BMTE-144

About the Course

Mathematical modelling of physical/biological problems generally give rise to ordinary or partial differential equations or integral equations or in terms of a set of such equations. A number of these problems can be solved exactly by using analytical methods but most of the time it is not possible to get the exact solution. Thus, a need arises to devise numerical methods to solve these problems. These methods of solving such problems may give rise to a system of algebraic equations or a non-linear equation or system of non-linear equations. The numerical solutions of these systems of equations are quantitative in nature but when interpreted give qualitative results and are very useful. Numerical analysis deals with the development and analysis of the numerical methods. We are offering this course of numerical analysis to students entering the B.Sc. (G)/B.A. (G) as an elective course in sixth semester.

It was in the year 1624 that the English mathematician Henry Briggs used a numerical procedure to construct his celebrated table of logarithms. The interpolation problem was first taken up by Briggs but was solved by the 17^th century mathematicians and physicists, Sir Isacc Newton and James Gregory. Later on, other problems were considered and solved by more and more efficient methods. In recent years the invention and development of electronic calculators/computers have strongly influenced the development of numerical analysis.

This course assumes the knowledge of the course BMTC-131 on calculus. Number of results from linear algebra is also used in this course. These results have been stated wherever required. For details of these results our linear algebra course (BMTE-141) may be referred. This course is divided into 2 volumes. First volume consists of 2 blocks. The first block, deals with the problem of finding approximate roots of a non-linear equation in one unknown. We have started the block with a recall of several important theorems from calculus which are referred to throughout the course. After introducing the concept of ‘error’ that arise due to approximations, we have discussed two basic approximation methods, namely, bisection and fixed point iteration methods and commonly used chord methods, namely, regula-falsi, secant and Newton-Raphson methods. In Block 2, we have considered the problem of finding the solution of systems of linear equations. We have discussed both direct and iterative methods of solving systems of linear equations.

Second volume consists of 2 blocks. Block 3 deals with the theory of interpolation. Here, we are concerned only with polynomial interpolation. The existence and uniqueness of interpolating polynomials are discussed. Several forms of interpolating polynomials like Lagrange’s and Newton’s divided difference forms with error terms are discussed. This block concludes with a discussion on Newton’s forward and backward difference form. In Block 4, using interpolating polynomials we have obtained numerical differentiation and integration formulae together with their error terms. After a brief introduction to difference equations the numerical solution of the first order ordinary differential equation is dealt with. More precisely, Taylor series, Euler’s method of second order and Runge Kutta method of order two, three and four are derived with error terms for the solution of differential equations.

Each block consists of 3 to 4 units. All the concepts given in the units are followed by a number of examples as well as exercises. These will help you get a better grasp of the techniques discussed in this course. We have used a non-programmable scientific calculator for doing computations throughout the course. While attempting the exercises given in the units, you would also need a calculator. The solutions/answers to the exercises in a unit are given at the end of the unit. We suggest that you look at them only after attempting the exercises. A list of symbols and notations are also given in for your reference.

Now, a few words about the way you should study the material! We have presented this course with the assumption that you have already studied the courses ‘Calculus’ (BMTC-131) and ‘Linear Algebra’ (BMTE-141). Do not simply read the material in these blocks. You must actually interact with every line, idea, example and question in it. As you know, whenever we introduce concepts, we give you a lot of concrete examples to help you understand it. We also include a list of exercises to help you strengthen your understanding of the concepts and processes concerned. You must solve every exercise as you come to it, to benefit from it.

Now, a word about the layout of a block. In each block, you will first find a block introduction, followed by a list of symbols that are used in the block. And then come the units of the block. Every unit starts with an introduction, where we also list the precise learning objectives of the unit. Each unit has been divided into sections. Since the material in the different units is heavily interlinked, we will be doing a lot of cross-referencing. For this purpose we will be using the notation Sec. to mean Section y of Unit x. As in your earlier mathematics courses, you will find the examples, exercises and important equations numbered sequentially throughout a unit.

Further, the exercises in each unit are interspersed within the text. They are meant to help you check your progress. The solutions, or solution outlines, or answers to the exercises in a unit are given at the end of the unit. After you finish studying a unit, please go back to the objectives of the unit (given in the introduction of the unit), and see if you are confident that you have achieved them.

As part of the tutorial component, at the end of every block you will find several miscellaneous examples and exercises for you to do. These are based on all the units you would have studied upto that point. Further, we will send you an assignment. It is meant to be a teaching aid, apart from an assessment aid. Your academic counsellor, at the study centre, will assess this, and return it to you with suitable detailed remarks.

You may like to look up some more books on the subject and try to solve some exercises given in them. This will help you get a better grasp of the techniques discussed in this course. We are giving you a list of titles which will be available in your study centre for reference purposes. Also, we are listing a few useful websites where you can access the content related to your course.

Some Useful Books

1. Numerical Methods for Scientific and Engineering Computation by M.K. Jain, S.R.K. Iyengar, R.K. Jain, from New Age International Publishers.

2. Elementary Numerical Analysis: An Algorithmic Approach, by Samuel D. Conte and Garl De Boor, McGraw Hill Publications.

Some Useful Websites

1. https://freevideolectures.com/course/3597/numerical-analysis

2. https://nptel.ac.in/courses/111/107/111107105/

3. https://ocw.mit.edu/courses/mathematics/18-335j-introduction-to-numerical-methods-spring-2019/

4. https://ocw.mit.edu/courses/mathematics/18-330-introduction-to-numerical-analysis-spring-2012/

Wishing you a happy learning experience!