So far we have studied the operators Δ, and δ. Here we will discuss some more operators which play a vital role in numerical analysis.
1. Shifting Operator:
If h is the interval of differencing in the argument x then the operator E is defined as
Sometimes it is also called translation operator due to it results value of the function for the next argument. Furhter we observe that;
In general, for all integral values of n
2. The D Operator:
The differential coefficient of y with respect to x is denoted by Dy, where D ≡ d/dx. Here D is called as differential operator or simply operator D. We may denote nth derivative of y with respect to x as Dny. Here Dn is called nth differential operator.
3. The Mean Operator:
The mean operator is denoted by μ and is defined as;
Some Important Relations:
The shift operator E is fundamental operator. All other derivatives may be derive from it. In this section we discuss some representation of other operators in terms of E. We have;
Which implies Δ ≡ E − 1, i.e.
E ≡ 1 Δ
Similarly for the backward operator, we have
By using relation above and rearranging, we get
Δ ≡ ehD −1.