hmatrix-0.12.0.2: Linear algebra and numerical computation

Portability uses ffi provisional Alberto Ruiz (aruiz at um dot es)

Numeric.GSL.Minimization

Description

Minimization of a multidimensional function using some of the algorithms described in:

The example in the GSL manual:

```
f [x,y] = 10*(x-1)^2 + 20*(y-2)^2 + 30

main = do
let (s,p) = minimize NMSimplex2 1E-2 30 [1,1] f [5,7]
print s
print p

> main
[0.9920430849306288,1.9969168063253182]
0.000  512.500  1.130  6.500  5.000
1.000  290.625  1.409  5.250  4.000
2.000  290.625  1.409  5.250  4.000
3.000  252.500  1.409  5.500  1.000
...
22.000   30.001  0.013  0.992  1.997
23.000   30.001  0.008  0.992  1.997
```

The path to the solution can be graphically shown by means of:

``Graphics.Plot.mplot` \$ drop 3 (`toColumns` p)`

Taken from the GSL manual:

The vector Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is a quasi-Newton method which builds up an approximation to the second derivatives of the function f using the difference between successive gradient vectors. By combining the first and second derivatives the algorithm is able to take Newton-type steps towards the function minimum, assuming quadratic behavior in that region.

The bfgs2 version of this minimizer is the most efficient version available, and is a faithful implementation of the line minimization scheme described in Fletcher's Practical Methods of Optimization, Algorithms 2.6.2 and 2.6.4. It supercedes the original bfgs routine and requires substantially fewer function and gradient evaluations. The user-supplied tolerance tol corresponds to the parameter sigma used by Fletcher. A value of 0.1 is recommended for typical use (larger values correspond to less accurate line searches).

The nmsimplex2 version is a new O(N) implementation of the earlier O(N^2) nmsimplex minimiser. It calculates the size of simplex as the rms distance of each vertex from the center rather than the mean distance, which has the advantage of allowing a linear update.

Synopsis

# Documentation

Arguments

 :: MinimizeMethod -> Double desired precision of the solution (size test) -> Int maximum number of iterations allowed -> [Double] sizes of the initial search box -> ([Double] -> Double) function to minimize -> [Double] starting point -> ([Double], Matrix Double) solution vector and optimization path

Minimization without derivatives

Arguments

 :: MinimizeMethod -> Double desired precision of the solution (size test) -> Int maximum number of iterations allowed -> Vector Double sizes of the initial search box -> (Vector Double -> Double) function to minimize -> Vector Double starting point -> (Vector Double, Matrix Double) solution vector and optimization path

Minimization without derivatives (vector version)

Constructors

 NMSimplex NMSimplex2

Arguments

 :: MinimizeMethodD -> Double desired precision of the solution (gradient test) -> Int maximum number of iterations allowed -> Double size of the first trial step -> Double tol (precise meaning depends on method) -> ([Double] -> Double) function to minimize -> ([Double] -> [Double]) gradient -> [Double] starting point -> ([Double], Matrix Double) solution vector and optimization path

Minimization with derivatives.

Arguments

 :: MinimizeMethodD -> Double desired precision of the solution (gradient test) -> Int maximum number of iterations allowed -> Double size of the first trial step -> Double tol (precise meaning depends on method) -> (Vector Double -> Double) function to minimize -> (Vector Double -> Vector Double) gradient -> Vector Double starting point -> (Vector Double, Matrix Double) solution vector and optimization path

Minimization with derivatives (vector version)

Constructors

 ConjugateFR ConjugatePR VectorBFGS VectorBFGS2 SteepestDescent