(HP71B) Newton's method

[Albert Chan]

(10-01-2023 09:47 PM)Albert Chan Wrote:  W(b=1) example, newton iteration formula:  x = (x + 1) / (1 + e^x/b)

Because of e^x term in denominator, this setup is not good for W(huge b)
Convegence is slow and inaccurate, even with good guess.

>2 B=1E10 @ DEF FNF(X)=B*EXP(-X)-X ! root = W(B)
>RUN
Guess, Accuracy = LN(B), RES*1E-6
 23.0258509299
 12.0135445721
 13.0133779957
 14.0128061672
 15.0110431755
 16.005828047
 16.9907276945
 17.9478731089
 18.8306249224
 19.5363990109
 19.9283981583
 20.0242147693
 20.0286862103
 20.0286854427
FNF calls = 28

>X*EXP(X) ! expected to be close to B
 10000000308.6

---

It is better is to setup Newton formula with less curvy f (e^-x is as bad as e^x)

x * e^x = b
ln(x) + x = ln(b)
x = f(x)      → f = ln(b) - ln(x)      → f' = -1/x

Newton: x = (x - f/f') / (1 - 1/f') = (1 + ln(b/x)) / (1 + 1/x)

>2 B=1E10 @ DEF FNF(X)=LN(X/B)+X ! root = W(B)
>RUN
Guess, Accuracy = LN(B), RES*1E-6
 23.0258509299
 20.0197856819
 20.0286854133
 20.0286854133
FNF calls = 8

>X*EXP(X) ! very close to B. no other 12-digits X can get closer!
 9999999999.95