This chapter describes routines for finding minima of arbitrary one-dimensional functions.
The minimization algorithms begin with a bounded region known to contain
a minimum. The region is described by a
lower bound a and an upper bound
b
, with an estimate of the location of the minimum x
.
The value of the function at x
must be less than the value of the
function at the ends of the interval,
f(a) > f(x) < f(b)
This condition guarantees that a minimum is contained somewhere within the
interval. On each iteration a new point x'
is selected using one of the
available algorithms. If the new point is a better estimate of the minimum,
f(x') < f(x)
, then the current estimate of the minimum x
is
updated. The new point also allows the size of the bounded interval to be
reduced, by choosing the most compact set of points which satisfies the
constraint f(a) > f(x) < f(b)
. The interval is reduced until it
encloses the true minimum to a desired tolerance. This provides a best
estimate of the location of the minimum and a rigorous error estimate.
Several bracketing algorithms are available within a single framework. The user provides a high-level driver for the algorithm, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,
s
, for algorithm T
s
using the iteration T
s
for convergence, and repeat iteration if necessaryThe state for the minimizers is held in a GSL::Min::FMinimizer
object .
The updating procedure uses only function evaluations (not derivatives).
The function to minimize is given as an instance of the GSL::Function class to the minimizer.
GSL::Min::FMinimizer.new(t)
GSL::Min::FMinimizer.alloc(t)
These method create an instance of the GSL::Min::FMinimizer
class of
type t. The type t is given by a Ruby constant,
ex1)
include GSL s1 = Min::FMinimizer.new(Min::FMinimizer::GOLDENSECTION)
ex2)
include GSL::Min s2 = FMinimizer.new(FMinimizer::BRENT)
GSL::Min::FMinimizer#set(f, xmin, xlow, xup)
This method sets, or resets, an existing minimizer self to use
the function f (given by a GSL::Function
object) and the initial search interval [xlow, xup],
with a guess for the location of the minimum xmin.
If the interval given does not contain a minimum, then the
method returns an error code of GSL::FAILURE
.
GSL::Min::FMinimizer#set_with_values(f, xmin, fmin, xlow, flow, xup, fup)
Fminimizer#set
but uses the values
fmin, flowe and fup instead of computing
f(xmin), f(xlow) and f(xup).GSL::Min::FMinimizer#name
GSL::Min::FMinimizer#iterate
This method performs a single iteration of the minimizer self. If the iteration encounters an unexpected problem then an error code will be returned,
GSL::EBADFUNC
: the iteration encountered a singular point where the
function evaluated to Inf
or NaN
.GSL::FAILURE
: the algorithm could not improve the current best
approximation or bounding interval.The minimizer maintains a current best estimate of the position of the minimum at all times, and the current interval bounding the minimum. This information can be accessed with the following auxiliary methods
GSL::Min::FMinimizer#x_minimum
GSL::Min::FMinimizer#x_upper
GSL::Min::FMinimizer#x_lower
GSL::Min::FMinimizer#f_minimum
GSL::Min::FMinimizer#f_upper
GSL::Min::FMinimizer#f_lower
GSL::Min::FMinimizer#test_interval(epsabs, epsrel)
GSL::Min.test_interval(xlow, xup, epsabs, epsrel)
These methoeds test for the convergence of the interval
[xlow, xup] with absolute error epsabs and relative
error epsrel. The test returns GSL::SUCCESS
if the following condition is achieved,
|a - b| < epsabs + epsrel min(|a|,|b|)
when the interval x = [a,b]
does not include the origin.
If the interval includes the origin then min(|a|,|b|)
is
replaced by zero (which is the minimum value of |x| over the interval).
This ensures that the relative error is accurately estimated for minima
close to the origin.
This condition on the interval also implies that any estimate of the minimum x_m in the interval satisfies the same condition with respect to the true minimum x_m^*,
|x_m - x_m^*| < epsabs + epsrel x_m^*
assuming that the true minimum x_m^* is contained within the interval.
To find the minimum of the function f(x) = cos(x) + 1.0:
#!/usr/bin/env ruby require("gsl") include GSL::Min fn1 = Function.new { |x| Math::cos(x) + 1.0 } iter = 0; max_iter = 500 m = 2.0 # initial guess m_expected = Math::PI a = 0.0 b = 6.0 gmf = FMinimizer.new(FMinimizer::BRENT) gmf.set(fn1, m, a, b) printf("using %s method\n", gmf.name) printf("%5s [%9s, %9s] %9s %10s %9s\n", "iter", "lower", "upper", "min", "err", "err(est)") printf("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n", iter, a, b, m, m - m_expected, b - a) begin iter += 1 status = gmf.iterate status = gmf.test_interval(0.001, 0.0) puts("Converged:") if status == GSL::SUCCESS a = gmf.x_lower b = gmf.x_upper m = gmf.x_minimum printf("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n", iter, a, b, m, m - m_expected, b - a); end while status == GSL::CONTINUE and iter < max_iter