What comes next?

[Albert Chan]

(10-19-2024 09:09 PM)Albert Chan Wrote:  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

For each digit, we can stop if (F + ∇F + ∇²F) > 0, eliminate routine BACK

Stack Y = (-F) = -K*(x^2 - a)
Stack X = (∇F + ∇²F) = K*((2x-1) + 2) = K*(2x+1) = ΔF

Stop routine NEXT if stack X > Y
Initial setup, K=0.5, x=0 --> Stack Y, X = a/2, 1/2

Code:

00 { 100-Byte Prgm }
01▸LBL "ROOT"
02 12       
03 STO 00
04 10↑X     
05 STO 01
06 X<>Y
07 XEQ "INIT"
08▸LBL 00
09 XEQ "NEXT"
10 XEQ "ZOOM"
11 DSE 00
12 GTO 00
13 IP
14 RCL÷ 01
15 RTN

16▸LBL "INIT"
17 0.5
18 ×
19 LASTX
20 RTN

21▸LBL "NEXT"
22 X>Y?
23 RTN
24 -
25 LASTX
26 1
27 +
28 GTO "NEXT"

29▸LBL "ZOOM"
30 10
31 STO× ST Z
32 STO× ST Z
33 ×
34 4.5
35 -
36 END

41 XEQ "ROOT"      → 6.40312423743

Here, we do √41 manually, digit by digit.

Code:

41
XEQ "INIT"      --> Y=20.5  , X=0.5
XEQ "NEXT"      --> Y=2.5   , X=6.5
XEQ "ZOOM"      --> Y=250   , X=60.5
XEQ "NEXT"      --> Y=2     , X=64.5
XEQ "ZOOM"      --> Y=200   , X=640.5
XEQ "NEXT"      --> Y=200   , X=640.5
XEQ "ZOOM"      --> Y=20000 , X=6400.5
XEQ "NEXT"      --> Y=795.5 , X=6403.5
...