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Brents Algorithm Assignment Help


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What is Brents Algorithm?


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:

  • Compute ‘f(x)’, ‘f(y)’, ‘f(z)’
  • Compute A, B, C, D, E
  • Let [y -> y + (A/B)]
  • Repeat as [f(y) -> 0]

      

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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