Programming Challenge: a classic trigonometry problem

[Albert Chan]

(01-07-2014 06:03 AM)Thomas Klemm Wrote:  Wikipedia provides yet another solution: Crossed ladders problem
\[ x^3(x-c)=1 \]
Which leads to:
\[ x=\frac{1}{x^3}+c \]

Note: X ≈ C if C is big; X ≈ 1 if C is tiny.      → X = [C, C+1]

lua> a, b, c = 40, 30, 15
lua> D = sqrt(a*a-b*b)
lua> C = 4*c / D
lua> S = require'solver'
lua> S.secant(fn'X: (X-C)-X^-3', C, C+1, 1e-9, true)
3.2677868380553634      2.3488985640799114
2.3488985640799114      2.3451475080781123
2.3451475080781123      2.345304807136498
2.345304807136498        2.345304763754943
2.345304763754943        2.345304763754418
2.345304763754418
lua> X = _
lua> u = D*(X+1/X)/2      -- FYI: u+v = D*X; u-v = D/X
lua> sqrt((a+u)*(a-u))     -- = x
15.987649008568585