Newton Interpolation Formula

A number of different formulae were given by Newton, however, two of these formulae are more common. One of these formulae is used when the independent variable assumes values with equal interval while the other is applicable when the interval is not equal. The first formula is referred to as “Newton’s formula for equal interval”, whereas the second formula is referred to as “Newton’s formula for unequal intervals”. In the subsequent tutorials we would present a brief discussion about what is a difference table and then the Newton’s formula for equal interval, thereafter we would learn about what are divided differences and finally the Newton’s formula for unequal intervals.

The Difference Table:
Suppose the initial value of the independence variable \left( X \right)  is “a” and the common class interval being “h”. The successive values of the independence variable could then be written symbolically as a , \left( {a + h} \right),\left( {a + 2h}  \right),\left( {a + 3h} \right), \cdots   etc. and the corresponding values of the dependent variable may be written symbolically as f\left( a \right), f\left( {a + h} \right), f\left( {a + 2h} \right),f\left( {a + 3h}  \right),...etc. the data thus looks as follows:      

X

a

a + h

a + 2h

a + 3h

a + 4h

f\left( X \right)

f\left( a \right)

f\left( {a + h} \right)

f\left( {a + 2h} \right)

f\left( {a + 3h} \right)

f\left( {a + 4h} \right)

 
In general the independence variable may be written as \left(  {a + Xh} \right) and the corresponding values of the dependent variable as f\left( {a + Xh} \right). The problem of interpolation here requires that, given {X_o} = a + Xh, what is the value of f\left( {{X_o}} \right)  or f\left(  {a + Xh} \right). In order to determine the value of f\left( {{X_o}} \right) we need to construct what is called the “Difference Table”.

The quantity f\left( {a + h} \right) - f\left( a \right)is denoted by \Delta f\left( a \right)and is called the first difference of f\left( a  \right). Similarly, the first difference of f\left( {a + h} \right)would be f\left( {a + 2h} \right) - f\left( {a + h} \right)and is denoted by \Delta f\left( {a + h} \right). Other first order differences are

\begin{gathered} \Delta f\left( {a + 2h} \right) = f\left( {a  + 3h} \right) - f\left( {a + 2h} \right) \\ \Delta f\left( {a + 3h} \right) = f\left( {a  + 4h} \right) - f\left( {a + 3h} \right) \\ ........................................................ \\ ........................................................\end{gathered}

NOTE: The symbol \Delta is read as delta, thus \Delta f\left( a \right)means delta f\left( a \right),{\Delta ^2}f\left( a  \right)is read as delta two f\left(  a \right) etc.
Moreover the quantity \Delta  f\left( {a + h} \right) - f\left( a \right)is denoted by {\Delta ^2}f\left( a \right)and is called the second order difference of f\left( a  \right). Other second order difference would be

\begin{gathered} \,\,{\Delta ^2}f\left( {a + h} \right) =  \Delta f\left( {a + 2h} \right) - \Delta f\left( {a + h} \right) \\ {\Delta ^2}f\left( {a + 2h} \right) = \Delta  f\left( {a + 3h} \right) - \Delta f\left( {a + 2h} \right) \\ ............................................................... \\ ...............................................................\end{gathered}

The second order difference is in fact the difference of the first order differences. The quantity

{\Delta  ^2}f\left( {a + h} \right) - {\Delta ^2}f\left( a \right)

is denoted by

{\Delta ^3}f\left( a \right)

  and is called the third order difference of f\left( a \right)and so on.

DIFFERENCE TABLE

X

f\left( X \right)

{\Delta ^1}

{\Delta ^2}

{\Delta ^3}

{\Delta ^4}

a

f\left( a \right)

\Delta f\left( a \right)

{\Delta ^2}f\left( a \right)

{\Delta ^3}f\left( a \right)

{\Delta ^4}f\left( a \right)

a + h

f\left( {a + h} \right)

\Delta f\left( {a + h} \right)

{\Delta ^2}f\left( {a + h} \right)

{\Delta ^3}f\left( {a + h} \right)

{\Delta ^4}f\left( {a + h} \right)

a + 2h

f\left( {a + 2h} \right)

\Delta f\left( {a + 2h} \right)

{\Delta ^2}f\left( {a + 2h} \right)

{\Delta ^3}f\left( {a + 2h} \right)

 \vdots

a + 3h

f\left( {a + 3h} \right)

\Delta f\left( {a + 3h} \right)

{\Delta ^2}f\left( {a + 3h} \right)

 \vdots

 \vdots

a + 4h

f\left( {a + 4h} \right)

\Delta f\left( {a + 4h} \right)

 \vdots

 \vdots

 \vdots

a + 5h

f\left( {a + 5h} \right)

 \vdots

 \vdots

 \vdots

 \vdots

Similarly differences of higher order than the fourth may be written. The first entry in the 2nd column (i.e. f\left( a \right)is called the leading term and the differences in the first row i.e. \Delta f\left( a \right), {\Delta ^2}f\left( a \right), … etc.) are called the leading differences. These are in fact the values which we will use in our formula of interpolation.

The presentation of the difference table in this form seems to be rather difficult, but, in actual practice it is very simple to construct. The entries in column 3 (1st differences) are obtained by subtracting each entry in column 2 from the entry immediately following (with the proper algebraic sign). In this way there will be one less entry in column 3 than in column 2. Similarly entries in column 4, are obtained by subtracting each entry of column 3 from the entry immediately following. Again, there will be one less entry in column 4 than in column 3. This process of taking difference continues till there is only one entry in some column. In fact there would be one less number of differences than the number of entries in the second column. Thus, if there are five entries in the second column, the difference table will be constructed only up to a fourth order i.e. {\Delta ^4}f\left( a \right) .

THE FORMULA:

\begin{gathered} f\left( {{X_o}}  \right) = f\left( {a + Xh} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \,f\left( a \right) + X\Delta f\left(  a \right) + \frac{{X\left( {X - 1} \right)}}{{2 \times 1}}{\Delta ^2}f\left( a  \right) + \frac{{X\left( {X - 1} \right)\left( {X - 2} \right)}}{{3 \times 2  \times 1}}{\Delta ^3}f\left( a \right) +  \cdots \, \\ \end{gathered}


The values of f\left( a \right),\Delta f\left( a \right),{\Delta  ^2}f\left( a \right) are the leading terms and the leading differences (given in the 1st row of the difference table). To find the value of X, equate the given value of {X_o} to a  + Xh. i.e. a + Xh = {X_o}
Where a and h are already known from the first column, “a” being the first term and h, the common class interval. Hence if {X_o}, a and h are known X may be obtained from the above equation. If someone finds it difficult to calculate X from the above equation, one may use the following formula to compute X thus

X =  \frac{{{X_o} - a}}{h}


It is to be noted that the formula for any specific problem will be written up to a difference of the order one less than the number of terms. 

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