Newton’s forward interpolation: representation of numerical data by a polynomial curve

In case of the interpolation by the existing formulae, the value of the dependent variable corresponding to each value of the independent variable is to be computed afresh from the used formula putting the value of the independent variable in it. That is if it is wanted to interpolate the values of the dependent variable corresponding to a number of values of the independent variable by a suitable existing interpolation formula, it is required to apply the formula for each value separately and thus the numerical computation of the value of the dependent variable based on the given data are to be performed in each of the cases. In order to get rid of these repeated numerical computations from the given data, one can think of an approach which consists of the representation of the given numerical data by a suitable polynomial and then to compute the value of the dependent variable from the polynomial corresponding to any given value of the independent variable. However, a method/formula is necessary for representing a given set of numerical data on a pair of variables by a suitable polynomial. One such formula has been developed in this study. The formula has been derived from Newton's forward interpolation formula. The formula obtained has been applied to represent the numerical data, on the total population of India since 1971, by a polynomial curve.

Newton's Forward Interpolation Formula:

Let ( ), ( ), ( )…………………………… ( ) be the set of (n + 1) pairs of values of x and y. Let the values of the independent variable x be at equal interval h i.e. for all i.

Further let u = Then Newton's forward interpolation formula is

This formula can be expressed as

Representation of Numerical Data by Polynomial Curve:

By algebraic expansion, one can obtain that

In general, one can obtain that

2), can be expressed as as defined above, is the required formula for representing a given set of numerical data on a pair of variables by a suitable polynomial we have aimed at.

Note:

The formula is valid for representing a given set of numerical data on a pair of variables by a suitable polynomial under the following two conditions:

1. The given values of the independent variable are at equal interval.

2. The value of the independent variable corresponding to which the value of the dependent variable is to be estimated lies in the first half of the series of the given values of the independent variable.

An Example of Application of the Formula:

The following table shows the data on total population of India corresponding to the years: Now here = 0, = 1, = 2, = 3, = 4

Conclusion:

The formula described by equation (3.2) can be used to represent a given set of numerical data on a pair of variables, by a polynomial. The degree of the polynomial is one less than the number of pairs of observations. The polynomial that represents the given set of numerical data can be used for interpolation at any position of the independent variable lying within its two extreme values. The approach of interpolation, described here, can be suitably applied in inverse interpolation also.

Newton's forward interpolation formula is valid for estimating the value of the dependent variable under the following two conditions: 1. The given values of the independent variable are at equal interval. 2. The value of the independent variable corresponding to which the value of the dependent variable is to be estimated lies in the first half of the series of the given values of the independent variable.

Therefore, the formula derived here is valid for representing a set of numerical data on a pair of variables by a polynomial under these two conditions only. Consequently, there is necessity of searching for some formula for representing a set of numerical data on a pair of variables by a polynomial if the value of the independent variable corresponding to which the value of the dependent variable is to be estimated lies in the last half of the series of the given values, which are at equal interval, of the independent variable. Moreover, there is also necessity of searching for some formula for representing a set of numerical data on a pair of variables by a polynomial if the given values of the independent variable are not at equal interval.

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