Version 5.0 The LEDA User Manual

5.17 Numerical Analysis Functions ( numerical
analysis )
We collect some functions of numerical analysis. The algorithms in this section are not
the best known and are not recommended for serious use. We refer the reader to the book
“Numerical Recipes in C: The Art of Scientific Computing” by B.P. Flannery, W.H. Press,
S.A. Teukolsky, and W.T. Vetterling, Cambridge University Press for better algorithms.
The functions in this section become available by including numerical analysis.h.
5.17.1 Minima and Maxima
double minimize function(double (∗f)(double), double& xmin, double tol = 1.0e − 10)
finds a local minimum of the function f of one argument. The
minimizing argument is returned in xmin and the minimal
function value is returned as the result of the function. xmin
is determined with tolerance tol, i.e., the true value of the
minimizing argument is contained in the interval [xmin(1 −
ɛ), xmin(1 + ɛ)], where ɛ = max(1, xmin) · tol. Please do not
choose tol smaller than 10 −15 .
Precondition: : If +∞ or −∞ is a local minimum of f, then
the call of minimize function may not terminate.
The algorithm is implemented as follows: First three arguments
are determined such that a < b < c (or a > b > c)
and f(a) ≥ f(b) ≤ f(c), i.e., a and c bracket a minimum.
The interval is found by first taking two arbitrary arguments
and comparing their function values. The argument with
the larger function value is taken as a. Then steps of larger
and larger size starting at b are taken until a function value
larger than f(b) is found. Once the bracketing interval is
found, golden-ratio search is applied to it.
template
double minimize function(const F & f, double& xmin, double tol = 1.0e − 10)
a more flexible version of the above. It is assumed that class
F offers the operator
double operator( )(double x). This operator is taken as the
function f.