: S0262-Analisis Numerik Tahun: 2010.
Introduction and Analysis of Error
Pertemuan 1
Matakuliah : S0262-Analisis NumerikTahun : 2010
Material Outline
• Introduction to Numerical Methods• Approximation and Errors• Propagation of Errors
19 Apr 2023 DR. Paston Sidauruk 4
Numerical Analysis: Methods for providing numerical answer when analytic
procedures are either computationally difficult or nonexistent.
Numerical Analysis: Always involving complicated arithmetic computations Need the help of computing machines (computers)
Numerical Analysis Numerical Methods
19 Apr 2023 DR. Paston Sidauruk 5
Why Numerical methods are important: In real life many problems can not be solved
analytically. Numerical Analysis is a universal procedure (It can be
applied to many different engineering fields such as CE, ME, EE, and IT)
The fast growing development of computing machines (computers)
19 Apr 2023 6
INTRODUCTIONIt is a fact that many mathematical or engineering
problems or or physics phenomena can not be solved analytically (exact solution) For this type of problems or phenomena, numerical analysis can give the solution.
Numerical Work in general will cover the following steps:
Modeling : formulating a given work (problem) into mathematical equations.
Choosing appropriate Numerical methods. Developing a program Executing the program Analyzing the results
Note: Numerical Methods is always involving serious arithmetical calculation Need the help of computer.
19 Apr 2023 7
Some Concerns About Numerical Analysis:– Numerical Value: is an approximation of true
value that never known, hence we have to concern about the errors that may occur.
19 Apr 2023 8
ERRORSIn numerical solution, the results we get is the
best approximation to true value. Hence the numerical solution is always associated to certain degree of errors. For this purpose, a true value of any parameter can be written as :
a ã - Et In which :
a = true value (exact) ã = approximation (derived from measurements,
calculations etc) Et = total error
Numerical Errors: arise from the use of approximations to represent exact mathematical operations and quantities.
19 Apr 2023 9
RELATIVE ERRORSIt is sometimes desired to normalize the error with the
true value such as to account the magnitudes of the quantities being evaluated, such error called relative error.
Relative error t=true error/true value Relative error of approximation:a= approximation error/approximation
In the approximation of using iteration ( current approximation is based on the previous approximation), then the relative error of approximation is given below:
a= (current approximation – previous approximation)/ current approximationIn numerical computations or iteration process, it is sometimes to pre specify the tolerance (s) such that the following eq is satisfied, | a|< tolerance (s) =(0,5 x 102-n) %n= significant figure
19 Apr 2023 10
Errors sources1. Experimental errors (errors arise from
experiments, measurements etc)2. Round off errors (errors because of
rounding)3. Truncation errors (errors arise from a
process of simplification an algorithms, computations, steps in algorithms)
4. Programming errors
19 Apr 2023 11
Example:A MacLaurin series of ex is given below:
If the 1st term is considered as 1st estimate, the first 2 terms as the second estimate of ex, how many do you have to include in the series such that the relative errors | a|< s. In which a pre specified tolerance conforming to 3 significant figures. Find the true error and approximation errors.
Note: e0,5= 1.648721271 Solution: s =(0,5 x 102-3)%= 0,05 %
..........!3!2
132
xx
xex
19 Apr 2023 12
• Example (Cont.):
e0,5= 1,648721271Solution: s =(0,5 x 102-3)%= 0,05 %
Therefore, the minimum of the first 6 terms have to be used to estimate ex such that the error of approximation is less than the pre-specified tolerance.
2nd column: Series value for x= 0.5, 3rd column= (2nd column)/1.648721271.
Ith tern Result t (%) a (%)
1. 1 39,3
2 1,5 9,02 33,3
3 1,625 1,44 7,69
4 1,645833333 0,175 1,27
5 1,648437500 0,0172 0,158
6 1,648697917 0,00142 0,0158
19 Apr 2023 13
FINITE NUMBERS: can be written in two ways
1. fixed- point system ( is written according to the
specified number of decimal place) Example: 62.358; 0.013; and 1.000
(3 decimal place)2. floating- point system is written
according to certain significant figures
Example:0.6238 * 103; 1.7130 * 10-13; 2000 * 104
19 Apr 2023 14
Significant Figures
The concept of significant figures designate the reliability of a numerical value.
All digits that can be used with confidence. Example: 4 digit significant figures 1.360 ; 1360 ; 0.001360All zeros that are needed only to locate the
decimal point are not counted as significant figures:
Example: all of the following numbers are in 4 significant figures 0.01845; 0.0001845; 0.001845
Also:4.53 * 104 (3 significant figures)4.530 * 104 (4 significant figures)4.5300 * 104 (5 significant figures)
19 Apr 2023 15
Rounding a number to certain significant figures
1. Those digits that are not significant will be omitted. The last digit that is saved will rounded up if the first digit in the omitted digits >5 and if the 1st in the omitted digits =5 and the last digit in the saved digits is odd number.
2. The final results of summation or subtraction will be rounded to the most significant figures of all the numbers (quantities) that are being operated.
3. The final results of multiplication or division will be rounded such that the number of significant figures will be equivalent with the least number of significant figures of all the numbers (quantities) that are being operated.
19 Apr 2023 16
Rounding a number to certain significant figures
• Examples:– Rounding
5.6723 5.67 (3 significant figures)10.406 10.41 (4 significant figures)7.3500 7.4 (2 significant figures)88.21650 88.216 (5 significant figures)1.25001 1.2 (2 significant figures)
– Summation/Subtraction Evaluate: 2.2 – 1.768
2.2-1.768= 0.432 0.44.68 x 10-7+8.3x10-4-228x10-6= ….? ……. (6.0x10-4)
– Multiplication/Division0.0642x 4,8= 0.30816 0.31945/0.3185= 2967.0329672970
19 Apr 2023 17
Error Propagation The purpose is to study how errors in numbers
propagates through mathematical functions
• Function of single Variable
xxxx
x
x
xfxfxfxf
xxfxf
xfy
~~ oferror of estimate ~ value true
ion valueapproximat ~)~()()~( oferror of estimate )~(
~)~(')~(
)(
19 Apr 2023 18
Error Propagation• Function of single VariableExample: Given a value of
Answer:
3)(function in theerror
resulting theestimate ,01.0~error an with 5.2~
xxf
xx
8125.15)5.2(4375.15
1875.0625.15)5.2(
1875.0)01.0()5.2(3~)~(')~(
3)(')(2
23
f
f
xxfxf
xxfxxf
19 Apr 2023 19
Error Propagation• Function of more than one variable
nn
n
n
nn
n
xxxxxx
xxx
xxxfxf
xx
fx
x
fx
x
fxf
xxxfy
~,,~,~ oferror of estimate ~,,~,~ion valuesapproximat ~,,~,~
)~,,~,~( oferror of estimate )~(
~~~)~(
),,,(
2121
21
21
22
11
21
19 Apr 2023 20
Error Propagation• Function of more than one variableExample/Exercises:
006.0~6.0~01.0~5.1~1.0~30~2~50~
given thaty in error resulting theestimate
8
),,,(4
ttzz
yyxx
zt
xytzyxfy
19 Apr 2023 21
Error PropagationEstimated Error for common mathematical
operations
Operation Estimated Error
Addition
Subtraction
Multiplication
Division 2~
~~~~
y
xyyx
y
x~
~
)~~( yx xyyx ~~~~
)~~( yx
)~~( yx yx ~~
yx ~~