Preface The objective of Numerical Methods is to solve complex numerical problems using simple arithmetic operations in order to develop and evaluate methods for computing numerical results from given data. Before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. In 20th century, after the advent of computer; numerical methods are the methods for solving problems on computers through calculations which results in table of numbers and/or graphical representations/ figures. This book is written in simple and easy language, in systematic manner, student- friendly and numerical problem solving orientation. Emphasis should be given on examples rather than pure theory. The theory part should be kept in minimum and presented in a heuristic and intuituve manner. Each concept can be justified with the help of examples (which is unavailable in other books) as student may come dilemma to find the solution of the concept from other books. So learning is with the help of examples as examples are the best source to learn and remember that particular problem. At the end of chapters, excercise questions will be given. This book is meant for undergraduate and graduate students of engineering, science and mathematics i.e., B.Tech (Chemical, Electrical, Production, Mechanical, Civil, Petroleum Engineering), BA (Math), B.Sc (Math), B.Sc (Computer Applications), BCA. The book is also useful for the instructors. In addition to this, the book is useful for engineers, managers and researchers to learn the basic principles and concepts of Numerical Methods for the purpose of design and analysis. The book can be adapted for short course on numerical methods.

1.1 Introduction to Numerical Method There are three types by which problem / system can be solved and are analytical methods, graphical methods and numerical methods. Analytical methods: This approach or methods are used to find the exact solution of simple problems / systems rather than complex systems. It provides the behavior and characteristic features of a particular system, but can only be used for systems which can be approximated through use of linear models. Another limitation is in the case of practical problems, consisting of complex and nonlinear equations / processes, as analytical methods cannot handle properly. Graphical methods: This approach or methods are used to find the approximate solution of problems / systems through drawing of graphs for each of the equation and then check for the intersection of graphs. Intersection of graphs represents solution to system of equations, and no interaction or parallel lines of graphs represents no solution. This is not useful because of non-precise nature due to graphs drawing through hands, which also makes the process tedious. Another limitation is its viability for problems involving three or lesser dimensions. Numerical methods: This approach or methods are used to find the approximate solution of problems / systems in a simpler way involving a large quantity of long arithmetic calculations, as compared to analytical methods. Numerical methods are the techniques by which mathematical problems are formulated to find the solution through arithmetic operations.

2.1 Introduction Consider two linear equations in two variables i.e., x and y, such that equations are

4.1 Introduction Numerical differentiation is the process of computing the function’s derivative values from the given set of numerical values when the actual form of the function is not known. Similar to the numerical interpolation, a number of formulae for differentiation are derived in this chapter and are Derivatives Based on Newton’s Forward Interpolation Formula: This is used to compute the derivative for some given x lying near the beginning of the data table. Derivatives Based on Newton’s Backward Interpolation Formula: This is used to compute the derivative for a point near the end of the data table. Derivatives Based on Stirling’s and Bessel’s Interpolation Formula: This is used to compute the derivative for some point lying in the middle of the data table. Derivatives Based on Newton’s Divided Difference Formula: This is used to compute the derivative for some point when the data table is not equi- spaced.

5.1 Introduction In applied mathematics the solution of problem generally consists of numbers which satisfy some kind of equation which are generally specified by numbers. But in practice it is found that these equations not always possible to express numerically or in exact decimal representative of the solution. Numerical methods are very important tools to provide practical methods for calculating the solutions of applied mathematics to a desired degree of accuracy. The wide use of electronic computers for solving problems in the fields of engineering, scientific, industry etc that further enhanced the scope of numerical methods. The calculus of finite differences deals with the changes in the values of the function (dependent variable) due to changes in the independent variables. Again through this the relation of the values of the function is studied whenever the independent variable changes by finite jumps whether equal or unequal.

6.1 Introduction Scientists and engineers have an experimental data of various variables in an experiment. To know the facts and to formulate related theories, they need exact relationship between such variables. So, they require the exact function which defines the outputs with introducing the inputs. For example, Ohm’s law: which states that voltage of a circuit is proportional to the current flowing in it i.e.

7.1 Introduction A Scottish mathematician David Gibb was firstly introduced a term ‘Numerical Integration’ in 1915. Like numerical differentiation, numerical integration is also a very useful tool for scientists and engineers especially when they have an experimental data for different variables without knowing any mathematical relationship between them. Beside it, there are also many other situations where it plays an important role for integrating the functions like:

8.1 Introduction In the fields like Physics, Chemistry, Biology, Engineering and Economics; their laws and principles contains various parameters and their derivatives. The mathematical relationships of such parameters with their derivatives constitute the differential equations. In calculus, differential equations were firstly introduced by Newton and Leibnitz. Newton listed the three kinds of differential equations:

Index A Absolute Errors 13 Accuracy 13 Adams-Bashforth Method 270 Adjoint of a Square Matrix 31 Aitken Exploration 123 Aitken’s D2 Method 123 Algebraic equation 93 Analytical methods 1 Argument 147 Augmented Matrix 42 Average operator 154