RationalCurve Class

public class RationalCurve : NonlinearCurve

Public Class RationalCurve
	Inherits NonlinearCurve

public ref class RationalCurve : public NonlinearCurve

type RationalCurve = 
    class
        inherit NonlinearCurve
    end

Inheritance

Object  →  Curve  →  NonlinearCurve  →  RationalCurve

Use the RationalCurve class to represent a rational function that can be used in non-linear least squares fitting. Rational functions are ratios of polynomials and can approximate a wide variety of functions with relatively few parameters.

Equation:

y = (a₀ + a₁x + a₂x² + ... + aₙx^n) / (1 + b₁x + b₂x² + ... + b_dx^d)

Parameters:

A rational curve has n+d+1 parameters, where n is the degree of the numerator polynomial, and d is the degree of the denominator polynomial. The first n+1 parameters are the coefficients of the numerator polynomial (a₀, a₁, ..., aₙ), starting with the constant term. The remaining d parameters are the coefficients of the denominator polynomial (b₁, b₂, ..., b_d), starting with the linear term. The denominator always has an implicit constant term of 1.

Degree Selection Guidelines:

• Start with low degrees (e.g., (1,1) or (2,1)) and increase only if needed
• Higher degrees provide more flexibility but risk overfitting
• Use cross-validation or information criteria (AIC, BIC) to select optimal degrees
• Numerator degree typically equals or exceeds denominator degree for most applications

Applications:

• Approximating complex mathematical functions
• Transfer function modeling in control systems
• Padé approximants
• Modeling asymptotic behavior with oblique asymptotes
• Rational interpolation

Numerical Considerations:

• Poles:
• Overfitting:
• Initial values:
• Scaling:

Asymptotic Behavior:

As x → ±∞, the behavior depends on the degree relationship:

• If n < d: y → 0 (horizontal asymptote at y = 0)
• If n = d: y → aₙ/b_d (horizontal asymptote at ratio of leading coefficients)
• If n > d: y → ±∞ (oblique or higher-order asymptote)

.

For details on how to fit a curve of this type, see the NonlinearCurveFitter class.

Reference