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 * Copyright (c) 2010-2020 Haifeng Li. All rights reserved.
 *
 * Smile is free software: you can redistribute it and/or modify
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package smile.math.rbf;

/**
 * Inverse multiquadric RBF. φ(r) = (r² + r²₀)^-1/2
 * where r₀ is a scale factor. Although it sounds odd, the inverse
 * multiquadric gives results that are comparable to the multiquadric,
 * sometimes better. The reason is what really matters is smoothness, and
 * certain properties of the function's Fourier transform that are not very
 * different between the multiquadric and its reciprocal. The fact that one
 * increases monotonically and the other decreases turns out to be almost
 * irrelevant. Besides, inverse multiquadric will extrapolate any function to
 * zero far from the data.
 * 
 * In general, r₀ should be larger than the typical separation of
 * points but smaller than the "outer scale" or feature size of the function
 * to interplate. There can be several orders of magnitude difference between
 * the interpolation accuracy with a good choice for r₀, versus a
 * poor choice, so it is definitely worth some experimentation. One way to
 * experiment is to construct an RBF interpolator omitting one data point
 * at a time and measuring the interpolation error at the omitted point.
 *
 * @author Haifeng Li
 */
public class InverseMultiquadricRadialBasis implements RadialBasisFunction {
    private static final long serialVersionUID = 1L;

    private double r02;

    public InverseMultiquadricRadialBasis() {
        this(1.0);
    }

    public InverseMultiquadricRadialBasis(double scale) {
        r02 = scale * scale;
    }

    @Override
    public double f(double r) {
        return 1 / Math.sqrt(r*r + r02);
    }

    @Override
    public String toString() {
        return String.format("Inverse Multi-quadric Radial Basis (r0^2 = %.4f)", r02);
    }
}