TVM solve for interest rate, revisited

[Albert Chan]

(06-12-2022 05:07 PM)Albert Chan Wrote:  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

To simplify, assume f1^p really mean |f1|^p
https://www.mathsisfun.com/calculus/product-rule.html

g0 = f0 * f1^(-1/2)

g1 = f0 * (-1/2*f1^(-3/2)*f2) + f1 * f1^(-1/2)
     = (f1^2 - f0*f2/2) * f1^(-3/2)

g2 = (f1^2 - f0*f2/2) * (-3/2*f1^(-5/2)*f2) + (2*f1*f2 - (f1*f2+f0*f3)/2) * f1^(-3/2)
     = f0 * (f2^2 - 2/3*f1*f3) * (3/4*f1^(-5/2))

Note: g'' may have more than f roots (2nd factor may also have root)

BTW, we had also proved that Newton correction for g ≡ Halley's correction for f

-g0/g1 = -(f0 * f1^(-1/2)) * f1^(3/2) / (f1^2 - f0*f2/2) = -f0*f1 / (f1^2 - f0*f2/2)