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[Albert Chan]

(06-19-2024 12:20 PM)Albert Chan Wrote:  asymptote line slope is more extreme (in size) than at any other point,

Here is a trick that work with any rate guess, and have one-sided convergence.

lua> n,pv,pmt,fv = 10,50,-30,80 -- sample #23
lua> f = fn'x: npmt(n,x,pv,pmt,fv)'
lua> i = 0.01 -- any guess OK

lua> X = -pv -- go for big root
lua> for k=1,5 do eps=f(i)/X; i=i+eps; print(i, eps) end
0.3514866009669545      0.3414866009669545
0.5527206785990931      0.20123407763213863
0.5821388175291544      0.029418138930061274
0.5844404930347874      0.002301675505633014
0.5846067021783057      0.0001662091435182589

We can speed up convergence, with smaller slope.
What is the ideal slope? We assume constant rate of convergence.

lua> r = 0.0001662091435182589 / 0.002301675505633014
lua> X = X * (1-r)
lua> for k=1,5 do eps=f(i)/X; i=i+eps; print(i, eps) end
0.5846195592477864      1.2857069480758773e-05
0.5846195526588186      -6.588967862434568e-09
0.5846195526622116      3.3930050292746237e-12
0.5846195526622099      -1.6848616523122454e-15
0.5846195526622099      -0