BrentSearch Class

In numerical analysis, Brent's method is a complicated but popular root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less reliable methods. The idea is to use the secant method or inverse quadratic interpolation if possible, because they converge faster, but to fall back to the more robust bisection method if necessary. Brent's method is due to Richard Brent (1973) and builds on an earlier algorithm of Theodorus Dekker (1969).

The algorithms implemented in this class are based on the original C source code available in Netlib (http://www.netlib.org/c/brent.shar) by Oleg Keselyov, 1991.

References:

• R.P. Brent (1973). Algorithms for Minimization without Derivatives, Chapter 4. Prentice-Hall, Englewood Cliffs, NJ. ISBN 0-13-022335-2.
• Wikipedia contributors. "Brent's method." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 11 May. 2012. Web. 22 Jun. 2012.

The following example shows how to compute the maximum, minimum and a single root of a univariate function.

// Suppose we were given the function x³ + 2x² - 10x + 1 and 
// we have to find its root, maximum and minimum inside 
// the interval [-4, 2]. First, we express this function
// as a lambda expression:
Func<double, double> function = x => x * x * x + 2 * x * x - 10 * x + 1;

// And now we can create the search algorithm:
BrentSearch search = new BrentSearch(function, -4, 2);

// Finally, we can query the information we need
bool success1 = search.Maximize();  // should be true
double max = search.Solution;       // occurs at -2.61

bool success2 = search.Minimize();   // should be true  
double min = search.Solution;       // occurs at  1.28

bool success3 = search.FindRoot();  // should be true 
double root = search.Solution;      // occurs at  0.10
double value = search.Value;        // should be zero

Reference