Extreme Optimization – Curves (Function Mapping) Part 1

Posted by JoshReuben on Geeks with Blogs See other posts from Geeks with Blogs or by JoshReuben
Published on Fri, 28 Jan 2011 09:46:22 GMT Indexed on 2011/01/28 23:27 UTC
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Overview

·        a curve is a functional map relationship between two factors (i.e. a function - However, the word function is a reserved word).

·        You can use the EO API to create common types of functions, find zeroes and calculate derivatives - currently supports constants, lines, quadratic curves, polynomials and Chebyshev approximations.

·        A function basis is a set of functions that can be combined to form a particular class of functions.

 

The Curve class

·        the abstract base class from which all other curve classes are derived – it provides the following methods:

·        ValueAt(Double) - evaluates the curve at a specific point.

·        SlopeAt(Double) - evaluates the derivative

·        Integral(Double, Double) - evaluates the definite integral over a specified interval.

·        TangentAt(Double) - returns a Line curve that is the tangent to the curve at a specific point.

·        FindRoots() - attempts to find all the roots or zeroes of the curve.

·        A particular type of curve is defined by a Parameters property, of type ParameterCollection

 

The GeneralCurve class

·        defines a curve whose value and, optionally, derivative and integrals, are calculated using arbitrary methods. A general curve has no parameters.

·        Constructor params:  RealFunction delegates – 1 for the function, and optionally another 2 for the derivative and integral

·        If no derivative  or integral function is supplied, they are calculated via the NumericalDifferentiation  and AdaptiveIntegrator classes in the Extreme.Mathematics.Calculus namespace.

// the function is 1/(1+x^2)

private double f(double x)

{

    return 1 / (1 + x*x);

}

 

// Its derivative is -2x/(1+x^2)^2

private double df(double x)

{

    double y = 1 + x*x;

    return -2*x* / (y*y);

}

 

// The integral of f is Arctan(x), which is available from the Math class.

var c1 = new GeneralCurve (new RealFunction(f), new RealFunction(df), new RealFunction(System.Math.Atan));

// Find the tangent to this curve at x=1 (the Line class is derived from Curve)

Line l1 = c1.TangentAt(1);

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