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Motilal Nehru National Institute of Technology

Civil Engineering Department

Computer Based Numerical Techniques

CE-401

Introduction


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Introduction

Consider the following equations.


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Numerical Methods - Definitions


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Numerical Methods


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Analytical vs. Numerical methods


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Mathematician and Engineer


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Reasons to study numerical Analysis

Powerful problem solving techniques and can be

used to handle large systems of equations

It enables you to intelligently use the commercial

software packages as well as designing your ownalgorithm.

Numerical Methods are efficient vehicles in learning

to use computers

It Reinforce your understanding of mathematics;

where it reduces higher mathematics to basic

arithmetic operation.


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Course

Contents


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CE-401

Mathematical Modeling


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Chapter 1: Mathematical Modeling

Mathematical Model

A formulation or equation that expresses the essential

features of a physical system or process in mathematical

terms. Generally, it can be represented as a functional

relationship of the form


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Mathematical Modeling


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Simple Mathematical Model

Example: Newtons Second Law

(The time rate of change of momentum of a body is

equal to the resultant force acting on it)

a = acceleration (m/s2) .the dependent variable

m = mass of the object (kg) .the parameterrepresenting a property of the system.

f = forceacting on the body (N)


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Complex Mathematical Model

Example: Newtons Second Law

Where:

c = drag coefficient (kg/s),

v = falling velocity (m/s)


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Complex Mathematical Model

At rest: (v = 0 at t = 0),

Calculus can be used to solve the equation


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Analytical solution to Newton's Second Law


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Analytical solution to Newton's Second Law


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Numerical Solution to Newton's Second Law

Numerical solution: approximates the exactsolution by arithmetic operations.

Suppose


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Numerical Solution to Newton's Second Law

.


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Comparison between Analytical vs. Numerical Solution


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CE-401

Accuracy and Errors


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Figure 4.1


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27

Measured Numbers

A measuring tool

is used to determine aquantity such as height

or the mass of an

object.

provides numbers for ameasurement called

measured numbers.Copyright 2009 by Pearson Education, Inc.


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28

. l2. . . . l . . . . l3. . . . l . . . . l4. . cm

The markings on the meterstick at the end of the orangeline are read as:

the first digit 2

plus the second digit 2.7

The last digit is obtained by estimating.

The end of the line may beestimated between 2.72.8 ashalf way (0.5) or a little more (0.6), which gives a reportedlength of 2.75cm or 2.76cm.

Reading a Meterstick


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29

Known & Estimated Digits

If the length is reported as 2.76 cm,

the digits 2 and 7 are certain (known).

the final digit, 6, is estimated (uncertain). all three digits (2, 7, and 6) are significant, including the

estimated digit.


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31

. l8. . . . l . . . . l9. . . . l . . . . l10. . cm

The length of the orange line could be reported as

2) 9.04 cm

or 3) 9.05 cm

The estimated digit may be slightly different. Both readingsare acceptable.

Solution


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32

. l3. . . . l . . . . l4. . . . l . . . . l5. . cm

For this measurement, the first and second known digitsare 4 and 5.

When a measurement ends on a mark, the estimated digit

in the hundredths place is 0.

This measurement is reported as 4.50 cm.

Zero as a Measured Number


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33

Significant Figures in

Measured Numbers

Significant Figures

obtained from a measurement include all of

the known digitsplusthe estimated digit. reported in a measurement depend on the

measuring tool.


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34

Significant Figures


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35

All nonzero numbersin a measured number aresignificant.

Number of

Measurement Significant Figures38.15 cm 4

5.6 ft 2

65.6 lb 3

122.55 m 5

Counting Significant Figures


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36

Sandwiched Zeros

occur between nonzero numbers.

are significant.

Number of

Measurement Significant Figures

50.8 mm 3

2001 min 4

0.0702 lb 3

0.405 05 m 5

Sandwiched Zeros


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Trailing Zeros

follow nonzero numbers in numbers without decimalpoints.

are usually placeholders.

are not significant.

Number of

Measurement Significant Figures

25 000cm 2

200kg 148 600mL 3

25 005 000g 5

Trailing Zeros


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38

Leading Zeros precede nonzero digits in a decimal number.

are not significant.

Number ofMeasurement Significant Figures

0.008mm 1

0.0156 oz 3

0.0042 lb 20.000 262 mL 3

Leading Zeros


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39

State the number of significant figures in each of thefollowing measurements.

A. 0.030 mB. 4.050 L

C. 0.0008 g

D. 2.80 m

Learning Check


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Significant Figures in

Scientific Notation

In scientific notation all digits in the coefficient

including zeros are significant.

Number of

Measurement Significant Figures8 x 104 m 1

8.0 x 104m 2

8.00 x 104m 3


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42

Study Tip: Significant Figures

The significant figures in a measured number are

all the nonzero numbers.

12.56 m 4 significant figures

zeros between nonzero numbers.4.05 g 3 significant figures

zeros that follow nonzero numbers in a decimalnumber.

25.800 L 5 significant figures


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A. Which answer(s) contain 3 significant figures?

2) 0.00476 3) 4.76 x 103

B. All the zeros are significant in

2) 25.300. 3) 2.050 x 103.

C. The number of significant figures in 5.80 x 102 is

3) three (3).

Solution


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45

In which set(s) do both numbers contain the same

number of significant figures?

1) 22.0 and 22.00

2) 400.0 and 40

3) 0.000 015 and 150 000

Learning Check


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Solution

In which set(s) do both numbers contain the same

number of significant figures?

3) 0.000 015 and 150 000

Both numbers contain 2 significant figures.


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Examples of Exact Numbers

An exact numberis obtained

when objects are counted.

Counted objects

2 soccer balls4 pizzas

from numbers in a defined relationship.

Defined relationships

1 foot = 12 inches

1 meter = 100 cm


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Learning Check

A. Exact numbers are obtained by

1. using a measuring tool.

2. counting.

3. definition.

B. Measured numbers are obtained by


2. counting.3. definition.


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Solution

A. Exact numbers are obtained by

2. counting.

3. definition.

B. Measured numbers are obtained by



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Learning Check

Classify each of the following as (1) exact or (2) measured

numbers.

A.__Gold melts at 1064 C.

B.__1 yard = 3 feet

C.__The diameter of a red blood cell is 6 x 10-4 cm.

D.__There are 6 hats on the shelf.

E.__A can of soda contains 355 mL of soda.


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53

Number Representation

Do machines represent integers and floating-

point numbers using the same

representation?

How does computer represent integers?

How does computer represent floating-point

numbers?


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Representation of Integers

13 as 8-bit unsigned integers (no negative #)

1310

= 000011012

= 0 x 27+ 1 x 26+ 0 x 25 + 0 x 24 +

1 x 23+ 1 x 22+ 0 x 21 + 1 x 20

= 8 + 4 + 0 + 1


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Exercise

What is the equivalent decimal number

represented by the following binary number?

1101012= ?


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Representation of Floating-point Numbers

156.78 = 0.15678 x 103

in an "imaginary" base-10 floating-point system

3 15678+

N li d R t ti


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Normalized Representation

(and notations used in this course)

is the sign is the base, e is the exponent

binary : =2

decimal : =10

1/ m < 1 (i.e., a1 0) binary: 0.5 m < 1

decimal: 0.1 m < 1

e

e

m

aaaz

...).0( 321

i f l i i b


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Representation of Floating-point Numbers

form)Normalized(2)1101101.0()(

)101.1101(

212021

21202121625.13

4

2

2

321

0123

Sign Signed exponent (e) Mantissa (a)


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59

Exercise

What is the normalized floating-point

representation of 12.7510(for = 2)?

What is the normalized floating-point

representation of 2.210(for = 2)?

What is the equivalent decimal value of

(0.110111)2x 23 ?


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60

There are discrete points

on the number lines that

can be represented by our

computer.How about the space

between ?


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Exercise

Consider the following floating-point representation The mantissa has only 3 bits

Exponent, e, ranges from -4 to 4

Can you give an integer that cannot be representedby this representation?

Can you give an integer between 0 and 14 thatcannot be represented by this representation?

eaaaz 2).0( 2321


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Note on IEEE 754 Representation

Exponents of all 0's and 1's are reserved for specialnumbers.

Zero is a special value denoted with an exponentfield of zero and a mantissa field of zero, and we

could have +0 and -0.

+ an - are denoted with an exponent of all 1'sand a mantissa field of all 0's.

NaN (Not-a-number) is denoted with an exponent ofall 1's and a non-zero mantissa field.


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Errors and Significant Digits

I paid $10 for 7 oranges. What is unit price of

each orange?

Rs. 1.528571429(that is the exact output from

my computer !!)

Is there any difference between

Rs. 1.527571429 and Rs. 1.5?

Is there any difference between Rs. 1.5 and

Rs. 1.50?


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Significant figures, or digits

The significant digits of a number are thosethat can be used with confidence.

They correspond to the number of certain

digits plus one estimated digits.

x = 3.5(2 significant digits)3.45 x < 3.55

x = 0.51234(5 significant digits)

0.512335 x < 0.512345


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67

Excercise

Supposex = 3.141592658979323

Show the value ofxup to 4 significant digits.

Show the value ofxup to 10 significant digits.

Calculate 22/7 up to 5 significant digits.


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CE-401

Approximations and Errors


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Approximations and Errors

The major advantage of numerical analysis is thata numerical answer can be obtained even when a

problem has no analytical solution.

Although the numerical technique yielded close

estimates to the exact analytical solutions, there

are errors because the numerical methods involveapproximations.

A i ti d R d Off E


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by Lale Yurttas, Texas A&M

University Chapter 3 70

Approximations and Round-Off ErrorsChapter 3

For many engineering problems, we cannot obtain analyticalsolutions.

Numerical methods yield approximate results, results that areclose to the exact analytical solution. Only rarely given data are exact, since they originate from

measurements. Therefore there is probably error in the inputinformation.

Algorithm itself usually introduces errors as well, e.g., unavoidableround-offs, etc

The output information will then contain error from both of thesesources.

How confident we are in our approximate result? The question is how much error is present in our calculation

and is it tolerable?


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Significant Figures

Significant figuresof a number are those that can be used

with confidence.

Rules for identifying sig. figures:

All non-zero digits are considered significant. For example, 91has two significant digits (9 and 1), while 123.45 has fivesignificant digits (1, 2, 3, 4 and 5).

Zeros appearing anywhere between two non-zero digits aresignificant. Example: 101.12 has five significant digits.

Leading zeros are not significant. For example, 0.00052 has

two significant digits Trailing zeros are generally considered as significant. For

example, 12.2300 has six significant digits.


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Error Definition

Numerical errors arise from the use of approximations

Truncation errors Round-off errors

Errors

Result when

approximations are usedto represent exact

mathematical procedure.

Result when numbers

having limited significantfigures are used to

represent exact numbers.


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Round-off Errors

Numbers such as p, e, or cannot be expressed bya fixed number of significant figures.

Computers use a base-2 representation, they cannot

precisely represent certain exact base-10 numbers Fractional quantities are typically represented in

computer using floating point form, e.g.,

Example:

p = 3.14159265358 to be stored carrying 7 significant digits.

p = 3.141592 chopping

p = 3.141593 rounding

7


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Truncation Errors

Truncation errors are those that result using

approximation in place of an exact mathematical

procedure.

1

1

i i

i i

V t V t dv v

dt t t t


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True Error

t

True error (Et) or Exact value of error

= true valueapproximated value

True error (Et)

True percent relative error ( )

(%)100

(%)100

valuetruevalueedapproximatvaluetrue

valueTrue

errorTrueerrorrelativepercentTrue t

See Example 3.1P 54


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Example 3.1

l


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Example 3.1


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Approximate Error

The true error is known only when we deal with functions thatcan be solved analytically.

In many applications, a prior true value is rarely available.

For this situation, an alternative is to calculate an

approximation of the errorusing the best available estimate ofthe true value as:

(%)100 ionapproximat

erroreApproximat

errorrelativepercenteApproximata


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Approximate Error

In many numerical methods a present approximation iscalculated using previous approximation:

a t

(%)100

ionapproximatpresent

ionapproximatpreviousionapproximatpresenta

Note:

- The sign of or may be positive or negative

- We interested in whether the absolute value is lower

than a prespecified tolerance (s), not to the sign of error.

Thus, the computation is repeated until (stopping criteria):

sa


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Prespecified Error

We can relate (s) to the number of significantfigures in the approximation,

So, we can assure that the result is correct to atleast nsignificant figures if the following criteriais met:

See Example 3.2 p56

%)105.0( 2 n

s


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