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Finite Differences | Numerical Methods - Relations between the operators Δ, ∇ and E | 12th Business Maths and Statistics : Chapter 5 : Numerical Methods

Chapter: 12th Business Maths and Statistics : Chapter 5 : Numerical Methods

Relations between the operators Δ, ∇ and E

Relations between the operators : Forward Difference Operator(∆), Backward Difference operator(∇), Shifting operator (E)

Relations between the operators Δ, and E:


1. Δ ≡ E1

Proof:         From the definition of Δ we know that

Δ f (x) = f (x + h ) f (x) and   

E[ f (x )] = f (x + h)

where h is the interval of difference.

Δ f (x) = f (x + h ) f (x)

Δ f (x) = Ef (x ) f (x)

Δ f (x) = (E1) f (x)

Δ ≡ E1

    E 1 + Δ


2.   Δ E              Δ E

Proof:

E(Δf(x)) = E[f(x+h)- f(x)]

= E f(x+h) - E f(x)

= f(x+2h) - f(x+h)

=Δf(x+h)

=ΔEf(x)

ΔE ≡ ΔE


3. E1 / E

f (x) = f (x) - f (x-h)

= f (x) – E-1f (x)

= (1- E-1) f (x)

(1- E-1)

≡ 1 – 1/E

Hence ∇ ≡ [E – 1]/E



 

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