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package org.apache.commons.math3.analysis.interpolation;

import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;

import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;
import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableVectorFunction;
import org.apache.commons.math3.analysis.polynomials.PolynomialFunction;
import org.apache.commons.math3.exception.MathArithmeticException;
import org.apache.commons.math3.exception.NoDataException;
import org.apache.commons.math3.exception.ZeroException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.CombinatoricsUtils;

Polynomial interpolator using both sample values and sample derivatives.

The interpolation polynomials match all sample points, including both values
and provided derivatives. There is one polynomial for each component of
the values vector. All polynomials have the same degree. The degree of the
polynomials depends on the number of points and number of derivatives at each
point. For example the interpolation polynomials for n sample points without
any derivatives all have degree n-1. The interpolation polynomials for n
sample points with the two extreme points having value and first derivative
and the remaining points having value only all have degree n+1. The
interpolation polynomial for n sample points with value, first and second
derivative for all points all have degree 3n-1.

Since:
3.1/** Polynomial interpolator using both sample values and sample derivatives.
 * <p>
 * The interpolation polynomials match all sample points, including both values
 * and provided derivatives. There is one polynomial for each component of
 * the values vector. All polynomials have the same degree. The degree of the
 * polynomials depends on the number of points and number of derivatives at each
 * point. For example the interpolation polynomials for n sample points without
 * any derivatives all have degree n-1. The interpolation polynomials for n
 * sample points with the two extreme points having value and first derivative
 * and the remaining points having value only all have degree n+1. The
 * interpolation polynomial for n sample points with value, first and second
 * derivative for all points all have degree 3n-1.
 * </p>
 *
 * @since 3.1
 */
public class HermiteInterpolator implements UnivariateDifferentiableVectorFunction {

    
Sample abscissae. /** Sample abscissae. */
    private final List<Double> abscissae;

    
Top diagonal of the divided differences array. /** Top diagonal of the divided differences array. */
    private final List<double[]> topDiagonal;

    
Bottom diagonal of the divided differences array. /** Bottom diagonal of the divided differences array. */
    private final List<double[]> bottomDiagonal;

    
Create an empty interpolator.
/** Create an empty interpolator.
     */
    public HermiteInterpolator() {
        this.abscissae      = new ArrayList<Double>();
        this.topDiagonal    = new ArrayList<double[]>();
        this.bottomDiagonal = new ArrayList<double[]>();
    }

    
Add a sample point.

This method must be called once for each sample point. It is allowed to
mix some calls with values only with calls with values and first
derivatives.


The point abscissae for all calls must be different.

Params:
x – abscissa of the sample point
value – value and derivatives of the sample point
(if only one row is passed, it is the value, if two rows are
passed the first one is the value and the second the derivative
and so on)
Throws:
ZeroException – if the abscissa difference between added point
and a previous point is zero (i.e. the two points are at same abscissa)
MathArithmeticException – if the number of derivatives is larger
than 20, which prevents computation of a factorial/** Add a sample point.
     * <p>
     * This method must be called once for each sample point. It is allowed to
     * mix some calls with values only with calls with values and first
     * derivatives.
     * </p>
     * <p>
     * The point abscissae for all calls <em>must</em> be different.
     * </p>
     * @param x abscissa of the sample point
     * @param value value and derivatives of the sample point
     * (if only one row is passed, it is the value, if two rows are
     * passed the first one is the value and the second the derivative
     * and so on)
     * @exception ZeroException if the abscissa difference between added point
     * and a previous point is zero (i.e. the two points are at same abscissa)
     * @exception MathArithmeticException if the number of derivatives is larger
     * than 20, which prevents computation of a factorial
     */
    public void addSamplePoint(final double x, final double[] ... value)
        throws ZeroException, MathArithmeticException {

        for (int i = 0; i < value.length; ++i) {

            final double[] y = value[i].clone();
            if (i > 1) {
                double inv = 1.0 / CombinatoricsUtils.factorial(i);
                for (int j = 0; j < y.length; ++j) {
                    y[j] *= inv;
                }
            }

            // update the bottom diagonal of the divided differences array
            final int n = abscissae.size();
            bottomDiagonal.add(n - i, y);
            double[] bottom0 = y;
            for (int j = i; j < n; ++j) {
                final double[] bottom1 = bottomDiagonal.get(n - (j + 1));
                final double inv = 1.0 / (x - abscissae.get(n - (j + 1)));
                if (Double.isInfinite(inv)) {
                    throw new ZeroException(LocalizedFormats.DUPLICATED_ABSCISSA_DIVISION_BY_ZERO, x);
                }
                for (int k = 0; k < y.length; ++k) {
                    bottom1[k] = inv * (bottom0[k] - bottom1[k]);
                }
                bottom0 = bottom1;
            }

            // update the top diagonal of the divided differences array
            topDiagonal.add(bottom0.clone());

            // update the abscissae array
            abscissae.add(x);

        }

    }

    
Compute the interpolation polynomials.
Throws:
NoDataException – if sample is empty
Returns:
interpolation polynomials array/** Compute the interpolation polynomials.
     * @return interpolation polynomials array
     * @exception NoDataException if sample is empty
     */
    public PolynomialFunction[] getPolynomials()
        throws NoDataException {

        // safety check
        checkInterpolation();

        // iteration initialization
        final PolynomialFunction zero = polynomial(0);
        PolynomialFunction[] polynomials = new PolynomialFunction[topDiagonal.get(0).length];
        for (int i = 0; i < polynomials.length; ++i) {
            polynomials[i] = zero;
        }
        PolynomialFunction coeff = polynomial(1);

        // build the polynomials by iterating on the top diagonal of the divided differences array
        for (int i = 0; i < topDiagonal.size(); ++i) {
            double[] tdi = topDiagonal.get(i);
            for (int k = 0; k < polynomials.length; ++k) {
                polynomials[k] = polynomials[k].add(coeff.multiply(polynomial(tdi[k])));
            }
            coeff = coeff.multiply(polynomial(-abscissae.get(i), 1.0));
        }

        return polynomials;

    }

    
Interpolate value at a specified abscissa.
 Calling this method is equivalent to call the 
value methods of all polynomials returned by getPolynomials, except it does not build the intermediate polynomials, so this method is faster and numerically more stable. 
Params:
x – interpolation abscissa
Throws:
NoDataException – if sample is empty
Returns:
interpolated value/** Interpolate value at a specified abscissa.
     * <p>
     * Calling this method is equivalent to call the {@link PolynomialFunction#value(double)
     * value} methods of all polynomials returned by {@link #getPolynomials() getPolynomials},
     * except it does not build the intermediate polynomials, so this method is faster and
     * numerically more stable.
     * </p>
     * @param x interpolation abscissa
     * @return interpolated value
     * @exception NoDataException if sample is empty
     */
    public double[] value(double x)
        throws NoDataException {

        // safety check
        checkInterpolation();

        final double[] value = new double[topDiagonal.get(0).length];
        double valueCoeff = 1;
        for (int i = 0; i < topDiagonal.size(); ++i) {
            double[] dividedDifference = topDiagonal.get(i);
            for (int k = 0; k < value.length; ++k) {
                value[k] += dividedDifference[k] * valueCoeff;
            }
            final double deltaX = x - abscissae.get(i);
            valueCoeff *= deltaX;
        }

        return value;

    }

    
Interpolate value at a specified abscissa.
 Calling this method is equivalent to call the value methods of all polynomials returned by getPolynomials, except it does not build the intermediate polynomials, so this method is faster and numerically more stable. 
Params:
x – interpolation abscissa
Throws:
NoDataException – if sample is empty
Returns:
interpolated value/** Interpolate value at a specified abscissa.
     * <p>
     * Calling this method is equivalent to call the {@link
     * PolynomialFunction#value(DerivativeStructure) value} methods of all polynomials
     * returned by {@link #getPolynomials() getPolynomials}, except it does not build the
     * intermediate polynomials, so this method is faster and numerically more stable.
     * </p>
     * @param x interpolation abscissa
     * @return interpolated value
     * @exception NoDataException if sample is empty
     */
    public DerivativeStructure[] value(final DerivativeStructure x)
        throws NoDataException {

        // safety check
        checkInterpolation();

        final DerivativeStructure[] value = new DerivativeStructure[topDiagonal.get(0).length];
        Arrays.fill(value, x.getField().getZero());
        DerivativeStructure valueCoeff = x.getField().getOne();
        for (int i = 0; i < topDiagonal.size(); ++i) {
            double[] dividedDifference = topDiagonal.get(i);
            for (int k = 0; k < value.length; ++k) {
                value[k] = value[k].add(valueCoeff.multiply(dividedDifference[k]));
            }
            final DerivativeStructure deltaX = x.subtract(abscissae.get(i));
            valueCoeff = valueCoeff.multiply(deltaX);
        }

        return value;

    }

    
Check interpolation can be performed.
Throws:
NoDataException – if interpolation cannot be performed
because sample is empty/** Check interpolation can be performed.
     * @exception NoDataException if interpolation cannot be performed
     * because sample is empty
     */
    private void checkInterpolation() throws NoDataException {
        if (abscissae.isEmpty()) {
            throw new NoDataException(LocalizedFormats.EMPTY_INTERPOLATION_SAMPLE);
        }
    }

    
Create a polynomial from its coefficients.
Params:
c – polynomials coefficients
Returns:
polynomial/** Create a polynomial from its coefficients.
     * @param c polynomials coefficients
     * @return polynomial
     */
    private PolynomialFunction polynomial(double ... c) {
        return new PolynomialFunction(c);
    }

}