Lab 16: Newton Interpolation Lab

Type the x values (equally spaced) and the y values, then the point x. The table lists every term, exactly as a written solution would show it. Add a true f(x) to see the actual error.

The difference table behind the formulas.

Newton forward and backward

Try this

Run it. Compare every term with Problems 1 and 2.
Interpolate 1915 with both formulas. Do they agree?
Try 1960. What happens, and why?

Q1 Derive Newton's forward interpolation formula.

Put

x = x_0 + ph

. Then

y(x) = E^p y_0 = (1 + \Delta)^p y_0

; expanding by the binomial theorem gives

y_0 + p\Delta y_0 + \frac{p(p-1)}{2!}\Delta^2 y_0 + \cdots

.

Q2 When do you use the backward formula instead of the forward one?

When

x

lies near the end of the table: the backward formula uses differences at the end (

\nabla^k y_n

) and keeps

|p| < 1

, so fewer terms are needed and the error is smaller.

Q3 From

x

: 1, 2, 3, 4, 5 and

y

: 1, 8, 27, 64, 125 find

y(1.5)

.

Forward with

p = 0.5

:

1 + 3.5 - 1.5 + 0.375 = 3.375

(exactly

1.5^3

).

Q4 Why is extrapolation dangerous?

The error term contains

p(p-1)\cdots(p-n)

, which grows rapidly when

x

is outside the table; nothing in the data controls the polynomial there.

Q5 Find

y(0.5)

for

x

: 0, 1, 2, 3, 4 and

y

: 1, 7, 23, 55, 109.

\Delta y_0 = 6

,

\Delta^2 y_0 = 10

,

\Delta^3 y_0 = 6

,

\Delta^4 y_0 = 0

;

p = 0.5

:

1 + 3 - 1.25 + 0.375 = 3.125

.