Looking for TVM contributions

[Albert Chan]

(06-14-2024 10:11 PM)robve Wrote:  Agree. But for Newton running on this 10 BCD machine the longer f(i)/f'(i) expression seems more likely to introduce floating point inaccuracies than the simpler Secant. Whereas the IEEE754 double precision version I have is more well behaved. It's one possible explanation.

Perhaps you missed my post regarding small rate branch. It is required to allow edge rate to work.
It is just simple quadratic f(i) ≈ f(0) + f'(0)*i + f''(0)/2*i²      (see LINE 20, FNZ)

f(0) = (pv+fv)/n + pmt
f'(0) = (pv-fv + (pv+fv)/n)/2
f''(0) = (pv+fv)/n * (n*n-1)/6

(06-07-2024 01:45 AM)Albert Chan Wrote:  10 INPUT "B,N,P,M,F? ";B,N,P,M,F
12 IF B THEN P=P+M @ F=F-M ! end mode
14 IF ABS(P)>ABS(F) THEN X=F @ F=-P @ P=-X @ N=-N ! time reversed
20 DEF FNZ(I,A,B,C) @ D=(A+B)/2 @ B=A*C @ FNZ=A+M+(D+B)*I @ D=D+2*B @ END DEF
30 DEF FNF(I) @ IF 1+N*I*I=1 THEN FNF=FNZ(I,(P+F)/N,P-F,(N*N-1)/12*I) @ END
40 S=EXPM1(LOGP1(I)*N) @ D=(P+F)/S*(1-N*(S+1)/(S+S/I))+P
50 FNF=((P+F)/S+P)*I+M @ END DEF
60 I=M/F @ Y=FNF(I) ! "smallest" edge guess
70 DISP 100*I,Y @ H=-Y/D @ I=I+H @ Y0=Y @ Y=FNF(I)
80 IF SGN(Y0)=SGN(Y) AND ABS(Y0)>ABS(Y) THEN 70
90 DISP 100*I,Y @ J=I-Y/D/2 @ DISP 100*J ! half-correction

BTW, I think your Newton's code termination test is wrong.
It does not look like HP71B LINE 80 at all!

do { ... } while (y != 0 && y != w && g-- > 0)

Quote:But it's almost free, right? Just iterate \( i_{\rm new} := i-(i-i_{\rm old})f/(f-f_{\rm old}) \) with one evaluation of npmt to get f(i) to update i.

Newton's method is not that bad. Also, (P+F)/S factor may also be shared.

FNF=((P+F)/S+P)*I+M
D=(P+F)/S*(1-N*(S+1)/(S+S/I))+P

Quote:This is true for Newton that runs fewer iterations. But this benefit diminishes rapidly with more precise initial rate guesses, especially when we only use single precision or only 10 digits then only a few iterations suffice in both cases.

Yes, I played with many such rate guess formulas.
The problem is they are not guaranteed to always work, unlike edge rate.