Numerical Methods and Statistical Techniques Using C

The limitations of analytical methods in practical applications have led mathematicians to evolve numerical methods. We know that exact methods often fail in drawing plausible inferences from a given set of tabulated data or in finding roots of transcendental equations or in solving non-linear differential equations. Even if analytical solutions are available, they are not amenable to direct numerical interpretation.

Error Analysis and Estimation

Consider the equation of the form f(x) = 0 If f(x) is a quadratic, cubic or biquadratic expression then algebraic formulae are available for expressing the roots. But when f(x) is a polynomial of higher degree or an expression involving transcendental functions e.g., 1 + cos x – 5x, x tan x – cos hx, e–x – sin x etc., algebraic methods are not available. In this chapter, we shall describe some numerical methods for the solution of f(x) = 0, where f(x) is algebraic or transcendental or both.

According to Theile, ‘Interpolation is the art of reading between the lines of the table’. It also means insertion or filling up intermediate terms of the series.

Consider a function of a single variable y = f(x). If f(x) is defined as an expression, its derivative or integral may often be determined using the techniques of calculus. However, when f(x) is a complicated function or when it is given in a tabular form, we use numerical methods.

The systems of simultaneous linear equations arise, both directly in modelling physical situations and indirectly in the numerical solution of other mathematical models. Problems such as determining the potential in certain electrical networks, stresses in a building frame, flow rates in a hydraulic system etc., are all reduced to solving a set of algebraic equations simultaneously. Linear algebraic systems are also involved in the optimization theory, least squares fitting of data, numerical solution of boundary value problems for ordinary and partial differential equations, statistical inference etc.

Let A and B be two square matrices of order n then B is called the inverse of A if AB = BA = In where In is a unit matrix of order n. Inverse of A is denoted by A–1. Inverse of a square matrix is always unique. Also A is invertible iff it is non-singular. In this chapter, we will study and discuss various methods of finding inverse of a square non-singular matrix.

Computation of eigen values and the corresponding eigen vectors of a matrix is of practical importance. For example, in solid mechanics, when we consider an element in a continuum, subjected to normal and shear stresses, we usually find principal stresses which are the maximum and minimum stresses in an element.

A physical situation that concerns with the rate of change of one quantity with respect to another gives rise to a differential equation.

Let there be two variables x and y which give us a set of n pairs of numerical values (x1, y1), (x2, y2).......(xn, yn). In order to have an approximate idea about the relationship of these two variables, we plot these n paired points on a graph thus, we get a diagram showing the simultaneous variation in values of both the variables called scatter or dot diagram. From scatter diagram, we get only an approximate non-mathematical relation between two variables. Curve fitting means an exact relationship between two variables by algebraic equations, infact this relationship is the equation of the curve. Therefore, curve fitting means to form an equation of the curve from the given data. Curve fitting is considered of immense importance both from the point of view of theoretical and practical statistics.

We often encounter partial differential equations in science and engineering, especially in problems involving wave phenomena, heat conduction in homogeneous solids and potential theory. The analytical treatment of these equations requires application of advanced mathematical methods. On the other hand, it is easier to produce sufficiently approximate solutions by simple and efficient numerical methods. Of the various numerical methods available for solving partial differential equations, the method of finite differences is commonly used. In this method, the derivatives appearing in the equation and the boundary conditions are replaced by their finite difference approximations. Then the given equation is changed into a system of linear equations which are solved by iterative procedures.

Statistical methods are devices by which complex and numerical data are so systematically treated as to present a comprehensible and intelligible view of them. In other words, the statistical method is a technique used to obtain, analyse and present numerical data.