[VA] SRC#006- Pi Day 2020 Special: A New Fast Way to Compute Pi

[Albert Chan]

(03-15-2020 10:55 AM)Bernd Grubert Wrote:  My HP 50G says, it could be interpreted as an application of the newton iteration method to the function:

f(x) = c * sqrt((cos(x) + 1) / (cos(x) - 1))

This is the solution of y(x) / y'(x) = -sin(x).
The Newton iteration is then:

x_n+1 = x_n - y(x_n) / y'(x_n) = x_n + sin(x_n)

with x_0 = 3.

Is that correct?

May be there are functions with an even faster convergence.

This is not how iteration X+=SIN(X) is derived.

With guess X=3, X+ASIN(SIN(X)) = X+ASIN(SIN(PI-X)) = X+(PI-X) = PI

ASIN(ε) = ε + ε^3/6 + 3ε^5/40 + ... = ε + O(ε^3)

Thus, with guess X=3, X+=SIN(X), converge to Pi cubically.

We can improve convergence rate with better estimation for ASIN
Say, with Pade[2,2] of ASIN(ε) = ε/(1-ε*ε/6)

> X=3
> Y=SIN(X)
> X+Y
3.14112000806
> X+Y/(1-Y*Y/6)
3.14158996537