Newton Interpolation Formula for Unequal Intervals

When the values of the independent variable occur with unequal spacing the formula discussed earlier is no more applicable. In this situation another formula which is based on divided difference is used. Before presenting the formula let us first discuss about what are divided differences.

Divided Differences:
Let the values of the independent variable \left( X \right)are given as {a_o},{a_{1,}}{a_2},{a_3},...  etc. and the corresponding values of the function (dependent variable) as f\left(  {{a_o}} \right),f\left( {{a_1}} \right),f\left( {{a_2}} \right),f\left( {{a_3}}  \right),... etc. The data thus looks as follows:

X

{a_o}

{a_1}

{a_2}

{a_3}

{a_4}

f\left( X \right)

f\left( {{a_o}} \right)

f\left( {{a_1}} \right)

f\left( {{a_2}} \right)

f\left( {{a_3}} \right)

f\left( {{a_4}} \right)

 
Where \left( {{a_1} - {a_o}} \right),\left( {{a_2} -  {a_1}} \right)  etc. are not equal. The problem of interpolation here requires that, X = {X_o} , what is the value of f\left( {{X_o}} \right). In order to determine the value of f\left( {{X_o}}  \right)we need to computer what are called divided difference.  
The quantities


\frac{{f\left(  {{a_1}} \right) - f\left( {{a_o}} \right)}}{{{a_1} - {a_o}}}  denoted by f\left(  {{a_1},{a_o}} \right)
\frac{{f\left(  {{a_2}} \right) - f\left( {{a_1}} \right)}}{{{a_2} - {a_1}}}  denoted by f\left(  {{a_2},{a_1}} \right)
\frac{{f\left(  {{a_3}} \right) - f\left( {{a_2}} \right)}}{{{a_3} - {a_2}}}  denoted by f\left(  {{a_3},{a_2}} \right)
etc.
are called divided difference of first order. Moreover the quantities
\frac{{f\left(  {{a_2},{a_1}} \right) - f\left( {{a_1},{a_o}} \right)}}{{{a_2} - {a_o}}}  denoted by f\left(  {{a_2},{a_1},{a_o}} \right)
\frac{{f\left(  {{a_3},{a_2}} \right) - f\left( {{a_2},{a_1}} \right)}}{{{a_3} - {a_1}}}  denoted by f\left(  {{a_3},{a_2},{a_1}} \right)
etc.
are called divided differences of second order. Similarly, divided difference of order three and higher may be computed.
The divided differences may be put in a tabular form as follows: 

DIFFERENCES OF ORDER

X

f\left( X \right)

I

II

III

IV

{a_o}

f\left( {{a_o}} \right)

f\left( {{a_1},{a_o}} \right)

f\left( {{a_2},{a_1},{a_o}} \right)

f\left( {{a_3},{a_2},{a_1},{a_o}} \right)

f\left( {{a_5},{a_4},{a_3},{a_2},{a_1}} \right)

{a_1}

f\left( {{a_1}} \right)

f\left( {{a_2},{a_1}} \right)

f\left( {{a_3},{a_2},{a_1}} \right)

f\left( {{a_4},{a_3},{a_2},{a_1}} \right)

 

{a_2}

f\left( {{a_2}} \right)

f\left( {{a_3},{a_2}} \right)

f\left( {{a_4},{a_3},{a_2}} \right)

 

 

{a_3}

f\left( {{a_3}} \right)

f\left( {{a_4},{a_3}} \right)

 

 

 

{a_4}

f\left( {{a_4}} \right)

 

 

 

 

 

 

 

 

 

 

It is observed that divided differences of higher order either vanish or become negligible. We continue computing these differences until such order where they become more or less constant or significantly different.

THE FORMULA:
The Newton’s formula for unique interval may be stated as follows:

\begin{gathered} f\left( {{X_o}} \right) = f\left( {{a_o}} \right) + \left( {{X_o} -  {a_o}} \right)\,f\left( {{a_1},{a_o}} \right) + \left( {{X_o} - {a_o}}  \right)\left( {{X_o} - {a_1}} \right)f\left( {{a_2},{a_1},{a_o}} \right)  \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {{X_o} - {a_o}} \right)\left(  {{X_o} - {a_1}} \right)\left( {{X_o} - {a_2}} \right)f\left(  {{a_3},{a_2},{a_1},{a_o}} \right) + \cdots + \left( {{X_o} - {a_o}}  \right)... \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,..........\left( {{X_o} - {a_{n - 1}}}  \right)f\left( {{a_n},{a_{n - 1}},...,{a_o}} \right) \\ \end{gathered}


Where {a_o},{a_1},{a_2} etc. are the values of the independent variable, {X_o} the given value corresponding to which f\left(  {{X_o}} \right)is required, and f\left(  {{a_1},{a_o}} \right),f\left( {{a_2},{a_1},{a_o}} \right) etc. are successive divided difference of first, second and third orders.

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