Accurate Normal Distribution for the HP67/97

[Albert Chan]

(12-02-2018 10:08 PM)Dieter Wrote:  Let t be the Newton correction term (p–Q(x))/Z(x), just as in your post (except the sign).
Then the third-order correction uses expressions in t, t² and t³ to get a much better result:

x := x + t + x/2 · t² + (2x²+1)/6 · t³

Newton's method uses only the first term t.
Halley's method is comparable to the above series up to t².
And with terms up to t³ even better results are obtained.

That is a cool formula ! I was expecting something much worse.
To prove above correction work:

f(x + h) = f(x) + f'(x) h + f''(x)/2 h² + f'''(x)/6 h³ + ...
t = (f(x+h) - f(x)) / f'(x) = h + f''(x)/(2 f'(x)) h² + f'''(x)/(6 f'(x)) h³ + ...

f'(x) = Z(x) = 1/√(2 pi) * exp(-x²/2), we get f''(x) = -x Z(x), and f'''(x) = (x² - 1) Z(x)

t =           h + (-x/2) h² + (x²-1)/6 h³ + ...
(x/2) t² =         (x/2) h² + (-3x²)/6 h³ + ...
(2x²+1)/6 t³ =               (2x²+1)/6 h³ + ...

Sum it all, and ignore O(h⁴) terms, we get:

x correction = h = t + x/2 · t² + (2x²+1)/6 · t³