Lab 15: Difference Table Lab · Numerical Methods

Type equally spaced data. Switch between the forward diagonal (Δ, used near the start) and the backward diagonal (∇, used near the end).

Add an error ε to one entry and watch how it spreads. Then hide it: can a classmate find which entry you changed?

Try this

Q1 Show that

E = e^{hD}

.

By Taylor's series,

Ef(x) = f(x+h) = f(x) + hf'(x) + \tfrac{h^2}{2!}f''(x) + \cdots = (1 + hD + \tfrac{(hD)^2}{2!} + \cdots)f(x) = e^{hD}f(x)

.

Q2 Prove that

\Delta - \nabla = \Delta\nabla

.

Both sides equal

E - 2 + E^{-1}

:

\Delta - \nabla = (E - 1) - (1 - E^{-1})

and

\Delta\nabla = (E - 1)(1 - E^{-1})

.

Q3 Prove that

\mu^2 = 1 + \tfrac14\delta^2

.

\mu^2 = \tfrac14(E^{1/2} + E^{-1/2})^2 = \tfrac14(E + 2 + E^{-1})

and

\delta^2 = E - 2 + E^{-1}

, so

\mu^2 - \tfrac14\delta^2 = \tfrac14 \cdot 4 = 1

.

Q4 Find the missing term in

1, 3, 9, \;?, 81

assuming a cubic.

\Delta^4 y_0 = 0

:

81 - 4y_3 + 54 - 12 + 1 = 0

, so

y_3 = 31

.

Q5 Evaluate

\Delta^2(x^3)

with

h = 1

.

\Delta x^3 = (x+1)^3 - x^3 = 3x^2 + 3x + 1

;

\Delta^2 x^3 = 3(2x + 1) + 3 = 6x + 6

.