By PROFESSOR SOLOMON ADEWALE ADEBOLA OKUNUGA NUMBERS: THE LANGUAGE OF THE THINKERS An Inaugural Lecture Delivered at the University of Lagos Main Auditorium on Wednesday March 20, 2013 PROFESSOR SOLOMON ADEWALE ADEBOLA OKUNUGA B.Sc. (First Class Honours), M.Sc. (llorin), Ph.D. (Lagos) Professor of Computational Mathematics Director Quality Assurance & SERVICOM Unit Department of Mathematics University of Lagos University of Lagos Press © Solomon Adewale Adebola Okunuga, 2013 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the author. Published by University of Lagos Press Unilag P. O. Box 132, University of Lagos, Akoka, Yaba-Lagos, Nigeria. e-mail: unilagpress@yahoo.com ISSN 1119-4456 Protocols The Vice-Chancellor, Deputy Vice-Chancellor (Academic & Research), Deputy Vice-Chancellor (Management Services), The Registrar, Other Principal Officers, The Provost, College of Medicine, The Dean of Science and other Deans here present, Members of Senate, ' Members of Academic Community, Dear students, My family and friends, Distinguished guests, Ladies and Gentlemen Preamble I give all glory and honour to the Almighty God who made it possible for me to stand before this audience to deliver this inaugural lecture. I stand before you this day because the University found me worthy to be elevated to the post of a professor of Mathematics. Since my appointment I have thought of what will be the title of my inaugural lecture and while toiling with this, my mentor said, "you have worked very well with numbers why not consider something along that line". Also I have jointly written more papers with him than anyone else. For these reasons I hereby dedicate this inaugural lecture to my mentor and academic father, Late Professor Adetokunbo Babatunde Sofoluwe, whom God used greatly in my life for greater part of my academic pursuit, and this helped me to where I am today. May his soul rest in peace. I will not want to emphasise the need for newly appointed professors to give inaugural lecture, as this is a requirement that must be satisfied. To this end I decided to give my inaugural lecture this day as the 6th lecture series in 2012/2013 session. Prof A. B. Sofoluwe delivered his inaugural lecture on the 8th of November 2006 on a title, "Beyond Calculations". I observed that calculations are done everyday almost by every grown up persons unawares. These calculations are done not with any other thing other than numbers, however, no one thinks or border about the everyday calculation or usage of numbers that we engage in. This prompt me to title my own lecture as "Numbers: the Language of the Thinkers". Introduction Mathematics has been perceived as a difficult subject in schools. Young and old, adult and school children perceive Mathematics as a subject that is not easy to grasp or retain. The difficulty perceived by many is not far-fetched. It is simply because Mathematics talks with symbols. It is quite clear that Mathematics has its own language just as there are languages being spoken all over the world for the purpose of communication. Everything about Mathematics started from numbers. In fact, there are several types of numbers existing in the mathematical world. No man on the earth can avoid dealing with numbers such as 1,2,3, ...,9,0. These numbers are well known as they may appear, they are merely symbols with attached values but the use of symbols seems to be the beginning of problems people have with Mathematics. It is suffice to say that Mathematics unlike other languages, speaks "only the truth and nothing but the truth". It does not allow you to change truth to false or vice-versa. Whatever is mathematically true today remains true tomorrow and nothing can be done to it. For example, the number 4 will always be greater than the number 3. Without compounding the problem for those who cannot speak Mathematics language, we simply write 4 > 3 or 3 < 4. These two symbols being introduced here can give a nightmare to someone. However, these symbols are right there on the typewriter or computer keyboard which do not frighten while typing them, but may look inconvenient to some people when constants or numbers are included such as 4 > 3 or a > 5 or even y < 1. The above examples simply gave us the foundation that Mathematics has its language. The 2 purpose of this lecture is to prove to my audience that as long as you are born into this world you have everything to do with Mathematics and in particular with numbers, and secondly that Mathematics is a friend to all. There are different levels of friendship existing among people. Some people are your friends because you work in the same place or you attend the same school or belong to the same class. Some other develop closer friendships and do visit each other at home or become family friends. Some are friends because they have common faith. They are friends because they attend the same church or worship in the same place. Some are also friends because they belong to the same social club. Whatever it may be different degrees of friendship exist. In the same vein, everyone has to use numbers or can I say everyone develops some degree of friendship with numbers, then a higher degree of friendship with everyday Arithmetic or Mathematics as the case may be, and fewer numbers of people have deeper friendship with higher Mathematics while the minority have true friendship with what is called Abstract Mathematics. Although, Mathematics would necessarily include numbers, we can say that arithmetic is the mother of numbers, while Mathematics is the father of arithmetic. In other words, Mathematics contain Arithmetic and some other things, while Arithmetic itself is more than just numbers but putting those numbers together for use by introducing some mathematical operations such as addition, subtraction and so on. This lecture will attempt not to leave my audience behind, as this simple lecture cannot make you a mathematician in a day. It will however create in you all the essence of living and how impossible it is for any man to exist without numbers and Mathematics, because numbers happen to be everybody' business. 3 Types of Numbers There are various types of numbers. Numbers can broadly be classified into two. There are real numbers and there are complex numbers. The real numbers are as real as what you can appreciate in terms of concrete value or quantity, while the complex numbers are not so, they contain some imaginary numbers that may not be easily conceptualized. Complex numbers are not complex in terms of complexity as such, but in terms of creating another world of numbers which may not directly be used by all. They however have relevance in applications and in drawing conclusion from a mathematical problem. For this simple reason we may just concentrate on real numbers. Real numbers are in various stages and of different categories. We have the following types or classification of numbers: • The counting numbers • The natural numbers • The integers • The rational numbers • The irrational numbers • The real numbers, and • The complex numbers Apart from the last one, we often use all these numbers directly or indirectly, without knowing them. In fact numbers are better appreciated if one can master the classification, as this will help to put things right. Let me illustrate with this simple mistake which is common in our schools today, and which many are guilty of. We all know decimal numbers. Some of them are rational numbers and some are not. A number like 25.47 is pronounced in different ways because of poor understanding of decimal numbers. A number that is greater than a but less than 1 can be expressed as a decimal number, however such numbers that follow the decimal point is not pronounced together as if they are greater than ten. Thus 25.47 is pronounced as twenty-five point four seven, and not 4 twenty-five point forty- seven, neither is it right to call it as two- five point forty-seven. The 4 and the 7 after the decimal point is pronounced one by one, and not as a single number greater than 10. What about the mobile phone number? Many people always pronounce the numbers in an unfriendly and non-numeric manner. A phone number such as 08023043270 is sometimes pronounced as 0 8 0 2 3 0 4 3 2 7 O. All the zeros are pronounced as letter "0" instead of number "0" (zero). Figure 1: Tree diagram of Numbers Numbers----...:*- Real Number Complex Number ~ Rational Irrational ~ ~:...-er_s __ N_on_-_In_te_ge_rs ~ Real Part Imaginary part Natural Numbers (Counting Numbers) Negative integers plus zero It is assumed that almost all the audience present here have used real numbers and their subsets before now. You m y not remember them now, but the fact is that any time you n ed them you do make use of them. The use of these numb r will normally require a man to think before taking decision. 5 The Historical background Mathematics has always been part of human being from the day the first man landed on earth. As far as 3500BC, man has been using numbers to count. The shepherd cannot know tilt number of cattle he has except he uses numbers, or at leas; use a form of tally of things to keep his records. In the process of counting, he may use his fingers and toes, or make marks on the wall, or use broom sticks and so on. This completel shows that man cannot exist without numbers. Thus we ca say that the concept of numbers is the heart of Mathematics from which higher mathematical concepts developed. [RuSS Rowlett (2003)]. Interestingly, everybody needs number because it is the language for the living. Different people of different language background may call the number 100 different names, but the value of what they all may be saying in different languages wi be the same. All of them will be referring to the same size or quantity. Hence Mathematics is universal. I have, received in the past, series of questions from school children and students about numbers because they get frightened with Mathematics. Some of the questions are: what are numbers? Who started or introduced Mathematics? Who needs Mathematics? Where does Mathematics and numbers come from? Why do we need negative numbers, when positive numbers are more than enough headaches to handle? Is infinity a number and if so what is its value? And several other questions like these ones. All of these have simple answer. Numbers can be said to be figures or symbols given to a thing as its value or quantity so as to appreciate its worth or size. Of course, man did not invent numbers or Mathematics. Man only discovered that numbers are in existence and began to give names to them as he discovers more numbers. For example, in the early days, there is no number called a million. Everything is counted in tens, hundreds and later in thousands. But after a while, man did not only need larger numbers but move from millions to billions and then to trillions. Of course the names were given to these numbers but that is not without 6 some difficulties and universal agreement between users of numbers. You will observe the following numbers: 10=1x10 100 = 10 x 10 1,000 = 100 x 1a 1,000,000 = 1,000 x 1,000 1,000,000,000 = 1,000,000 x 1,000 1,000,000,000,000 = 1,000,000,000 x 1,000 You will notice that at the earlier and smaller numbers, the name changed after a multiple of ten. However, when we get to a million it is a thousand times a thousand. From there on, the name changes after a multiple of a thousand. Thus a million changed to a billion and a billion to a trillion. From this idea, we already know the next level of numbers as a trillion times a thousand. Numbers starting from a billion is rather confusing and does not have a general agreement, especially as some are termed as American style while another European. In all of this confusion of high numbers as per the name, it is out of place to ascribe the invention of a billion or trillion to one Mr X or Y. The scientific truth is that numbers have been in existence before man's existence, so man could not have invented numbers. The only option left to man is to agree that God is the creator and the inventor of numbers and Mathematics. Names for Large Numbers The English names for large numbers are coined from the Latin names for small numbers n by adding the ending -illion, suggested by the name "million." Thus billion and trillion are coined from the Latin prefixes bi- (n = 2) and tri- (n = 3), respectively. In the American system for naming large numbers, the name coined from the Latin number n applies to the number 103n + 3 . In a system traditional in many European countries, the same name applies to the number 106n . In particular, a billion is 109 = 1 000 000 000 in the American 7 system and 1012 = 1 000000000 000 in the European system. For 109 , Europeans say "thousand million" or "milliard." Although we describe the two systems today as American or European, both systems are actually of French origin. The French physician and mathematician Nicolas Chuquet (1445- 1488) apparently coined the words byl/ion and tryl/ion and used them to represent 1012 and 1018 , respectively, thus establishing what we now think of as the "European" system. However, it was also French mathematicians of the 1600's who used billion and trillion for 109 and 1012 , respectively. This usage became common in France and in America, while the original Chuquet nomenclature remained in use in Britain and Germany. The French decided in 1948 to revert to the Chuquet ("European") system, leaving the U.S. as the chief standard bearer for what then became clearly an American system. In recent years, American usage has eroded the European system, particularly in Britain and to a lesser extent in other countries. This is primarily due to American finance, because Americans insist that $1,000,000,000 be called a billion dollars. In 1974, the government of Prime Minister Harold Wilson announced that henceforth "billion" would mean 109 and not 1012 in official British reports and statistics. The Times of London style guide now defines "billion" as "one thousand million, not a million million." There is yet a greater number that its value is unimaginable. This number is called infinity (00). Infinity is a number that is so large that value cannot be assigned to it. 8 --,-- In = I 121~-' 131 ~'---:--::C-<"'---' r51["6'---:--:;-0--- American 1---1I European name name i I million I million I mega- I billion I milliard I giga-j I trillion I billion I tera- I quadrillion I billiard I peta- I quintillion I trillion I exa- I sextillion I trilliard , zetta- septillion I quadrillion , yotta- octillion I quadrilliard I. . .. SI prefix nonillion I quintillion ----.,-",....--.r--d-e-c-iII-io-n---, qui ntill iard Everyday Usage of Mathematics and Numbers Numbers are limited when they are not translated into mathematical tools. We observed. that Mathematics has been a tool used on daily basis to achieve the day's task. For example, if you are to buy a car, follow a recipe, go to work or decorate your home, you are bound to use Mathematics principles. People have been using these same principles for thousands of years, across countries and continents. Whether you are sailing a boat off the West African coast or driving on a bend road or building a house in Abuja or Yenogoa, you are using Mathematics to get things done. How can Mathematics be so universal? First, since human beings did not invent Mathematics concepts; but only discovered them, then the language of Mathematics is not English, French or Yoruba, neither is it German or Russian, though there are great mathematicians of old who were Russians and Germans. The language of Mathematics is number. If we are well versed in this language of numbers, it can help us to think rightly, make important decisions and perform everyday tasks. Thus, numbers or Mathematics is the language of the thinkers who is set to make a right decision. Mathematics can help one to shop wisely, spend prudently, 9 buy the right insurance, remodel a home within a budget, understand population growth, or even bet on the football team with the best chance of winning a match. Numbers are so important that we hardly exist without it. So much has been criticized of the difficulty that Mathematics poses to man, but the truth of the matter is that everyday living requires Mathematics. That reminds me of the story of Chinedu who was helping his uncle in a shop to sell vehicle spare parts. While in the class, the teacher asked Chinedu few questions such as; 20 times 8, Chinedu did not know the answer; 50 plus 120, Chinedu missed this sum, and finally 200 minus 60, he could not provide the correct answer. He was reported to the school's Principal who understood very well the background of this boy. The school's Principal said, "Chinedu, I bought goods of N60 and gave you W200", what is my change. He quickly answered, "W140 sir. What of W50 plus W120", he said "W170 sir", and finally the Principal said, "how much do you sell rear light bulb of a motor car in your shop", he said "W20". "If I buy 8 of them, how much will I pay you?" He said, "W160 sir". Then the teacher understood that Chinedu's kind of number or Mathematics is money-wise and not ordinary numbers. Let me take you through the lane on how Mathematics can help us in our daily lives. First of all you need to look at the language of numbers through common situations, such as playing games or cooking. Put your decision-making skills to the test by deciding whether buying or leasing a new car is right for you, and predict how much money you can save for your retirement. What about increasing your savings, you surely need some Mathematics understanding of simple and compound interests. The principles of simple and compound interest are the same whether you are calculating your earnings from a savings account or your gains from shares on stocks. Paying a little attention to these principles could mean big payoffs over time. 10 Mathematics on Phone calls In everyday phone calls, you want to flow along with the network provider and monitor what is the best offer. From MTN to Airtel or Globacom, you are made to create ten friends, call it family and friends. They vary the offer so that you can decide what plan you want to join. You have some 7.5k/sec for family and friends and 20k/sec for other networks or 10k/sec for family and friends and 15k/sec for other networks. You need a little but simple Mathematics to decide which one will payoff. This simply suggests that you will also need some thinking to arrive at the best offer. These network providers are forcing you to think as they are playing with numbers. So we can once again say that "Numbers" is the language of the thinkers. Mathematics of Interest on Savings Account In banking, interest is calculated and added at the end of a certain time period. You might have a savings account that offers a 3% interest rate annually. At the end of each year, the bank multiplies the principal (the amount in the account) by the interest rate of 3% to compute what you have earned in interest. If the interest will drop in your account at a particular date, you must ensure that you do not withdraw from that account before that day so as to earn your full interest. Areas of Rectangles and Curves Mathematics is equally useful in home decorating. Most home decorators need to work within a budget. But in order to figure out what you will spend, you first have to know what you need. How will you know how many rolls of wallpaper or number of yards or metre of decorating cloth to buy if you do not calculate how much wall space you have to cover? Understanding some basic geometry can help you stick to your budget. If your walls are mainly rectangles then it is easier to measure and calculate the amount of cloth needed for the decoration. However, if the walls include a curve then you will need further knowledge of area of a circle which will involve the use of pi. Pi is a constant relating to diameter of a circle and assumes an approximate value of 22/7 or 3.14159 11 Imagine you are planning to buy new carpeting for your home (see Figure 2a). You are going to put down carpeting in the living room, bedroom, and hallway, but not in the bathroom. A simple Mathematics shows that the whole house will need a dimension of 264 sq metres (that is, 22x12). However removing the bathroom from this figure we obtain 229 sq.m. 7m Sm 7m Bedroom Bath Sitting Room Passage Figure 2a Sm 12m 10m 7m Sm 7m Bedroom Bath Sitting Room Figure 2b Sm Passage 12m 10m 6m If your living room has a semi-circular alcove as shown in the floor plan of Figure 2 above, you will need to use additional formula of area of a circle to find its area. A calculation of this alcove part gives 56.5 square metres and this can be added to the earlier floor plan's area of 229 square metres to get the total area you want to carpet as 285.5 square metres. Thus by using geometry, you can buy exactly the amount of carpet you need. Numbers needed for cooking In following a recipe for food and edibles you will agree with me that not all people are chefs, but we are all eaters. Most of us need to learn how to follow a recipe at some point. To create dishes with good flavour, consistency, and texture, the various ingredients must have a kind of relationship to one another. For instance, to make cookies that both look and taste 12 like cookies, you need to make sure you use the right amount of each ingredient. Add too much flour and your cookies will be solid as a rock. Add too much salt and they will taste terrible. The way out of the trouble of not having too much of a particular ingredient is to have a knowledge of mathematical . term called ratios. To know that ingredients have relationships to each other in a recipe is an important concept in cooking. It is also an important Mathematics concept of ratio. If a recipe calls for 1 egg and 2 cups of flour, the relationship of eggs to cups of flour is 1 to 2. In mathematical language, that relationship can be written in two ways: Y2 or 1:2. Both of these express the ratio of eggs to cups of flour: 1 to 2. If you mistakenly alter that ratio, the results may not be edible Mathematics and Numerical Analysis Mr. Vice-Chancellor sir, as one advances in the study of Mathematics, various symbols are being introduced in other to solve many real life problems. For example a problem is posed on the output of a company as follows: The square of the present output of a company equals four times the output in the previous year plus a constant sum of five. What is the output of the company This scenario can be translated into a mathematical equation by writing x2 = 4x + 5 This is then solved by writing ;i2 - 4x - 5 ;; 0 or (x - 5)(x + 1);;:; 0 Which gives x = 5 or -1 Now we may have some equations of this kind which may not be easily solved the way this is done as many real IIf problems may not be as exact in figures as the coefficients of x looks like in this case. In real life we may have some qu tlons describing a phenomenon which may look like 13 3xZ -204.7x- 510.5 = 0 This equation will require the use of a quadratics formula and sometimes may be cumbersome. An alternative approach is to introduce an approximate method which can get the roots of this equation with a value very close to the exact solution. The process of using the alternative and approximate method is another branch of Mathematics called the Numerical Analysis, which is my main area of research. What then is Numerical Analysis? Numerical Analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from Discrete Mathematics). Numerical analysis is also defined as the area of Mathematics and Computer science that creates, analyzes, and implements algorithms for solving numerically the problems of continuous Mathematics. One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection (YBC 7289), which gives a sexagesimal numerical approximation of J2 , the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in carpentry and construction. Thus an approximate method is sort to find the value of 12 The square root of 2 is approximated by sum of fractions as .1 + 24/60 + 51/602 + 10/603 = 1.41421296 ... Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of 12, numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. 14 A numerical method is the same as an algorithm, the steps required to solve a numerical problem. Algorithms became very important as computers were increasingly used to solve problems. It was no longer necessary to solve complex mathematical problems with a single closed form equation. Numerical analysis naturally finds applications in all fields of engineering and the physical sciences and in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in the movement of heavenly bodies (planets, stars and galaxies); numerical linear algebra is important for data analysis. Before the advent of modern computers, numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead. These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations. (Wikipedia, the free encyclopaedia, 2004) Numerical analysis is a mathematical technique that is used to handle problems which originate generally from real-world applications of algebra, geometry, and calculus, and they involve variables which vary continuously. These problems occur throughout the natural sciences, social sciences, medicine, engineering, and business. Beginning in the 1940's, the growth in power and availability of digital computers has led to an increasing use of realistic mathematical models in science, medicine, engineering, and business; and numerical analysis of increasing sophistication has been needed to solve these more accurate and complex mathematical models of the world. The formal academic area of numerical analysis varies from highly theoretical mathematical studies to computer science issues involving the effects of computer hardware and software on the implementation of specific algorithms (Atkinson, 2007). 15 Today, numerical analysis now plays a central role in engineering and in the quantitative parts of pure and applied science. The tasks of numerical analysis include the development of fast and reliable numerical methods together with the provision of a suitable error analysis. The algorithms are developed as computer programs, taking full account of machine architectures such as parallelism (Oaintith, 2004). Thus, numerical analysis involves development of methods and determines the performance of a numerical method being used to solve a problem. It also examines its accuracy, stability and convergence and in general, the efficiency of the algorithm with respect to its order. Areas of Numerical Analysis Numerical analysis has grown so large and wide that it can be categorized into various classes because of the kind of problems they handle and the techniques that are required, although, there is often a great deal of overlaps between the listed areas. In addition, the numerical solution of many mathematical problems involves some combination of some of these areas, possibly all of them. A rough categorization of the principal areas of numerical analysis is given as follows: Systems of linear and nonlinear equations • Numerical solution of systems of linear equations: This refers to solving for x in the system of equation Ax=b with given matrix A and column vector b. The most important case has A a square matrix. There are both direct methods of solution (requiring only a finite number of arithmetic operations) and iterative methods (giving increased accuracy with each new iteration). • Numerical solution of nonlinear equations: This refers to root finding problems which are usually written as f(x)=O with x a vector with n components and f(x) a vector with m components .. • Optimization; This refers to minimizing or maximizing a real-valued function f(x). The permitted values for 16 X=(X1 ,X2, ... ,xn) can be either constrained or unconstrained. The 'linear programming problem' is a well-known and important case; f(x) is linear, and there are linear equality and/or inequality constraints on x. Approximation theory This is the use of computable functions p(x) to approximate the values of functions f(x) that are not easily computable or use approximations to simplify dealing with such functions. The most popular types of computable functions p(x) are polynomials, rational functions, and piecewise versions of them, for example spline functions. Trigonometric polynomials are also a very useful choice. • A given function f(x) is approximated within a given finite-dimensional family of computable functions. The quality of the approximation often depends on the technique adopted. • Numerical integration and differentiation: Furthermore, for approximation theory, most integrals cannot be evaluated directly in terms of elementary functions, and instead they must be approximated numerically. Although, most functions can be differentiated analytically, but there is still a need for numerical differentiation, both to approximate the derivative of numerical data and to obtain approximations for discretizing differential equations. Numerical Solution of Differential and Integral Equations These equations occur widely as mathematical models for the physical world, and their numerical solution is important throughout the sciences and engineering. • Ordinary differential equations: This refers to systems of differential equations in which the unknown solutions are functions of only a single variable. The most important cases are initial value problems and boundary value problems, and these are the subjects of a number of textbooks. Of more recent interest are 'differential- algebraic equations', which are mixed systems of algebraic equations and ordinary differential equations. 17 This is a part of my research problems in the last decade. Also we have delay differential equations', which also is receiving attention from the Numerical Analysts. • Solution of Partial differential equations: These equations occur in almost all areas of engineering, and many basic models of the physical sciences are given as partial differential equations. Thus such equations are a very important topic for numerical analysis. For example, the Navier-Stokes equations are the main theoretical model for the motion of fluids, and the very large area of 'computational fluid mechanics' is concerned with solving numerically these and other equations of fluid dynamics • Furthermore, Integral equations involve the integration of an unknown function, and linear equations probably occur most frequently. Some mathematical models lead directly to integral equations of which their solutions are difficult to obtain in closed form, hence numerical analysis is the best option to generate such solutions. The Concerns in Numerical Analysis Most numerical analysts specialize in small sub-areas of the areas listed above, but they share some common concerns and perspectives. These include the following. • Since numerical analysis is based on approximation, it makes use of every possible assumption in order to get a solution to a problem. Thus a numerical analyst believes that if you cannot solve a problem directly, then replace it with a 'nearby problem' which can be solved more easily. This is an important perspective which cuts across all types of mathematical problems. For example, to evaluate a definite integral numerically, begin by approximating its integrand using polynomial interpolation or a Taylor series, and then integrate exactly the polynomial approximation. • All numerical calculations are carried out using finite precision arithmetic, usually in a framework of floating- point representation of numbers. What are the effects of using such finite precision computer arithmetic? How are arithmetic calculations to be carried out? Using finite precision arithmetic will affect how we compute solutions 18 to all types of problems, and it forces us to think about the limits on the accuracy with which a problem can be solved numerically. • There is a concern with 'stability', a concept referring to the sensitivity of the solution of a given problem to small changes in the data or the given parameters of the problem. There are two aspects to this. First, how sensitive is the original problem to small changes in the data of the problem? Second, the numerical method should not introduce additional sensitivity that is not present in the original mathematical problem being solved. In developing a numerical method to solve a problem, the method should be no more sensitive to changes in the data than is true of the original mathematical problem. • There is a fundamental concern with error, its size, and its analytic form. When approximating a problem, a numerical analyst would want to understand the behaviour of the error in the computed solution. Understanding the form of the error may allow one to minimize or estimate it. The concern is to develop algorithms that will produce a very minimal error when compared with the analytical solution of a problem. In the process of doing such, the stability of the method and the step function evaluation is put into consideration. A 'forward error analysis' looks at the effect of errors made in the solution process. This is the standard way of understanding the consequences of the approximation errors that occur in setting up a numerical method of solution. Modern Numerical Analysis Modern numerical analysis can be credibly said to begin with the 1947 paper by John von Neumann and Herman Goldstine, "Numerical Inverting of Matrices of High Order" (Bulletin of the AMS, Nov. 1947). It is one of the first papers to study rounding error, and include discussion of what today is called scientific computing. Although numerical analysis has a longer and richer history, "modern" numerical analysis, as USed here, i characterized by the synergy of the programmable electronic computer, mathematical analysis, and the opportunity nd 19 need to solve large and complex problems in applications. The need for advances in applications, such as ballistics prediction, neutron transport, and non-steady, multidimensional fluid dynamics drives the development of the computer and depended strongly on advances in numerical analysis and mathematical modelling. Modern numerical analysis and scientific computing develops quickly and on many fronts. Our current focus is on numerical methods for solving ordinary differential equations, methods of approximation of functions and the impact of these developments on science and technology. Current interest is the impact of mathematical software packages, accuracy of the schemes being developed and their properties. Numerical Analysis and Differential Equations In this lecture, I will not be able to discuss all the works done on numerical analysis but at this point I will like to give some basic definitions of differential equations and their types which are useful to numerical method for solving ordinary differential equations. Any equation containing differential coefficients dy , d 2 ;;, (dy)2, etc is called a differential dx dx dx (Okunuga, 2008a) Where dy is the rate of change of y with respect to x dx Examples of some differential equations include: 1. dy -5y =0 dx a? (d )22~+xy 1 =0 dx? dx av + av = 1 as at such as equation 2. 3. 20 4. ? ? 2 O-ll + O-U + ~ = 0 ? 2 ?O;C oy Dz: The above examples show that there are different types of differential equations, as all of these do not belong to the same class. Hence differential equations are broadly classified into two groups: (i) Ordinary Differential Equation (ODE) (ii) Partial Differential Equation (PDE) A differential equation involving ordinary derivatives (or total derivatives) with respect to a single independent variable is called an Ordinary Differential Equation. Examples of ODEs are Nos 1 and 2 above. A differential equation involving partial derivatives of one or more dependent variables with respect to more than one independent variable is called a Partial Differential Equation (PDE). Examples 3 and 4 above are PDEs A general first order differential equation with a condition specified at the initial point is called an Initial Value Problem (IVP). Thus an IVP is written as dy =f(x,y) , y(xO)=yO (1) dx Where (xO, yO) are the initial values which permit the solution of the differential equation to be unique. A stiff differential equation is an equation with a solution involving fast decaying parameters. These basic definitions guide the areas of my research work. Numerical methods for solving differential equation Often we examine simple initial value problem of the form (1). This problem serves as the standard form for all first order IVPs from which numerical scheme can be developed. The simplest numerical method ever proposed to solve this IVP Is 21 the one-step method of Euler. Although Euler has other methods, the explicit Euler scheme given by Y:1-1 = 3'n '7 hIll This is known as the simplest numerical method for solving IVPs. There are other one step methods including the Runge- Kutta methods. The Euler method, though simple, is often used to generate starting values for several other methods, however, this method is not so accurate for the purpose of approximation. Several other numerical methods were developed in early days of numerical analysis but they were developed for desk computation prior to computer era. In the attempt to develop more numerical methods, it was more and more of using numbers, approximations and error analysis. This leads to the fact that human generally appreciates numbers rather than abstract terms which may not be easily comprehended and hence not appreciated. To this end we observe that numerical analysis tends to deal with numbers as a way of presenting concrete answers to mathematical problems. This idea of numbers which many easily comprehend is also part of God's method of dealing with man. It is noted that since Mathematics has many branches, starting with numbers, then arithmetic, later to symbols and development of equations and difficult terminologies, it got to a point that we started going back to numbers as many abstract Mathematics require meaning which can only be obtained via numbers. This is what gave birth to computer era, and today all of us use computer for our daily operations. Do you know that everything you type on a computer is not recognised the way you input them? What computer does is to translate whatever you type into it to Os and 1s. In other words computer codes are 0 and 1 and that is what computer uses to give you your answer. The computer re-codes everything you send into it, be it sentences or figure, into 0 and 1 and then process your request. The reason for this is that those who developed computer looked for simplest base numeral to code 22 it. The lowest number base to operate with is base 2 numeral which permits only 0 and 1 as its digits, rather than base ten which uses 0, 1,2, ... ,9 as its figures. If man can do this I want to prove to you that God is a better computational mathematician that can compute with any base number and I will show this briefly. God is the author of Mathematics Mr. Vice-Chancellor sir, as an Applied Mathematician and a Numerical Analyst in particular, I apply my algorithms to treat both physical and spiritual problems. Hence, I will dwell a bit on what God says about numbers. God created numbers for use and He also work by numbers. I will give some points from the Holy Bible about how numbers and Mathematics were greatly used. Point 1 Numbers are seen to be so important that the Psalmist says teach us to number our days so as to apply our hearts to wisdom. So my first submission is that wisdom emanates from numbers. Therefore everyone who loves number will necessarily be a thinker and thinkers are men who invest on wisdom. Point 2 It takes a great thinker to think like God thinks. I see the way God does his counting as against that of man, which makes numbers to be the language of the thinkers: "... that with the Lord one day is as a thousand years, and a thousand years as one day. "(2 Peter 3:8) It seems God is not bounded by time. Event of one thousand years can be accomplished in a day. If you can think like God you can be like Him. Point 3 Imagine this sequence of numbers: 1000, 10000, 19000, 28000, . .. This is A.P with d= 9000 103,104,105,106 , This is GP with r = 10 103, 104, 106, 109, This is also a sequence 23 Now, Deuteronomy 32:30 says "One will chase a thousand, and two shall put ten thousand to flight': God deals with ratio, sequences and series. The ratio is 1 to 103 and 2 to 104 . But the series in not clear since we do not know the third term of the series, it may even not be a geometric series. So it is better to call it "Godometric" series. (a series defined by God). Point 4 Do you know that God loves numbers and work specifically with certain numbers? May be the thinkers will work with such numbers. There are some numbers that are unique in the Holy Bible. I give a list of the prominent ones. They include: 1, 3, 7, 12, 40, 70, 120. Number "1" refers to one God or unity Number '3' is very essential; it implies Trinity (agreement) The number '3' describes the Trinity - The Father, the Son and the Holy Spirit. The bible says these Three are One. In God's Mathematics three equals one (3 = 1). The Bible says man is a triune being. He is a spirit, he has a soul and lives in a body (I Thessalonians 5:23). The combination of these three components make up what God calls man. Once again three entities equal one (3 = 1). In a simple mathematical equation we can write Spirit + Soul + Body = Man where spirit is God's breath in man, soul is the mind and intellect, and body is the shape or house for the spirit and soul to dwell. This is why God said "let us make man in our own image". The Triune God created a triune being called man (Genesis 1: 26; 2:7). This also linked with calculus idea as we can easily deduce that: J bo dv d(spirir) = man 24 d (77 ar ) 1 = dead bodvdsou . d (man) -'--- = lit,"ing soul dspir it d (77 an) .. ---=sp!nt dbody Do you know that calculus is in the Holy book? I will not dWE on this in this lecture but it suffice to say that John 15: 1- gave an example of integration into the Vine. God declared himself as a triune God. God also declare himself as God of Abraham, God of Isaac and God of Jaco (no more). The 3 are the people that God made 1-1 covenant with an declared to his people "I am the God of Abraham, Isaac an Jacob". So number "3" seems to be important to God Furthermore, among all the disciples of Jesus Christ, '3' c them belong to what we can call the Inner Circle. Peter, Jame and John were the three closest to Jesus (Luke 8: 51). Thes three were the ones present when special miracles were done They were the ones at transfiguration. They were the one when Christ went to pray at Gethsemane and more ... In th transfiguration, only 3 people appeared - Jesus, Elijah an Moses. A three cord is not easily broken. ' The number 3 is for completeness. Hence by mathematic, induction we can state a corollary that: The necessary and sufficient condition is that a comp/et man must be a three in one man (that is called Okunri. meta in Yoruba) The number "7" stands for perfection. Seven describe perfection. The Bible talks of the '7' spirits of God and thes seven spirits rests on Jesus for completeness (see Isaiah 11 2, Revelation 5: 6, 4: 5). The Lamb that was slain received 7 virtues of God for mankinc Revelations 5: 12 says: "Worthy is the Lamb that was slain t. 25 receive (i) power, and (ii) riches, and (iii) wisdom, and (iv strength, and (v) honour, and (vi) glory, and (vii) blessing" Have you considered number 12? The number twelve (12) i: unique in the sense that God loves working with 12. Jacob b: divine agenda (I do not think it is by accident), had 12 son: which became 12 tribes of Israel till today. Exodus 15:27: And they came to Elim, where there were 12 wells of water, ... and they encamped there by the waters. Solomon had 12 officers over all Israel, (1Kings 4:7). Jesus had 12 disciples. (Luke 6:13) In one of the miracles where Jesus fed 5000 people, we wen told that " ... they took up 12 baskets full ofthe fragments, ani of the fishes" after everyone had eaten. In the Book of Revelation alone, the number 12 was mentionei ten times, which talks of 12 gates, 12 stars 12 pearls, etc. Point 5: The Number "Forty" The number 40 is so significant in the Bible that we need tl pay close attention to it. It is applicable to individual lives ani the nation. The events of children of Israel were severall described with number 40 (a) Moses life time was partitioned or collocated into thre parts. The first collocation point is when he was 40 years when he came to visit his brethren. 40 years later Go called him to lead his people. And for 40 years he lei them through the wilderness. 40 years later after he wa called by God, he died. Moses life was divided into thre significant and equal 40 years. The thinkers will thin about number 40. (b) God fed the children of Israel directly from heaven wit manna in the wilderness for 40 years. Miracles! (c) For 40 years the children of Israel grieved God in th wilderness (Hebrews 3:9, 17; 8:2). These numbers call fc thinking that is why this lecture is titled, Numbers: Th language of the thinkers. 26 (d) David reigned as king for 40 years. Solomon reigned as king for 40 years. Joash reigned as king for 40 years. (1 Kings 2: 11; 1 Kings 11:42; 2 Kings 12: 1) Moses, Elijah and Jesus fasted for 40 days. 40 days plan and 40 day agenda can lead to some giant results, even the vision 20 - 2020 talks of 20 in the vision with any of the 20s to give 40. Pont 6 Do you know that God uses simple mathematical operations like addition, multiplication and division? There are simple operations we use every day to get to our destination. Consider words like 'addition', 'multiplication'. They sound like mathematical terms. God like using those two words very well. The word add or added appears in the Bible 47 times. The word "multiply" or similar words to it appears 87 times. The following are examples of these words: But seek ye first the kingdom of God, and his righteousness; and all these things shall be added unto you. (Mat 6:33) Grace and peace be multiplied unto you through the knowledge of God, and of Jesus our Lord, (2Peter 1:2) And beside this, giving all diligence, add to your faith virtue; and to virtue knowledge; (2Peter 1:5) My son, forget not my law; but let thine heart keep my commandments: For length of days, and long life, and peace, shall they add to thee. (Proverbs 3:1-2) I have heard thy prayer, I have seen thy tears: behold, I will add unto thy days fifteen years. (Isaiah 38:5 I 2Kings 20:6) And the angel of the LORD said unto her. I will multiply thy seed exceedingly, that it shall not be numbered for multitude. (Genesis 16:10) 27 Saying, Surely blessing I will bless thee, and multiplying I will multiplv thee. (Hebrews 6:14) Point 7 God is a computational Mathematician In human Mathematics, there are topics such as Differentiation and Integration. Every function that is differentiated and then integrated cannot return to the same function except under certain conditions. John 12:24-25 "Verily, verily, I say unto you, Except a corn of wheat fall into the ground and die, it abideth alone: but if it die, it bringeth forth much fruit. He that loveth his life shall lose it; and he that hateth his life in this world shall keep it unto life eterna/". Jesus looked at this word not only from agricultural point but in a mathematical way. He is saying here that a corn will remain a corn and even will not last long before it withers away if it's not planted. A corn that is planted is in the ground and dies but with all the nutrients of the soil around it is integrated to become hundreds of seeds of corn. One corn becomes hundreds of corns. One grain of corn that dies comes alive to feed a family and a nation. This is calculus at work. The variable here is the corn which is differentiated with respect to the soil and under certain limits and condition. The process of getting the corn at harvest can be represented purely by a differential equation, which is either solved analytically or numerically. The solution that God provides here is that of a computational mathematician who will like to count the seed at the harvest because we deal with numbers as a numerical analyst. My Contributions to the field of Numerical Analysis I will like to discuss some of my contributions in the field of numerical methods, which are developed as suitable schemes for solving different classes of ordinary differential equations. I will like to divide my work into 5 segments, which will summarize most of the work which I have done to date. 28 The Collocation-Tau Methods Lanczos [1956] introduced the Standard Collocation method with some selected points. However, Fox and Parker introduced the use of Chebyshev polynomials in collocating the existing method which was captioned as the Lanczos-Tau method (Fox and Parker, 1968). Also, Ortiz (1969) went on to discuss the general Tau method which was later extended by Onumanyi and Ortiz (1984), to a method known as the Collocation-Tau method. The Standard Collocation method with method of selected points provides a direct extension of the Tau method to linear ODEs with non-polynomial coefficients. The Collocation -Tau method however uses the Chebyshev perturbation terms to select the collocation points. Okunuga and Onumanyi (1985, 1986) gave the generalized Tau method which permits exact fractional values in the computation with more than one T- term as perturbation on the right hand side of the linear differential equation. The novel approach introduced by the authors developed accurate collocation methods by various types for the solution of ordinary differential equations. For a higher order scalar differential equation, numerical results show that the proposed method is more accurate than the Standard Collocation method for the same degree of polynomial approximation. We observed that when differential equation with polynomial coefficients is involved, the new method gives identical results with the Lanczos Tau method. Furthermore, this method provides a direct approach of extending it to non-linear differential equations. Thus, Okunuga and Sofoluwe (1990) modified the Tau method to accommodate non-linear differential equations and by Newton linearization processes, this new Collocation method was applied on the problem describing the meridian of the dropped shaped tank. The problem has some singularities at the initial point which make it more difficult to solve by many existing numerical methods. However our Collocation method applied on this problem 9 ve some high accuracy as reported in our work. 29 Exponential Fitted Schemes with Composite Formulas The author over the years introduced the exponentially fitted formulas which were found to be quite suitable for solution of stiff initial problems. Stiff differential equations usually pose some difficulties for several numerical schemes, as many of them usually fail to cope with the fast decaying nature of the solution of stiff problems. These often are problems that emanate from modern Physics and astronomical problems. Also some are due to chemical kinetic reactions. The author developed orders 2 to 6 exponentially fitted methods which is a form of the Multiderivative Linear Multistep Method (MLMM) and suitable for handling the stiff problem that permits exponential fitting. (Okunuga, 1999 (a), (b)) The methods with low orders were derived to give a unified model that combines both the predictor and the corrector methods during the process of exponential fitting. The higher order of the methods introduced some other variants such as Pade approximation so as to obtain a better result and satisfying some good stability criteria (Okunuga & Sofoluwe 2008). The methods developed in all of these papers were seen to be very accurate, convergent and satisfied the A-stability and zero stability conditions. In further research carried out, some composite integration formula were developed and tested on systems of non-linear IVPs resulting from chemical kinetic reaction problem. The results show that order 4 scheme exhibits the highest accuracy and was recommended for use to all users of numerical methods when solving stiff problems. In Okunuga 1998, the stability conditions for which all classes of Multiderivative Linear Multistep Method (MLMM) must satisfied were discussed including the convergence of the methods. The author showed in his work that all the orders 2 to 6 methods of the MLMM are A- stable, a condition which is difficult for many numerical schemes to satisfy. Furthermore, the schemes satisfied the stiff stability condition and they also satisfied the definitions of zero and absolute 30 stabilities. The article shed some light on how accuracy can be improved upon when solving stiff IVPs .. The results obtained by using our schemes when compared with results of other authors were found to be more accurate {Okunuga, 2009). Abhulimen & Okunuga (2008) developed a more difficult scheme by extending the earlier work done by Okunuga (1999(a)). The authors developed a fifth order of the MLMM with higher derivatives that excludes the Pade approximation in lower derivative scheme earlier proposed by Okunuga & Sofoluwe (2008). The price paid for including the higher derivatives in the development of the new scheme, was justified by the accuracy of the results obtained when implemented on standard problems. - A class of two-step second derivative Linear Multistep Method with some exponential fittings for order two was developed for generating solutions for Stiff IVPs (Okunuga, 2009). The resulting integration formula is applied on systems of stiff problems in Predictor-Corrector mode. The predictor formula corresponding to the case k = 2 with one free parameter is obtained as Yn+2- Yn =h{sf1+2 +(2-s)fJ (3) while the corrector method obtained is given by Yn+2- Yn =h{rf,,+3 +t(2-3r)f1+2 +t(2+r)fJ (4) By introducing exponential fitting, we write the two formulas as a single formula to get Yn+2= 1+rqR*(q)+Xq(r+2) =R(q). (5) y" 1+ Xq(3r-2) Equation (5) unites both the predictor and the corrector formula. The formula (5) together with sand r are used to generate solutions to stiff problems for which exponential fittings are applicable. The method derived was implemented on some standard problems and the result obtained gave a very high accuracy 31 compared to some known methods. This is partly due to the choice of the free parameters with the advantage of the exponential fitting. The peculiarity of order 2 formula derived in this paper is that it is simple compared to higher order of the MLMM for example the fourth order given by Okunuga (1999a). Furthermore its accuracy is high enough compared to the exact solution. To illustrate some of our results, the Pade Exponential scheme was used to solve some Stiff problem discussed by Oalquist. Problem 1 Oalquist and Bjock (1974) showed that there are some stiff problems for which Runge-Kutta (R-K) method is unsuitable. One of such problems is the second order stiff differential equation y" + 1001y' + 1000y = 0 (6) y(O) = 1, y'(O) = -1 This problem has a general solution given by Y(x) = Ae-x + Be-10oox And for solution in [0,1], the exact solution is given as y(x) = e-x By using the explicit fourth order R-K method, Oalquist showed that the method explodes for a step length h > 0.0027. A similar argument is adduced by Shampine and Gladwell (2003) despite that this is unsatisfactory step size for describing the function e-x . On the contrary, this same problem was solved using our fifth order method with step- lengths greater than 0.0027. The results obtained at x = 1 using both the R-K method and the Pace-Exponential Formula (PEF) are given in Table 1 below. 32 T bl 1 E fIR It S d d ODEa e xpenmen a esu on econ o~ er h Formula y(1) Error 0.0027 R-K 0.367885000 5.6 x 10-6 PEF 0.367879435 2.3 x 10.13 0.05 R-K 4608.72 4.6 x 103 PEF 0.367879435 5.6 x 10.12 0.1 PEF 0.367879435 9.6 x 10.12 R-K - Explodes Exact Solution 0.367879435 - It was observed that the explicit R-K method could not cope with this problem for h>0.0027. However, the order five Pade- exponential formula gave a far better accuracy than the explicit R-K scheme. Problem 2 Also, we considered the Chemical Kinetic Problem (7). Gear (1971) discussed the application of stiffly-stable integration formulae based on backward difference approximation of the derivative and used the following example to illustrate the method v. = - 0.013 Y1+ 1000 Y1Y3 y1 (0) = 1 } y'2 = 2500 Y2Y3 y2(0) = 1 y'3 = 0.013 Y1-1000 Y1Y3- 2500 Y2Y3 Y3(0) = 0 (7) This problem is an application of chemical reaction kinetic in which Yi represents the concentration of a very reactive species which is an intermediate in the course of the reaction and always stays small. Y1 and Y2 are monotonically decreasing and increasing respectively while Y3 increases to a maximum and thereafter is monotonically decreasing. This problem is considered using our fifth order Pade-exponential formula and the result obtained shows that the accuracy is high compared to the exact solution (see Table 2 below). Table 2: Experimental results on non-linear stiff problem·using the 5th order PEF steplength Absolute Error Absolute Error Absolute Error v1(1) vi1) V3(1) 0.0625 4.4 x 10.14 4.4 x 10.14 1.3 X 10.13 0.05 4.4 x 10.14 -4.4 x 10.14 -8.9 x 10.15 0.1 6.7 x 10-13 6.7 X 10.13 1.8 X 10-12 It will be observed from Table 2 that for any of the step-lengths used, the method is quite accurate. Hence the integration formula is very efficient. Developments of Multiderivative Explicit Runge-Kutta (MERK) Methods The author together with his research group, also did some work on the popular Runge-Kutta methods in a bid to improve on the existing methods. It was noted that explicit Runge-Kutta (ERK) methods have some limitations among which are the maximum attainable orders of the methods. Traditionally, the maximum attainable order of an s-stage Explicit Runge-Kutta method is not greater than s. However, we developed a 2- stage ERK Method of Order 4, which is twice as high as the expected order for a 2-stage method. The paper also discussed the stability of the method (Akanbi et aI., 2008 (a), (b)). We deduced that ~ Four of our methods with stability functions P1, P3, P4, P5 have the same region of absolute stability (-2.78529 < h < 0) with the 4 stage classical methods of order 4; ~ Methods with stability functions P8 (-3.1195 < h < 0), P, (- - - 2.93842 < h < 0), P11 and P12 (- 2.78529 < h < 0) have wider regions of absolute stability than the 4 stage classical methods of order 4; ~ Whereas 4 - stage classical methods of order 4 require four internal function evaluations, these newly derived methods require two internal function evaluations to advance from xn to xn + 1 because they are 2-stage methods, thus preferred. 34 The newly derived schemes given were implemented on a system of Differential - Algebraic Equation (DAE) having some singularities. Butcher (2003) discussed the difficulties that are inherent in DAEs and their solutions. It is a known fact that rigorous researches suitable for these class of problems and probably simpler methods of handling the DAEs are still on. Thus, we considered the DAE (9) given by Lambert (1991). v' + yz = 0 y2 + yz + 1 = 0 } y(O) = 0 The analytical solution of the DAE (1) is y(x)=tan(x+~)" z(x)=-y-f, This problem requires a scheme of a very high order of accuracy due to the singularity occurring at X = ~. The computational results using the methods derived in this paper were compared with the existing methods. (9) As far as our results showed, the newly developed algorithm was seen to be the most efficient when compared with other known ones. This is illustrated in Figures 4 below. 10 Graph of Numerical .1 _. -RK2s2p 8 -!-------~I/C-------__l -- Goeken2s3 p 6-i-----~--------__1---P_1 I __ NeW_RK_21 >- /---- s4fJ.1 4 -1----#------,.-.::''-.' -------1 - P_8 New_RK_2 ~ ,,~~~~~~~::::::::::~ s4fJ.2 2 - -P_9 • . . - • Nevl_RK_2 o ------- ~'5~ 1 5 2 2 New_RK_2. x 1 s~4 Graph of Numerical Experiment for h = 0.1 ... -- Figure 4: 35 The research group went further to examine some of the properties of RK methods which were developed. One of the characteristics of selecting a good numerical algorithm for Initial Value Problems is the error bounds on the method. We examined the error bounds of the 2 - stage Multiderivative Explicit Runge-Kutta methods. The choice of step lengths for this new class of methods is also discussed. Interestingly this new class of methods has higher bounds when compared with the standard explicit Runge-Kutta methods (See Okunuga & Akanbi 2008; Akanbi et al. 2008(a)). In our work, we discussed these errors in relation to the new 2 - stage Multidervative Explicit Runge-Kutta (MERK) methods. Consequently, we tested our claims on two problems. The first one is the IVP (Fatunla, 1988; Lambert, 1973). y'(x)=-lO(y(x)-lY,y(O)=2,0