Find a real minimum of a real function f(x) via bracketing. Given a function f and a range (ax..bx), returns the value of x in the range which is closest to a minimum of f(x). f is never evaluted at the endpoints of ax and bx. If f(x) has more than one minimum in the range, one will be chosen arbitrarily. If f(x) returns NaN or -Infinity, (x, f(x), NaN) will be returned; otherwise, this algorithm is guaranteed to succeed.
Function to be analyzed
Left bound of initial range of f known to contain the minimum.
Right bound of initial range of f known to contain the minimum.
ax and bx shall be finite reals.
relTolerance shall be normal positive real.
absTolerance shall be normal positive real no less then T.epsilon*2.
A tuple consisting of x, y = f(x) and error = 3 * (absTolerance * fabs(x) + relTolerance).
The method used is a combination of golden section search and successive parabolic interpolation. Convergence is never much slower than that for a Fibonacci search.
References: "Algorithms for Minimization without Derivatives", Richard Brent, Prentice-Hall, Inc. (1973)
auto ret = findLocalMin((double x) => (x-4)^^2, -1e9, 1e9); assert(ret.x.approxEqual(4.0)); assert(ret.y.approxEqual(0.0));