numerical computation lab1 practical oau

7/28/2019 numerical computation LAB1 Practical oau

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COMPUTER SCIENCE & ENGINEERING DEPARTMENT

O. BAFE. MI AWOLO. WO. UNIVERSITY

ILE-IFE. , O. S. UN STATE, NIGERIA

Harmattan Semester, 2012-2013 Session

PRACTICAL LAB 1

THIS DOCUMENT IS NOT FOR SALE!

CSC 307: Numerical Computations I

March 29, 2013

Contents

1 Background 2

2 Introduction to the laboratory 3

2.1 Purpose of the Laboratory . . . . . . . . . . . . . . . . . . . . . . . 4

3 Programming 5

3.1 Why FORTRAN 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2 Why Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3 Programming Works . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Laboratory Reports 6

4.1 Assignment Submission Dates . . . . . . . . . . . . . . . . . . . . . 7

5 Laboratory Assignment 1 9

5.1 EXPERIMENT I: Error and Error Propagation in Numerical Compu-

tations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.2 Aim of Experiment I . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.3 Experiment 1A: Inherent Error in single precision arithmetic . . . . . 95.4 Machine epsilon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.5 Task set 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.6 Machine Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.7 Task set 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.8 Experiment 1B: Inherent Error in Double Precision Arithmetic . . . . 12

5.9 Task set 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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1 Background

This is a 3 unit course. You are expected to have two hours of lectures and three

hours of practical classes every week, throughout the duration of this course. Note

that CSC201: Introduction to Programming I is a prerequisite to this course. What this

indicates is that the knowledge of CSC 201 is required for you to do well in this course.

Therefore, you are expected to have offered and passed CSC 201 before registering for

this course.

This document has been written to assist you during the practical classes. It is

important that you are familiar with the contents of this document before coming for

your practical classes.

As discussed in our lectures, the aim in numerical computations is to develop accu-

rate, reliable and economic solution to problems. A number of factors militate against

the computation of accurate results. They are highlighted as follows.

Conceptual errors: These errors result from wrong conception of the problem or its

solution approach. An example is the use of a wrong mathematical model or

scientific theory to describe the problem and/or its solution.

Data error: Numerical computations are carried out with numbers which are usually

approximated values. The measuring instruments, e.g. a thermostat in the case of

temperature, will only return results accurate to a specified number of significant

digits (this depends on the instruments resolution). The exact values is never

obtained particularly when dealing with real numbers. Blunder can also lead

to data errors.

Algorithmic error: Algorithms underlying the solution methods in numerical compu-

tations sometime uses mathematical series with infinite terms. Only a finite num-ber of terms can be taken into account when designing the solution algorithm for

a problem. This requires that some terms will be ignored in the computation,

hence the error. Taking the first neglected term as the error does not remove this

inexactness.

Computational tool error: There are errors inherent in the computational tool due to

the limited capacity of the computer. This depends on what the computer can

represent, stores, process and/or transmit. Most data cannot be stored exactly so

a compromise must be made vis-a-vis the capability of the computer.

Despite the above, note that the computer does not make mistakes. The computer

carries out only what it is designed and programmed to do. A general conceptualisa-

tion of errors that militates against obtaining accurate computational results is depictedin Fig. 1.

Our aim in numerical computations is to find an answer which is correct to a speci-

fied level or standard of accuracy while taking into account the above sources of error.

Hence, numerical computations comprises the artand science of obtaining practical so-

lutions to computational problems. The solution obtained to a problem may not be the

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Solution

Conception

Problem

Data

ComputationalModel

RunningProgram

ComputedSolution

Measuring instrumentInherent error

Data round off/ Chopping

Algorithmic error

Mathematical

Function truncation

Concept

representation error

Data entry error

(Blunder)

Computing

Machine

Inherent error

Solution

Presenation error

Solution

Interpretation error

Fig.1: Context of Computational Errors

most accurate. However, a solution is expected to be within a reasonable range of ac-

cepted values. The value will serve to inform an engineeror a scientistin making prac-

tical decisions. Therefore, the purpose of computation is not numeration. Computation

is done to gain informed scientific insight into real world situation or phenomenon.

The design and implementation of computational solutions are in the context of

what is practical. This is constrained by what can be measured and how it is measured

as well as the data, algorithm and computing device used to represent and manipulate

them.

The issue of accuracy, or error, in numerical computation and the computing pro-cess are meant to facilitate the construction of an acceptable expression of the extent

to which computation results can be reasonably applied. Acceptability, however, is a

function of the technology, cost and application environment. Therefore, the emphasis

in our practical classes will not be on merely generating the correct results, but also

on knowing the degree or extent to which the result can be relied upon in engineering

decision making.

The background information provided above will underpin all the exercises and

tasks that will be assigned to you in this laboratory.

2 Introduction to the laboratory

The practical classes in this course will, among other things, provide insights into the

design of accurate numerical algorithms and their implementation on digital com-

puters. You will be expected to gain experiences on:

i. The fundamentals of numerical rendering of problems.

ii. Data representation and manipulation on digital computing machines.

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iii. Numerical algorithm construction methodology.

iv. Algorithm implementation as program (Plato Fortran and Octave).

v. Error management techniques: detection, reduction or avoidance strategies.

vi. Issues in the use of modern numerical software and tools.

vii. Hints on numerical software development and deployment processes.

Skill development through practices with programming exercises will provide a

background for much of the course work. The programming exercises will focus on

the utilization of numerical concepts and methods which have engineering applications.

The practical exercises will emphasis good programming style as well as adequate

and accurate documentation procedures. The fundamental questions that underlies the

activities in the laboratory include the following:

1. How can we accurately approximate continuous or infinite processes by finite

discrete processes?

2. How do we cope with the errors arising from these approximations?

3. How rapidly can a given class of equations be solved for a given level of accu-

racy?

4. How can symbolic manipulations on equations, such as integration, differentia-

tion, and reduction to minimal terms be most accurately performed?

5. Given assorted numerical solution methods, how can we select or synthesis the

best solution method for a target problem?

6. How can the solutions to problems be incorporated into efficient, reliable, high-

quality mathematical software packages when such is required?

2.1 Purpose of the Laboratory

The laboratory assignments in this course have four (4) basic goals;

1. They help you to understand the general concepts and principles discussed in

CSC-307 course lectures, as well as during your private studies, by allowing you

to experiment with and confirm specific concepts expressed in some of the ideas

presented.

2. They help to improve your proficiency in algorithm design and computationalproblem solving. This proficiency will be useful in modeling and simulation of

engineering concepts and systems.

3. They help you to understand the principles of scientific and engineering experi-

ment documentation.

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4. They allow your instructor(s) to evaluate your practical skills in Numerical Com-

putations.Accordingly, you are advised strongly to make sure that all the work which you

submit for assessment arise out of your efforts and ideas. You are permitted, and en-

couraged, to discuss general algorithm design with the course lecturers as well as the

laboratory coordinators. You may also receive help with specific debugging problems

during the practical sessions from the laboratory coordinators. However, you are ex-

pected to work independently or only within your own team (where applicable) when in

the laboratory. Remember that the motto of this university is for learning and culture.

So do not let your behaviour suggest a lack of culture or home training, inside or

outside the laboratory.

3 ProgrammingWe will be using a programming language, FORTRAN with Plato IDE (Integrated De-

velopment Environment) and a programming tool, Octave modelling and simulation

software. You will ask, why we have selected these two platform given the fact that

there are more popular programming tools such as Java and simulation and mod-

elling software such as MatLab, Mapel and Mathematical. One of the reasons

is that programming languages such as Java are designed for witting general-purpose

software. They are not developed specifically for engineering and scientific comput-

ing as is the case for FORTRAN. Modelling and simulation tool such as MatLab,

Mapel and Mathematical are very expensive and outside our reach. Plato IDE

and Octave are open-source software, thus they are freely available for teaching and

research. The more technical reasons for our choices are as follows.

3.1 Why FORTRAN 95

The FORTRAN (FORmular TRANslator) programming language was originally writ-

ten for engineering and scientific programming. It was therefore design to provide the

necessary programming structure for expressing and solving engineering and scientific

problems.

Specifically:

1. It provides greater expressive power. FORTRAN statements are intuitive to en-

gineers and scientist.

2. It enhances safety (i.e. provide tools for computational errors detection).

3. It enhance regularity since it uses standard syntax (i.e. rule for writing program

instruction).

4. It provides extra fundamental features (such as dynamic storage).

5. It exploits modern computer hardware better, and

6. It has better portability between different machines.

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7. There are a lot of free supports, in terms of program code, suggestion, patches,

etc., from research groups and online open source for FORTRAN.8. FORTRAN produces more efficient codes as codes are compilednot interpreted

like Java.

In addition, the Plato IDE (Integrated Development Environment) that we will be

using in this course is freely available and can be downloaded from the Internet. The

IDE is also easy to use as it provides a lot of supporting documents and examples

online.

The main disadvantage of FORTRAN is that it was developed before several im-

portant advances in modern programming paradigm. This limitation nonetheless, the

drill provided by the FORTRAN programming language platform facilitates the kind

of environment suitable for educational development for improving students learning

experiences. Also, several modern features have been added to FORTRAN in the lastdecade. One of such addition is the object orientation programming concept.

3.2 Why Octave

We will be using the Octave platform to plot graphs and write simple simulation

programs in this course. Although this tasks could be achieved using FORTRAN it will

take more programming efforts to achieve comparative results. Note that, FORTRAN

will allow us to access the features of the computer system directly in a manner that

modelling as simulation languages like Octave or even MatLab will not.

Octave package provides a close compatibility with MatLab. This gives us the

opportunity to learn the syntax and power of both packages without financial and/or

licence restrictions associated with MatLab.

3.3 Programming Works

You will be required to carry out your programming works inside the laboratory. You

may design your algorithms and test your code before coming to the laboratory. You

may also create the source code and make some corrections by removing errors (debug-

ging). You will, however, be required to explain the design procedure and the contents

of your program. Familiarity with the contents of your program code and your creativ-

ity and ingenuity at problem solving will be reckoned and rewarded accordingly.

Documentation on Plato and Octave can be obtained from the Internet. The

respective URL will be provided and a pointer to the website to download them is

provided on the Computing and Intelligent Systems Research Group webpage

www.ifecisrg.org. If you are having issues with these packages please seek assis-tance from the laboratory coordinators.

4 Laboratory Reports

The documentation of your experiments is very important and will be given much

attention during the grading of your laboratory work. You should, therefore, ensure

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that your laboratory reports follow the format stated in Table 1. It should include

information about your observations as much as possible. Also note that the items listedas 1, 2, 3, 6, 7, 8, and 13 must be included in your reports. Items listed as 5, 9, 10, 11are important. While items listed as 4 and 12 should be included when used. Thepresentation of your laboratory report should be clean, clearand legible.

4.1 Assignment Submission Dates

The dates when you will be required to submit the reports of your assignments will

be announced during lectures. You should endeavour to submit your reports on or

before such dates. Late submission may attract stiff penalties, for example rejection of

submission or mark deductions.

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Table 1: Laboratory Document Contents

Ser. Item names Item Descriptions

No.

1. Name and Title of Report (i) NAME: Numerical Computations, Labo-ratory 1.

(ii) TITLE: Error and Error Propagation2. Author and date (i) Your full name(s), Identification numbers,

and Department of major (this must be listed

for all group members if it is a group work)

3. Tables of Contents (i) Table of Contents(ii) List of Figures (If any)(iii) List of Tables (If any)

4. Glossary of Notation andTerms

(i) List all uncommon symbols and terms andspecify their meanings.

5. Introduction (i) Brief discussions, stating the backgroundof the present work.

6. Problem statement (i) Statement of the problem(ii) Supporting theory and physical laws (ifany).

(iii) Mathematical models of problem andsolution.

7. Objectives of Experiment (i) General Objective or aim of the lab. work(ii) Specific Objective to be accomplished inthis experiment

8. Experiment procedure(

i)

Initial setting/material and tools.

(ii) Experiment processes.(iii) Measurement and recording processes.(iv) Algorithm design.(v) Program development and running.(vi) Final setting (if any)

9. Discussion of results (i) Analysis of method used, significance ofresult obtained.

10. Technological interpreta-

tion and Application of

results

(i) Illustrate, with examples, the real-life in-terpretation and applications of the result of

your experiment.

11. Conclusion and Sugges-

tions for further works

(i) Summary and general observation, nextcourse of action.

12. Appendixes (i) Tables of values and results(ii) Program listings.

13. References (i) List all the literature cited in your work.

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5 Laboratory Assignment 1

5.1 EXPERIMENT I: Error and Error Propagation in Numerical

Computations

The aim of most activities in numerical computations is to develop and deploy accu-

rate, effective and efficient numerical solution. Therefore, numerical algorithms are

designed with the almost aim of eliminating errors that can affect the computations of

exact result. Numerical analysis have provided us with recipe of mathematical meth-

ods that can be employed in solving general numerical problems. Computational tech-

niques are required for crafting out a solution which can give accurate result when

implemented on a computer. Built-in mechanisms for estimating the error expected

are also used in this process.

The task of minimizing error in a computational process is, therefore, a relative one

which depends on many factors. Some of the factors include:

1. Software environment for implementing the computation algorithm.

2. Hardware capability of the computational tool used: This will affect how num-

bers are stored.

3. Features of the problem being solved, e.g. the kind of mathematical models used.

An adequate understating of how these three factors culminate to affect the re-

sults of computational processes is necessary for the proper interpretation the resulting

model. Familiarity with subtle, but critical, factors which may give rise to computa-

tional errors, is also important. For example how a programming language built-in

functions are implemented. This will help in putting in place adequate measured to

reduce the occurrence ofavoidable

errors.It must, however, be noted that this course only focus on errors which may arise

at the algorithm design and implementation level of numerical solution to problems.

Errors which have to do with the conception of the problems and/or it solution methods

are not of primary concern.

5.2 Aim of Experiment I

This laboratory experiment is meant to familiarise students with the sources of some

of errors in computational methods, implemented as algorithms on digital computer

system.

5.3 Experiment 1A: Inherent Error in single precision arithmetic

The accuracy of any real number stored on a computer system depend heavily on the

word length of the system. The word length is the number of signal communication

lines on the processors data bus. Errors related to the limited capability of a compu-

tational tool such as this are called machine inherent error. The format for storing real

numbers 1 on the computer can also influence the accuracy of numeric data.

1Note that No real number is real. All numbers are representation for an abstract entity or object

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The representation of real numbers is similar to that used in scientific notations. A

real number is represented as

F = M E (1)

Where:

M is called the mantissa of the number.

is the base of the number.

E is the exponent.

The number of bits in M determines the precision of the number. It determines

the smallest number that can be represented by the system. Therefore, the more the

number of bits that is assigned to the mantissa, the more precise the data that can be

stored on the computer system. is the base of the number. In the case of digital

computers, = 2. This is because, digital computers uses the binary number system. = 10 for the decimal number system.

Since is same for all the numbers that will be stored on a machine, it is generally

not store, but implied. The exponent E, determines the magnitude of the number.

The number of bits for representing E determines the magnitude of the number that

can be store. For example, the decimal number 23.567635 105 is smaller but moreprecise than 23.56 1010. This is because, there are more digits in the mantissa butthe exponent is smaller, i.e. 5 < 10.

Generally, the more the number of bits assigned to the mantissa, i.e. M, the better

the precision of the number stored.

5.4 Machine epsilon

The epsilon (represented using the symbol ) of a computer is the smallest number it

can store, using the machines floating-pointarithmetic. Any number smaller than the

machine epsilon of a machine will be stored as zero. Determining the epsilon of a ma-

chine is critical to the implementation of most numerical algorithms and the interpre-

tation of their results. Also, the accuracy of real numbers and the results of arithmetic

operations on it are constrained by machine epsilon. Algorithms for determining the

machine epsilon are meant to find the smallest number such that when added to the

value one (1.0) will be greater than one, that is 1.0 + epsilon > 1.0.A way to do this is to start with a small number and repeatedly divide it by 2 until

the expression 1.0 + epsilon > 1.0 becomes false. The last number that was obtainedbefore this is taken as the machine epsilon. The algorithm in Table 2 implements this

computation process for machine epsilon.

5.5 Task set 1

1. Write a program to implement the algorithm in Table 2

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Table 2: Algorithm: Machine Epsilon

START:REAL epsilon

REAL Testeps = 1.0

5 Testeps := Testeps/2.0

IF ((1.0 + Testeps) > 1.0) THEN

epsilon = Testeps

GO TO 5

ENDIF

WRITE epsilon

END:

2. Experiment with various division for the epsilon by modifying line 5 of the al-

gorithm, i.e. 5 Testeps := Testeps/2.0, using various divisor such

as 3.0, 5.0, 10.00

3. Plot a graph (use Octave) relating the epsilon you obtained against the divisor

you used.

4. Discuss your observations and results.

5.6 Machine Gamma

The Machine Gamma () of a computer is the largest real number that it can correctlystore. Any number bigger than the machine Gamma is considered as Not a Numbers

and represented as N aN. The machine Gamma is related to the machine epsilon by

the formula:

=1

2(2)

5.7 Task set 2

1. Write a program to computer machine Gamma based on the machine epsilon you

obtained in Task 1.

2. Plot a graph to relate the machine epsilon and Gamma.


The numerical space of the computer is defined by the closed interval [, ]. Anycomputation that results in values outside this range cannot be handled by the computer

and the result is said to be out of scope for the respective machine. If a computation

produces a result smaller than the machine epsilon (i.e. ), the operation is said to have

resulted in underflow error. If a computation produces a result bigger than the machine

gamma (i.e. ), the operation is said to have resulted in overflow error.

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Overflow and underflow operations are outside the scope of the a computing de-

vice. However, wider numerical scope can be achieved on the same computer systemsby increasing the precision of data and operations using programming language data

declaration facilities.

5.8 Experiment 1B: Inherent Error in Double Precision Arithmetic

The values for machine epsilon and gamma obtained in Experiment 1A are for sin-

gle precision arithmetic. This is the native precision of the computer. It is possible to

extend this precision using program instruction. When a variable is defined as double

precision, the mantissa used is doubled. This has the effect of increasing the preci-

sion of the number that can be processes. This however, comes at the cost of slow

computation.

5.9 Task set 3

1. Repeat Task 1 and Task 2 using double precision arithmetic.


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numerical computation lab1 practical oau

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