Small Solver Program

[Albert Chan]

Just realized rate formula is equivalent to secant's method !

Post #6 showed rate formula is equivalent to Aitken's formula.
Post #27 showed we can visualize sequence running backwards, getting the same formula.

Assuming iteration formula f, with b = f(a), c = f(b):

\(Aitken(a,b,c) = Aitken(c,b,a) = a - {(a-b)^2 \over (a-b)-(b-c)}\)

If we make 2 points: \((x_1,y_1) = (a,a-b) \;,\; (x_2, y_2) = (b,b-c)\), we have:

\(Aitken(a,b,c) = x_1 - y_1 \left({x_1 - x_2 \over y_1 - y_2}\right)\)

This is the formula for secant's method, interpolate for y = x - f(x) = 0

QED