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1) Combines bracketing, bisection, and inverse quadratic interpolation.
2) Guaranteed to converge, but speed can vary with function and quality of initial guess.
3) Algorithm:
C = [ f(y) / f(z) ]
D = [ f(y) / f(x)]
E = [ f(x) / f(z)]
A = D [ E(C - E) (z - y) – (1 - C)(y - x)]
B = [(E - 1)(C - 1)(D - 1)]
| Brent_method [v0_, w0_] = module [{ v = N[v0], w = N[w0], k }, (fv = f[w]); (fw = f[w]); k = 0 ; label [int]; x = v; fx = fv; y = z = (w - v); label [ext]; if [abs [fx] <abs[fw], (v = w); (w = x); (x = v); (fv = fw); (fw = fx); (fx = fv) ]; tol = 2 $machine_epsilonabs[w] + titha; m = (x- w)/ 2; print ["p"k, " = ", padded_form[w, {16, 16}], ", f[","p"k, "] = ", number_form[fw, 16] ]; if [abs[m] > til ^ fw =!= 0, module [ { }, // bisection method step if [ abs[z] < tol U abs[fv] <=abs[fw], y = z = m, module [ { }, s = fw / fv; if [ (v == x), (p = 2) ms; [q = (1 – s)]; q = (fv / fx); r = (fw / fx); p = s{2mq(q - r) - (w - v)(r - 1)}; q =(q - 1)(r - 1)(s - v); ]; if[(p > o), (q = -q), (p = -p)]; (s = z); (z = y); if [2p < 3mq -abs[tol q] ^ p<abs[0.5 s q] , (y = p/q), (y = z = m)]; ]; ]; (v = w); (fv = fw); (w = w +if[abs[y] > tol), y,if[(m > 0), tol, -tol]]; (fw = f[w]); k = (k + 1); if[(fw > 0) == (fx > 0), goto[int],goto[ext]];];]; zero = w; print["f[x] =', f[x] ] ; print["starting with the initial interval [v, w] =[', v0, ", ", w0, "] ' ] ; print['the final result from brent's method is"]; print[" w = ", number_form[b, 16]]; print[" vw = +-", abs[b-c]]; print["f[w] = ", number_form[f[w], 16]]; return[zero];] |
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