Interpolation, numerical differentiation and numerical integration

UNIT - IV

INTERPOLATION, NUMERICAL DIFFERENTIATION AND NUMERICAL INTEGRATION

INTRODUCTION

The estimation of values between well-known discrete points are called interpolation.

Interpolation is the process of finding the most appropriate estimate for missing data. For making the most probable estimate, it requires the following assumptions.

(1) The frequency distribution is normal and is not marked by sudden ups and downs.

(2) The changes in the series are uniform within a period.

It is used to fill in the gaps in the statistical data for the sake of continuity of information.

Many famous mathematicians have their names associated with procedures for interpolation: Gauss, Newton, Bessel, Stirling. The need to interpolate began with the early studies of astronomy, when the motion of heavenly bodies was to be determined from periodic observations. Interpolation technique is used in various disciplines like, business, economics, population studies, price determination, etc.

Interpolating function

Let a set of tabular values of a function y = f (x), where the explicit nature of the function is not known, then f (x) is replaced by a simpler function ϕ (x), such that f ϕ (x) and ϕ (x) agree with the set of tabulated points. Any other value may be calculated from ϕ (x). This function ϕ (x) is known as an interpolating function.i