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### Locating Function Minima: Brent's algorithm

##### synopsis

    #include <boost/math/tools/minima.hpp>

    template <class F, class T>
    std::pair<T, T> brent_find_minima(F f, T min, T max, int bits);

    template <class F, class T>
    std::pair<T, T> brent_find_minima(F f, T min, T max, int bits, boost::uintmax_t& max_iter);

##### Description

These two functions locate the minima of the continuous function f using Brent's algorithm. Parameters are:

f

  The function to minimise. The function should be smooth over the range [min,max], with no maxima occurring in that interval.

min

  The lower endpoint of the range in which to search for the minima.

max

  The upper endpoint of the range in which to search for the minima.

bits

  The number of bits precision to which the minima should be found. Note that in principle, the minima can not be located to greater accuracy than the square root of machine epsilon (for 64-bit double, sqrt(1e-16)≅1e-8), therefore if bits is set to a value greater than one half of the bits in type T, then the value will be ignored.

max_iter

  The maximum number of iterations to use in the algorithm, if not provided the algorithm will just keep on going until the minima is found.

Returns: a pair containing the value of the abscissa at the minima and the value of f(x) at the minima.

##### Implementation

This is a reasonably faithful implementation of Brent's algorithm, refer to:

Brent, R.P. 1973, Algorithms for Minimization without Derivatives (Englewood Cliffs, NJ: Prentice-Hall), Chapter 5.

Numerical Recipes in C, The Art of Scientific Computing, Second Edition, William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Cambridge University Press. 1988, 1992.

An algorithm with guaranteed convergence for finding a zero of a function, R. P. Brent, The Computer Journal, Vol 44, 1971.