TVM solve for interest rate, revisited

[Albert Chan]

(06-10-2022 08:25 PM)Albert Chan Wrote:  If solution exist, Halley's method seems not to over-shoot, but not yet proved.

1-sided convergence observation only apply to Halley's rational formula.
Halley's Irrational formula does overshoot, even if guesses not bracketed by 2 roots.

Irrational Halley's modifed slope = f' * (1 + √(1 - 2*f*f''/(f')^2)) / 2

(1 + √(1-2x))/2 = 1 - x/2 - x^2/4 - x^3/4 - ...

Assume correction x is small, cutting off O(x^2) terms made slope correction factor bigger.
Bigger slope made Halley's rational formula correction less aggressive.

Rational Halley's modified slope = f' * (1 - (f*f'')/(f')^2/2) = f' - (f''/2)*(f/f')

In exchange for slightly slower convergence, it is more stable.

(01-07-2020 06:13 PM)Albert Chan Wrote:  Example, for N=30, PV=6500, PMT=-1000, FV=50000, we have I = 8.96% or 11.10%

Does anyone knows what returns should be reported for above investments ?

This is the example I found that Halley's Irrational formula iterations over-shoot.

Code:

Rational    Irrational
0.075       0.075
0.0878885   0.0906481
0.0895961   0.0896056
0.0896059   0.0896059

0.08        0.08 
0.0888735   0.0898506
0.0896050   0.0896059
0.0896059   0.0896059