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.
Relative tolerance.
Absolute tolerance.
number of addition inner points of uniform grid.
The algorithm computes function value for the N points.
Then reassing ax to the point that preceeds the grid's argmin,
reassing bx to the point that follows the the grid's argmin.
The search interval reduces (N + 1) / 2 times.
Preconditions:
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 FindLocalMinResult 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)
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.