TVM solve for interest rate, revisited

[Albert Chan]

Examples taken from previous post, where we have 2 rate solutions.

XCAS> tmv := ((pv+fv)/((1+x)^n-1) + pv)*x + pmt
XCAS> f0 := tmv(n=30, pv=6500., pmt=-1000., fv=50000); f1 := f0' :;
XCAS> [a, b] := [limit(x-f0/f1,x=-1), limit(x-f0/f1,x=inf)]

[-1/50, 2/13]            // = 2 edges, [pmt/fv, -pmt/pv]

XCAS> m := fsolve(f1, x=a); f0(x=m)

(0.100019883274, -6.52249630654)              // f extremum

XCAS> [fsolve(f0, x=a), fsolve(f0, x=b)]

[0.0896058562612, 0.111003286405]            // f roots

Extremum(f) = f(x=m) < 0, and f have 2 roots ⇒ f has a concaved-up V shape.

Halley's method, derived from g = f / sqrt(|f'|), Newton correction for g ≡ Halley correction for f

XCAS> g0 := f0 / sqrt(abs(f1)):; g1 := g0':; g2 := g1':;

g have the same roots as f (by its definition)
Interestingly, g'' also have same roots as f

XCAS> fsolve(g2, x=a..b)

[0.0896058562612, 0.111003286405]            // g'' roots = f roots

XCAS> g2(x=a), g2(x=m), g2(x=b)

(796.773155573, -2.55834266547e+38, 371.946352558)

g is concave-down, for intervals between 2 roots, concaved-up otherwise.
⇒ Newton's method on g have 1-sided convergence, for whatever rate guess.
⇒ Halley's method on f have 1-sided convergence, for whatever rate guess.

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Paradoxically, f and g'' roots matched guaranteed 1-sided convergence is not true.
(at least, not when f has only 1 root)

From the extreme edge (-1,∞), f'' and g'' must have same sign.
We had proved f'' have same sign throughout; g'' signs must be +/-/+, or -/+/-

But, what happens if f has only 1 root ? g'' must still have 2 roots !

Disprove of Halley's 1-sided convergence, by counter-example

We flip above exmple fv sign; f now has only 1 root. (overshoot rate in bold)

lua> iter = loan_rate2(30,6500,-1000,-50000)
lua> for i=1,5 do print(i, iter()) end
1      0.1511929109248802     0.1311929109248802
2      0.16524185831684263    0.014048947391962436
3      0.1652186592665784     -2.319905026425022e-005
4      0.16521865926667817    9.978337255768468e-014
5      0.1652186592666782     2.8509535016486494e-017


This counter-example, g'' roots = [0.0789877153657, 0.165218659267]
g'' second root matched f (only) root.