Computer Based Numerical and Statistical Techniques

The document contains the preface.

The document contains the preface.

The document contains the syllabus.

The document contains the salient features.

The document contains the foreword.

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.

Following are the broad sources of errors in numerical analysis:
(1) Input errors: The input information is rarely exact since it comes from the experiments and any experiment can give results of only limited accuracy. Moreover, the quantity used can be represented in a computer for only a limited number of digits.
(2) Algorithmic errors: If direct algorithms based on a finite sequence of operations are used, errors due to limited steps don’t amplify the existing errors but if infinite algos are used, ideally exact results are expected only after an infinite number of steps. As this cannot be done in practice, the algorithm has to be stopped after a finite number of steps and as a consequence thereof the results are not exact.

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 – cosh x, e–x – sin x etc. algebraic methods are not available.
Here, we shall describe some numerical methods for the solution of f(x) = 0, where f(x) is algebraic or transcendental or both.

1. There are no sudden jumps or falls in the values during the period under consideration.
2. The rise and fall in the values should be uniform.
e.g., if we are given data regarding deaths in various years in a particular town and some of the observations are for the years in which epidemic or war overtook the town then interpolation methods are not applicable in such cases.

1. There are no sudden jumps or falls in the values during the period under consideration.
2. The rise and fall in the values should be uniform.
e.g., if we are given data regarding deaths in various years in a particular town and some of the observations are for the years in which epidemic or war overtook the town then interpolation methods are not applicable in such cases.

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.

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.

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

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.

The document contains the tables.