TVM solve for interest rate, revisited

[Albert Chan]

(06-15-2024 03:29 PM)Albert Chan Wrote:  I recently translated find_rate() for HP71B (with test samples)

Replaced with NPMT = (D-1)*(PV+FV) + PV*N*I + F0 --> FNF = NPMT/n, we have:

10 INPUT "B,N,P,M,F? ";B,N,P,M,F @ F0=(P+F+N*M)/N
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=F0+(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) @ K=(P+F)/S @ D=K*(1-N*(S+1)/(S+S/I))+P
50 FNF=K*(I-S/N)+P*I+F0 @ 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

Running test samples shown similar result, except for Problem 5

Code:

B,N,P,M,F? 0,365*24*60,0,-0.01,5260.382426
-1.90100247286E-4     5.82743536295E-3
-2.255093023E-5       6.13183770435E-4
-1.232943034E-7       1.15811690244E-5
 3.16928060096E-7     4.46510114E-9
 3.17097918102E-7     3.945E-14
 3.17097919603E-7     1.586E-14
 3.17097920206E-7     3.172E-14
 3.17097920809E-7
>res * 60*n
 10.0000000306

It is interesting we have a one solution case, but FNF have no sign changes.
Final FNF value get worse, on the same side of root!

I expect this behavior from a 2 solutions to 0 solution borderline case, but not here!

The problem here is F0 is tiny (if true rate is 0, so does F0), much smaller than M
Without needed cancellation, FNF is less stable. Sometimes, cancellation is good!

If we do an extra iteration, indeed we have sign changes, i.e. guaranteed solution.

>FNF(J)
-1.546E-14
>(J-RES/D/2) * 100 ! half-correction
 3.17097920515E-7