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NAME;OMURWA DAVID
UNIT;NUMERICAL ANALYSIS
DEPARTMENT;MATHEMATICS
INTRODUCTION
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Algorithm
Definition
Is a finite sequence of effective statements that when applied to the problem will solve it
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EERROR AND THEIR ANALYSIS
1
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2
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3
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i
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Data
iii
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Representation of numbers e
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14 etc
v
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Mistakes due to unpredicted human errors
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g input through computers
vii
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Inherent errors-This is the quantity which is already present in the statement of a problem
due to errors in the physical measurement of the parameters of a problem
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Round off errors
3
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Absolute and relative errors

5
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Truncation errors
7
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This introduces an error which is a machine dependent and is called machine epsilon
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The greater the precision the more accurate the computation
Example;
4
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32,4
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Rounding off errors
They arise because it is imposibble to represent all real numbers exactly on a machine with finite
memory(practical digital computers have finite memory)
It involves the process by which decimal points and significant figures are dropped out to represent
ordinary numbers approximately, in two ways,
I
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Chopping-This involves dropping extra digits i
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g
3
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1428
To 4 significant figures 3
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i
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If the (n+1)th is less than 5 then nth digit is;
i
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if odd it is increased by 1
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4/3
1/30
2/3
4
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168551

ROUND OFF TO
4s
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333
0
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6667
4
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169

ROUND OFF TO
4 dp
1
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0333
0
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1676
4
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f
1
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03333
0
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167
4
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TRUNCATION ERRORS
These can be caused by;
i
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Replacing an infinite series by a finite one
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e
by using the maximum terms
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Find the truncation error for ex at x=1/5 if first threeterms are retained in
expansion
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Error=true value –approximate value
Using three terms
=(1+x+x2/2! + x3/3! +)……… -(1+x +x2/2!)
=x3/3! + X4/4! + x5/5! + x6/6! +…
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2
Error =(0
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2)4/4! + (0
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2)6/6!
=0
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p) or 5 s
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Absolute error=absolute difference between no
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X*
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Let ∆x be such that eabs =/X-X*/≤ ∆x ,where ∆x →upper bound
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4(1 d
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35,2
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Max
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05
So X Σ 2
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05 i
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35≤X≤2
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35-2
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45-2
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05/ X-Xx/0
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05
eabs = 0
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Relative error= /X-Xx/ X ,error of the true alue
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er and percentage error are independent of the units used while eabs is expressed in units
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is correct to n d
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0005
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if the first significant digit of a correct to n significant figures ,then
er <1/(k*10n-1)
e
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73
K=9, n=5
er<1/(9*104)
ans;=1/9 *10-4
exercise
1
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414 is used to approximate √2, find the absolute value and relative error
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Round off the numbers 865250 and 37
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f and compute eabs , er , ep in each case
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The total accumulating error at any step of the procedures is called accumulating error
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X=X*± Σ and Y=Y*±П
Let Ш be an operation
X Ш Y –X* Ш Y*
(X Ш Y – X* Ш Y*) +( X* Ш Y* - X* Ш* Y*)
(X Ш Y – X* Ш Y*) is a propagation error while ( X* Ш Y* - X* Ш* Y*) is a generated error
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Addition and substraction(X±Y)
Error =(X+Y) –(X*+Y*)
Error =(X*± Σ) + (Y*± П)-(X*+Y*)
When you open the brackets you get and simplify;
e=±/Σ/ +/П/
/e/≤/Σ/ + /П/
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45 + 4
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Solution
X* =3
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87

X=3
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005

Y=4
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005

3
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45

4
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87

3
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865≤X+Y≤3
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87
8
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33

X+Y=8
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Exercise
Evaluate as accurate as possible
X-Y where X=3
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005, Y=2
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005



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