What comes next?

[Albert Chan]

(10-19-2024 08:12 AM)Thomas Klemm Wrote:  Now this is a bit off topic, but the same method can be used to calculate square roots.
To calculate \(\sqrt{a}\), we use NEXT to calculate consecutive values of \(f(x)=x^2-a\).

Just to clear up confusion, f(x) = k*(x^2-a), where k start at 0.5, scale-up by 10 for each correct x digit.

Initially, f = 0.5*(x^2-a), ∇f = x-0.5, ∇²f = 1

This is the reason for INIT setup, at x=0:
M03 = f = -0.5*a
M02 = ∇f = -0.5
M01 = ∇²f = 1

Example, if we do √41 manually, digit by digit, we have:

f(x=6) = 0.5*(6^2 - 41) = -0.5*41 + 0.5 + 1.5 + 2.5 + 3.5 + 4.5 + 5.5 = -2.5
f(x=6.4) = 5*(6.4^2 - 41) = -25 + 6.05 + 6.15 + 6.25 + 6.35 = -0.2
f(x=6.40) = 50*(6.40^2 - 41) = -2
f(x=6.403) = 500*(6.403^2 - 41) = -20 + 6.4005 + 6.4015 + 6.4025 = -0.7955
...

This is similar to HP calculator, but not quite.
Calculator sqrt algorithm scale F by 100 for each correct digit, to keep ∇²F = 1, all the time.

F(X=6) = -2.5
F(X=64) = -250 + 60.5 + 61.5 + 62.5 + 63.5 = -2
F(X=640) = -200
F(X=6403) = -20000 + 6400.5 + 6401.5 + 6402.5 = -795.5
...

https://www.hpmuseum.org/forum/thread-17...#pid152220