(HP71B) Newton's method

[Albert Chan]

Inspired by Csaba Tizedes videos about Newton's method

(09-23-2023 05:19 PM)Csaba Tizedes Wrote:  https://www.youtube.com/shorts/1R3IMq227dM

(09-23-2023 05:21 PM)Csaba Tizedes Wrote:  Same, without d/dx()

https://www.youtube.com/shorts/-8pGxTXSk40

f'(x) ≈ (f(x+ε) - f(x-ε)) / (2ε)      → f/f' needed 3 function calls per iteration.

We may reduce this to about 2 calls per iteration.
Redo same example in HP71B, for full convergence.

Quote:1 DESTROY ALL @ C=0
2 DEF FNF(X)=X^4-4*X^3
3 INPUT "Guess, Accuracy = ";X,A
10 FOR I=1 TO 30 @ DISP X
20 E=.00001
30 F1=FNF(X-E) @ F2=FNF(X+E)
40 R=2*E/(F2-F1) ! reciprocal slope
50 D=(F1+F2)/2*R ! secant root = X-D
60 C=C+2 @ IF ABS(D)<E THEN D=D+FNF(X-D)*R @ C=C+1
70 X=X-D @ IF ABS(D)<A THEN EXIT I
80 NEXT I
90 DISP X @ DISP "FNF calls =";C

>RUN
Guess, Accuracy = 10, 1e-6
 10
 7.85714477037
 6.29727032561
 5.20041359757
 4.4911553721
 4.12133235435
 4.00984649554
 4.00007201239
 4.00000000386
 4
FNF calls = 19

For comparison, this is using video's formula for Newton's correction.
There is no gain in rate of convergence ... just more function calls.

>50 D = FNF(X)*R
>60 C = C+3
>RUN
Guess, Accuracy = 10, 1e-6
 10
 7.8571447704
 6.29727032563
 5.20041359761
 4.49115522316
 4.12133220071
 4.00984642245
 4.0000719974
 4.00000000388
 4
FNF calls = 27