MathArrays.java

/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

package org.apache.commons.math4.legacy.core;

import java.lang.reflect.Array;
import java.util.Arrays;
import java.util.Iterator;
import java.util.TreeSet;

import org.apache.commons.numbers.core.Precision;
import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
import org.apache.commons.math4.legacy.exception.MathArithmeticException;
import org.apache.commons.math4.legacy.exception.MathIllegalArgumentException;
import org.apache.commons.math4.legacy.exception.MathInternalError;
import org.apache.commons.math4.legacy.exception.NoDataException;
import org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException;
import org.apache.commons.math4.legacy.exception.NotANumberException;
import org.apache.commons.math4.legacy.exception.NotPositiveException;
import org.apache.commons.math4.legacy.exception.NotStrictlyPositiveException;
import org.apache.commons.math4.legacy.exception.NullArgumentException;
import org.apache.commons.math4.legacy.exception.NumberIsTooLargeException;
import org.apache.commons.math4.legacy.exception.NotFiniteNumberException;
import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
import org.apache.commons.math4.core.jdkmath.JdkMath;

/**
 * Arrays utilities.
 *
 * @since 3.0
 */
public final class MathArrays {

    /**
     * Private constructor.
     */
    private MathArrays() {}

    /**
     * Real-valued function that operate on an array or a part of it.
     * @since 3.1
     */
    public interface Function {
        /**
         * Operates on an entire array.
         *
         * @param array Array to operate on.
         * @return the result of the operation.
         */
        double evaluate(double[] array);
        /**
         * @param array Array to operate on.
         * @param startIndex Index of the first element to take into account.
         * @param numElements Number of elements to take into account.
         * @return the result of the operation.
         */
        double evaluate(double[] array,
                        int startIndex,
                        int numElements);
    }

    /**
     * Create a copy of an array scaled by a value.
     *
     * @param arr Array to scale.
     * @param val Scalar.
     * @return scaled copy of array with each entry multiplied by val.
     * @since 3.2
     */
    public static double[] scale(double val, final double[] arr) {
        double[] newArr = new double[arr.length];
        for (int i = 0; i < arr.length; i++) {
            newArr[i] = arr[i] * val;
        }
        return newArr;
    }

    /**
     * <p>Multiply each element of an array by a value.</p>
     *
     * <p>The array is modified in place (no copy is created).</p>
     *
     * @param arr Array to scale
     * @param val Scalar
     * @since 3.2
     */
    public static void scaleInPlace(double val, final double[] arr) {
        for (int i = 0; i < arr.length; i++) {
            arr[i] *= val;
        }
    }

    /**
     * Creates an array whose contents will be the element-by-element
     * addition of the arguments.
     *
     * @param a First term of the addition.
     * @param b Second term of the addition.
     * @return a new array {@code r} where {@code r[i] = a[i] + b[i]}.
     * @throws DimensionMismatchException if the array lengths differ.
     * @since 3.1
     */
    public static double[] ebeAdd(double[] a, double[] b) {
        checkEqualLength(a, b);

        final double[] result = a.clone();
        for (int i = 0; i < a.length; i++) {
            result[i] += b[i];
        }
        return result;
    }
    /**
     * Creates an array whose contents will be the element-by-element
     * subtraction of the second argument from the first.
     *
     * @param a First term.
     * @param b Element to be subtracted.
     * @return a new array {@code r} where {@code r[i] = a[i] - b[i]}.
     * @throws DimensionMismatchException if the array lengths differ.
     * @since 3.1
     */
    public static double[] ebeSubtract(double[] a, double[] b) {
        checkEqualLength(a, b);

        final double[] result = a.clone();
        for (int i = 0; i < a.length; i++) {
            result[i] -= b[i];
        }
        return result;
    }
    /**
     * Creates an array whose contents will be the element-by-element
     * multiplication of the arguments.
     *
     * @param a First factor of the multiplication.
     * @param b Second factor of the multiplication.
     * @return a new array {@code r} where {@code r[i] = a[i] * b[i]}.
     * @throws DimensionMismatchException if the array lengths differ.
     * @since 3.1
     */
    public static double[] ebeMultiply(double[] a, double[] b) {
        checkEqualLength(a, b);

        final double[] result = a.clone();
        for (int i = 0; i < a.length; i++) {
            result[i] *= b[i];
        }
        return result;
    }
    /**
     * Creates an array whose contents will be the element-by-element
     * division of the first argument by the second.
     *
     * @param a Numerator of the division.
     * @param b Denominator of the division.
     * @return a new array {@code r} where {@code r[i] = a[i] / b[i]}.
     * @throws DimensionMismatchException if the array lengths differ.
     * @since 3.1
     */
    public static double[] ebeDivide(double[] a, double[] b) {
        checkEqualLength(a, b);

        final double[] result = a.clone();
        for (int i = 0; i < a.length; i++) {
            result[i] /= b[i];
        }
        return result;
    }

    /**
     * Calculates the L<sub>1</sub> (sum of abs) distance between two points.
     *
     * @param p1 the first point
     * @param p2 the second point
     * @return the L<sub>1</sub> distance between the two points
     * @throws DimensionMismatchException if the array lengths differ.
     */
    public static double distance1(double[] p1, double[] p2) {
        checkEqualLength(p1, p2);
        double sum = 0;
        for (int i = 0; i < p1.length; i++) {
            sum += JdkMath.abs(p1[i] - p2[i]);
        }
        return sum;
    }

    /**
     * Calculates the L<sub>1</sub> (sum of abs) distance between two points.
     *
     * @param p1 the first point
     * @param p2 the second point
     * @return the L<sub>1</sub> distance between the two points
     * @throws DimensionMismatchException if the array lengths differ.
     */
    public static int distance1(int[] p1, int[] p2) {
        checkEqualLength(p1, p2);
        int sum = 0;
        for (int i = 0; i < p1.length; i++) {
            sum += JdkMath.abs(p1[i] - p2[i]);
        }
        return sum;
    }

    /**
     * Calculates the L<sub>2</sub> (Euclidean) distance between two points.
     *
     * @param p1 the first point
     * @param p2 the second point
     * @return the L<sub>2</sub> distance between the two points
     * @throws DimensionMismatchException if the array lengths differ.
     */
    public static double distance(double[] p1, double[] p2) {
        checkEqualLength(p1, p2);
        double sum = 0;
        for (int i = 0; i < p1.length; i++) {
            final double dp = p1[i] - p2[i];
            sum += dp * dp;
        }
        return JdkMath.sqrt(sum);
    }

    /**
     * Calculates the L<sub>2</sub> (Euclidean) distance between two points.
     *
     * @param p1 the first point
     * @param p2 the second point
     * @return the L<sub>2</sub> distance between the two points
     * @throws DimensionMismatchException if the array lengths differ.
     */
    public static double distance(int[] p1, int[] p2) {
        checkEqualLength(p1, p2);
        double sum = 0;
        for (int i = 0; i < p1.length; i++) {
            final double dp = (double) p1[i] - p2[i];
            sum += dp * dp;
        }
        return JdkMath.sqrt(sum);
    }

    /**
     * Calculates the L<sub>&infin;</sub> (max of abs) distance between two points.
     *
     * @param p1 the first point
     * @param p2 the second point
     * @return the L<sub>&infin;</sub> distance between the two points
     * @throws DimensionMismatchException if the array lengths differ.
     */
    public static double distanceInf(double[] p1, double[] p2) {
        checkEqualLength(p1, p2);
        double max = 0;
        for (int i = 0; i < p1.length; i++) {
            max = JdkMath.max(max, JdkMath.abs(p1[i] - p2[i]));
        }
        return max;
    }

    /**
     * Calculates the L<sub>&infin;</sub> (max of abs) distance between two points.
     *
     * @param p1 the first point
     * @param p2 the second point
     * @return the L<sub>&infin;</sub> distance between the two points
     * @throws DimensionMismatchException if the array lengths differ.
     */
    public static int distanceInf(int[] p1, int[] p2) {
        checkEqualLength(p1, p2);
        int max = 0;
        for (int i = 0; i < p1.length; i++) {
            max = JdkMath.max(max, JdkMath.abs(p1[i] - p2[i]));
        }
        return max;
    }

    /**
     * Specification of ordering direction.
     */
    public enum OrderDirection {
        /** Constant for increasing direction. */
        INCREASING,
        /** Constant for decreasing direction. */
        DECREASING
    }

    /**
     * Check that an array is monotonically increasing or decreasing.
     *
     * @param <T> the type of the elements in the specified array
     * @param val Values.
     * @param dir Ordering direction.
     * @param strict Whether the order should be strict.
     * @return {@code true} if sorted, {@code false} otherwise.
     */
    public static  <T extends Comparable<? super T>> boolean isMonotonic(T[] val,
                                      OrderDirection dir,
                                      boolean strict) {
        T previous = val[0];
        final int max = val.length;
        for (int i = 1; i < max; i++) {
            final int comp;
            switch (dir) {
            case INCREASING:
                comp = previous.compareTo(val[i]);
                if (strict) {
                    if (comp >= 0) {
                        return false;
                    }
                } else {
                    if (comp > 0) {
                        return false;
                    }
                }
                break;
            case DECREASING:
                comp = val[i].compareTo(previous);
                if (strict) {
                    if (comp >= 0) {
                        return false;
                    }
                } else {
                    if (comp > 0) {
                        return false;
                    }
                }
                break;
            default:
                // Should never happen.
                throw new MathInternalError();
            }

            previous = val[i];
        }
        return true;
    }

    /**
     * Check that an array is monotonically increasing or decreasing.
     *
     * @param val Values.
     * @param dir Ordering direction.
     * @param strict Whether the order should be strict.
     * @return {@code true} if sorted, {@code false} otherwise.
     */
    public static boolean isMonotonic(double[] val, OrderDirection dir, boolean strict) {
        return checkOrder(val, dir, strict, false);
    }

    /**
     * Check that both arrays have the same length.
     *
     * @param a Array.
     * @param b Array.
     * @param abort Whether to throw an exception if the check fails.
     * @return {@code true} if the arrays have the same length.
     * @throws DimensionMismatchException if the lengths differ and
     * {@code abort} is {@code true}.
     * @since 3.6
     */
    public static boolean checkEqualLength(double[] a,
                                           double[] b,
                                           boolean abort) {
        if (a.length == b.length) {
            return true;
        } else {
            if (abort) {
                throw new DimensionMismatchException(a.length, b.length);
            }
            return false;
        }
    }

    /**
     * Check that both arrays have the same length.
     *
     * @param a Array.
     * @param b Array.
     * @throws DimensionMismatchException if the lengths differ.
     * @since 3.6
     */
    public static void checkEqualLength(double[] a,
                                        double[] b) {
        checkEqualLength(a, b, true);
    }


    /**
     * Check that both arrays have the same length.
     *
     * @param a Array.
     * @param b Array.
     * @param abort Whether to throw an exception if the check fails.
     * @return {@code true} if the arrays have the same length.
     * @throws DimensionMismatchException if the lengths differ and
     * {@code abort} is {@code true}.
     * @since 3.6
     */
    public static boolean checkEqualLength(int[] a,
                                           int[] b,
                                           boolean abort) {
        if (a.length == b.length) {
            return true;
        } else {
            if (abort) {
                throw new DimensionMismatchException(a.length, b.length);
            }
            return false;
        }
    }

    /**
     * Check that both arrays have the same length.
     *
     * @param a Array.
     * @param b Array.
     * @throws DimensionMismatchException if the lengths differ.
     * @since 3.6
     */
    public static void checkEqualLength(int[] a,
                                        int[] b) {
        checkEqualLength(a, b, true);
    }

    /**
     * Check that the given array is sorted.
     *
     * @param val Values.
     * @param dir Ordering direction.
     * @param strict Whether the order should be strict.
     * @param abort Whether to throw an exception if the check fails.
     * @return {@code true} if the array is sorted.
     * @throws NonMonotonicSequenceException if the array is not sorted
     * and {@code abort} is {@code true}.
     */
    public static boolean checkOrder(double[] val, OrderDirection dir,
                                     boolean strict, boolean abort) {
        double previous = val[0];
        final int max = val.length;

        int index;
        ITEM:
        for (index = 1; index < max; index++) {
            switch (dir) {
            case INCREASING:
                if (strict) {
                    if (val[index] <= previous) {
                        break ITEM;
                    }
                } else {
                    if (val[index] < previous) {
                        break ITEM;
                    }
                }
                break;
            case DECREASING:
                if (strict) {
                    if (val[index] >= previous) {
                        break ITEM;
                    }
                } else {
                    if (val[index] > previous) {
                        break ITEM;
                    }
                }
                break;
            default:
                // Should never happen.
                throw new MathInternalError();
            }

            previous = val[index];
        }

        if (index == max) {
            // Loop completed.
            return true;
        }

        // Loop early exit means wrong ordering.
        if (abort) {
            throw new NonMonotonicSequenceException(val[index],
                                                    previous,
                                                    index,
                                                    dir == OrderDirection.INCREASING,
                                                    strict);
        } else {
            return false;
        }
    }

    /**
     * Check that the given array is sorted.
     *
     * @param val Values.
     * @param dir Ordering direction.
     * @param strict Whether the order should be strict.
     * @throws NonMonotonicSequenceException if the array is not sorted.
     * @since 2.2
     */
    public static void checkOrder(double[] val, OrderDirection dir, boolean strict) {
        checkOrder(val, dir, strict, true);
    }

    /**
     * Check that the given array is sorted in strictly increasing order.
     *
     * @param val Values.
     * @throws NonMonotonicSequenceException if the array is not sorted.
     * @since 2.2
     */
    public static void checkOrder(double[] val) {
        checkOrder(val, OrderDirection.INCREASING, true);
    }

    /**
     * Throws DimensionMismatchException if the input array is not rectangular.
     *
     * @param in array to be tested
     * @throws NullArgumentException if input array is null
     * @throws DimensionMismatchException if input array is not rectangular
     * @since 3.1
     */
    public static void checkRectangular(final long[][] in) {
        NullArgumentException.check(in);
        for (int i = 1; i < in.length; i++) {
            if (in[i].length != in[0].length) {
                throw new DimensionMismatchException(
                        LocalizedFormats.DIFFERENT_ROWS_LENGTHS,
                        in[i].length, in[0].length);
            }
        }
    }

    /**
     * Check that all entries of the input array are strictly positive.
     *
     * @param in Array to be tested
     * @throws NotStrictlyPositiveException if any entries of the array are not
     * strictly positive.
     * @since 3.1
     */
    public static void checkPositive(final double[] in) {
        for (double x : in) {
            if (x <= 0) {
                throw new NotStrictlyPositiveException(x);
            }
        }
    }

    /**
     * Check that no entry of the input array is {@code NaN}.
     *
     * @param in Array to be tested.
     * @throws NotANumberException if an entry is {@code NaN}.
     * @since 3.4
     */
    public static void checkNotNaN(final double[] in) {
        for (double x : in) {
            if (Double.isNaN(x)) {
                throw new NotANumberException();
            }
        }
    }

    /**
     * Check that all the elements are real numbers.
     *
     * @param val Arguments.
     * @throws NotFiniteNumberException if any values of the array is not a
     * finite real number.
     */
    public static void checkFinite(final double[] val) {
        for (double x : val) {
            if (!Double.isFinite(x)) {
                throw new NotFiniteNumberException(x);
            }
        }
    }

    /**
     * Check that all entries of the input array are &gt;= 0.
     *
     * @param in Array to be tested
     * @throws NotPositiveException if any array entries are less than 0.
     * @since 3.1
     */
    public static void checkNonNegative(final long[] in) {
        for (long i : in) {
            if (i < 0) {
                throw new NotPositiveException(i);
            }
        }
    }

    /**
     * Check all entries of the input array are &gt;= 0.
     *
     * @param in Array to be tested
     * @throws NotPositiveException if any array entries are less than 0.
     * @since 3.1
     */
    public static void checkNonNegative(final long[][] in) {
        for (int i = 0; i < in.length; i++) {
            for (int j = 0; j < in[i].length; j++) {
                if (in[i][j] < 0) {
                    throw new NotPositiveException(in[i][j]);
                }
            }
        }
    }

    /**
     * Returns true iff both arguments are null or have same dimensions and all
     * their elements are equal as defined by
     * {@link Precision#equals(float,float)}.
     *
     * @param x first array
     * @param y second array
     * @return true if the values are both null or have same dimension
     * and equal elements.
     */
    public static boolean equals(float[] x, float[] y) {
        if (x == null || y == null) {
            return (x == null) == (y == null);
        }
        if (x.length != y.length) {
            return false;
        }
        for (int i = 0; i < x.length; ++i) {
            if (!Precision.equals(x[i], y[i])) {
                return false;
            }
        }
        return true;
    }

    /**
     * Returns true iff both arguments are null or have same dimensions and all
     * their elements are equal as defined by
     * {@link Precision#equalsIncludingNaN(double,double) this method}.
     *
     * @param x first array
     * @param y second array
     * @return true if the values are both null or have same dimension and
     * equal elements
     * @since 2.2
     */
    public static boolean equalsIncludingNaN(float[] x, float[] y) {
        if (x == null || y == null) {
            return (x == null) == (y == null);
        }
        if (x.length != y.length) {
            return false;
        }
        for (int i = 0; i < x.length; ++i) {
            if (!Precision.equalsIncludingNaN(x[i], y[i])) {
                return false;
            }
        }
        return true;
    }

    /**
     * Returns {@code true} iff both arguments are {@code null} or have same
     * dimensions and all their elements are equal as defined by
     * {@link Precision#equals(double,double)}.
     *
     * @param x First array.
     * @param y Second array.
     * @return {@code true} if the values are both {@code null} or have same
     * dimension and equal elements.
     */
    public static boolean equals(double[] x, double[] y) {
        if (x == null || y == null) {
            return (x == null) == (y == null);
        }
        if (x.length != y.length) {
            return false;
        }
        for (int i = 0; i < x.length; ++i) {
            if (!Precision.equals(x[i], y[i])) {
                return false;
            }
        }
        return true;
    }

    /**
     * Returns {@code true} iff both arguments are {@code null} or have same
     * dimensions and all their elements are equal as defined by
     * {@link Precision#equalsIncludingNaN(double,double) this method}.
     *
     * @param x First array.
     * @param y Second array.
     * @return {@code true} if the values are both {@code null} or have same
     * dimension and equal elements.
     * @since 2.2
     */
    public static boolean equalsIncludingNaN(double[] x, double[] y) {
        if (x == null || y == null) {
            return (x == null) == (y == null);
        }
        if (x.length != y.length) {
            return false;
        }
        for (int i = 0; i < x.length; ++i) {
            if (!Precision.equalsIncludingNaN(x[i], y[i])) {
                return false;
            }
        }
        return true;
    }

    /**
     * Normalizes an array to make it sum to a specified value.
     * Returns the result of the transformation
     * <pre>
     *    x |-&gt; x * normalizedSum / sum
     * </pre>
     * applied to each non-NaN element x of the input array, where sum is the
     * sum of the non-NaN entries in the input array.
     * <p>
     * Throws IllegalArgumentException if {@code normalizedSum} is infinite
     * or NaN and ArithmeticException if the input array contains any infinite elements
     * or sums to 0.
     * <p>
     * Ignores (i.e., copies unchanged to the output array) NaNs in the input array.
     *
     * @param values Input array to be normalized
     * @param normalizedSum Target sum for the normalized array
     * @return the normalized array.
     * @throws MathArithmeticException if the input array contains infinite
     * elements or sums to zero.
     * @throws MathIllegalArgumentException if the target sum is infinite or {@code NaN}.
     * @since 2.1
     */
    public static double[] normalizeArray(double[] values, double normalizedSum) {
        if (Double.isInfinite(normalizedSum)) {
            throw new MathIllegalArgumentException(LocalizedFormats.NORMALIZE_INFINITE);
        }
        if (Double.isNaN(normalizedSum)) {
            throw new MathIllegalArgumentException(LocalizedFormats.NORMALIZE_NAN);
        }
        double sum = 0d;
        final int len = values.length;
        double[] out = new double[len];
        for (int i = 0; i < len; i++) {
            if (Double.isInfinite(values[i])) {
                throw new MathIllegalArgumentException(LocalizedFormats.INFINITE_ARRAY_ELEMENT, values[i], i);
            }
            if (!Double.isNaN(values[i])) {
                sum += values[i];
            }
        }
        if (sum == 0) {
            throw new MathArithmeticException(LocalizedFormats.ARRAY_SUMS_TO_ZERO);
        }
        final double scale = normalizedSum / sum;
        for (int i = 0; i < len; i++) {
            if (Double.isNaN(values[i])) {
                out[i] = Double.NaN;
            } else {
                out[i] = values[i] * scale;
            }
        }
        return out;
    }

    /** Build an array of elements.
     * <p>
     * Arrays are filled with field.getZero()
     *
     * @param <T> the type of the field elements
     * @param field field to which array elements belong
     * @param length of the array
     * @return a new array
     * @since 3.2
     */
    public static <T> T[] buildArray(final Field<T> field, final int length) {
        @SuppressWarnings("unchecked") // OK because field must be correct class
        T[] array = (T[]) Array.newInstance(field.getRuntimeClass(), length);
        Arrays.fill(array, field.getZero());
        return array;
    }

    /** Build a double dimension  array of elements.
     * <p>
     * Arrays are filled with field.getZero()
     *
     * @param <T> the type of the field elements
     * @param field field to which array elements belong
     * @param rows number of rows in the array
     * @param columns number of columns (may be negative to build partial
     * arrays in the same way <code>new Field[rows][]</code> works)
     * @return a new array
     * @since 3.2
     */
    @SuppressWarnings("unchecked")
    public static <T> T[][] buildArray(final Field<T> field, final int rows, final int columns) {
        final T[][] array;
        if (columns < 0) {
            T[] dummyRow = buildArray(field, 0);
            array = (T[][]) Array.newInstance(dummyRow.getClass(), rows);
        } else {
            array = (T[][]) Array.newInstance(field.getRuntimeClass(),
                                              rows, columns);
            for (int i = 0; i < rows; ++i) {
                Arrays.fill(array[i], field.getZero());
            }
        }
        return array;
    }

    /**
     * Calculates the <a href="http://en.wikipedia.org/wiki/Convolution">
     * convolution</a> between two sequences.
     * <p>
     * The solution is obtained via straightforward computation of the
     * convolution sum (and not via FFT). Whenever the computation needs
     * an element that would be located at an index outside the input arrays,
     * the value is assumed to be zero.
     *
     * @param x First sequence.
     * Typically, this sequence will represent an input signal to a system.
     * @param h Second sequence.
     * Typically, this sequence will represent the impulse response of the system.
     * @return the convolution of {@code x} and {@code h}.
     * This array's length will be {@code x.length + h.length - 1}.
     * @throws NullArgumentException if either {@code x} or {@code h} is {@code null}.
     * @throws NoDataException if either {@code x} or {@code h} is empty.
     *
     * @since 3.3
     */
    public static double[] convolve(double[] x, double[] h) {
        NullArgumentException.check(x);
        NullArgumentException.check(h);

        final int xLen = x.length;
        final int hLen = h.length;

        if (xLen == 0 || hLen == 0) {
            throw new NoDataException();
        }

        // initialize the output array
        final int totalLength = xLen + hLen - 1;
        final double[] y = new double[totalLength];

        // straightforward implementation of the convolution sum
        for (int n = 0; n < totalLength; n++) {
            double yn = 0;
            int k = JdkMath.max(0, n + 1 - xLen);
            int j = n - k;
            while (k < hLen && j >= 0) {
                yn += x[j--] * h[k++];
            }
            y[n] = yn;
        }

        return y;
    }

    /**
     * Returns an array representing the natural number {@code n}.
     *
     * @param n Natural number.
     * @return an array whose entries are the numbers 0, 1, ..., {@code n}-1.
     * If {@code n == 0}, the returned array is empty.
     */
    public static int[] natural(int n) {
        return sequence(n, 0, 1);
    }
    /**
     * Returns an array of {@code size} integers starting at {@code start},
     * skipping {@code stride} numbers.
     *
     * @param size Natural number.
     * @param start Natural number.
     * @param stride Natural number.
     * @return an array whose entries are the numbers
     * {@code start, start + stride, ..., start + (size - 1) * stride}.
     * If {@code size == 0}, the returned array is empty.
     *
     * @since 3.4
     */
    public static int[] sequence(int size,
                                 int start,
                                 int stride) {
        final int[] a = new int[size];
        for (int i = 0; i < size; i++) {
            a[i] = start + i * stride;
        }
        return a;
    }
    /**
     * This method is used
     * to verify that the input parameters designate a subarray of positive length.
     * <ul>
     * <li>returns <code>true</code> iff the parameters designate a subarray of
     * positive length</li>
     * <li>throws <code>MathIllegalArgumentException</code> if the array is null or
     * or the indices are invalid</li>
     * <li>returns <code>false</code> if the array is non-null, but
     * <code>length</code> is 0.</li>
     * </ul>
     *
     * @param values the input array
     * @param begin index of the first array element to include
     * @param length the number of elements to include
     * @return true if the parameters are valid and designate a subarray of positive length
     * @throws MathIllegalArgumentException if the indices are invalid or the array is null
     * @since 3.3
     */
    public static boolean verifyValues(final double[] values, final int begin, final int length) {
        return verifyValues(values, begin, length, false);
    }

    /**
     * This method is used
     * to verify that the input parameters designate a subarray of positive length.
     * <ul>
     * <li>returns <code>true</code> iff the parameters designate a subarray of
     * non-negative length</li>
     * <li>throws <code>IllegalArgumentException</code> if the array is null or
     * or the indices are invalid</li>
     * <li>returns <code>false</code> if the array is non-null, but
     * <code>length</code> is 0 unless <code>allowEmpty</code> is <code>true</code></li>
     * </ul>
     *
     * @param values the input array
     * @param begin index of the first array element to include
     * @param length the number of elements to include
     * @param allowEmpty if <code>true</code> then zero length arrays are allowed
     * @return true if the parameters are valid
     * @throws MathIllegalArgumentException if the indices are invalid or the array is null
     * @since 3.3
     */
    public static boolean verifyValues(final double[] values, final int begin,
                                       final int length, final boolean allowEmpty) {

        if (values == null) {
            throw new NullArgumentException(LocalizedFormats.INPUT_ARRAY);
        }

        if (begin < 0) {
            throw new NotPositiveException(LocalizedFormats.START_POSITION, Integer.valueOf(begin));
        }

        if (length < 0) {
            throw new NotPositiveException(LocalizedFormats.LENGTH, Integer.valueOf(length));
        }

        if (begin + length > values.length) {
            throw new NumberIsTooLargeException(LocalizedFormats.SUBARRAY_ENDS_AFTER_ARRAY_END,
                    Integer.valueOf(begin + length), Integer.valueOf(values.length), true);
        }

        return !(length == 0 && !allowEmpty);
    }

    /**
     * This method is used
     * to verify that the begin and length parameters designate a subarray of positive length
     * and the weights are all non-negative, non-NaN, finite, and not all zero.
     * <ul>
     * <li>returns <code>true</code> iff the parameters designate a subarray of
     * positive length and the weights array contains legitimate values.</li>
     * <li>throws <code>IllegalArgumentException</code> if any of the following are true:
     * <ul><li>the values array is null</li>
     *     <li>the weights array is null</li>
     *     <li>the weights array does not have the same length as the values array</li>
     *     <li>the weights array contains one or more infinite values</li>
     *     <li>the weights array contains one or more NaN values</li>
     *     <li>the weights array contains negative values</li>
     *     <li>the weights array does not contain at least one non-zero value (applies when length is non zero)</li>
     *     <li>the start and length arguments do not determine a valid array</li></ul>
     * </li>
     * <li>returns <code>false</code> if the array is non-null, but
     * <code>length</code> is 0.</li>
     * </ul>
     *
     * @param values the input array
     * @param weights the weights array
     * @param begin index of the first array element to include
     * @param length the number of elements to include
     * @return true if the parameters are valid and designate a subarray of positive length
     * @throws MathIllegalArgumentException if the indices are invalid or the array is null
     * @since 3.3
     */
    public static boolean verifyValues(
        final double[] values,
        final double[] weights,
        final int begin,
        final int length) {
        return verifyValues(values, weights, begin, length, false);
    }

    /**
     * This method is used
     * to verify that the begin and length parameters designate a subarray of positive length
     * and the weights are all non-negative, non-NaN, finite, and not all zero.
     * <ul>
     * <li>returns <code>true</code> iff the parameters designate a subarray of
     * non-negative length and the weights array contains legitimate values.</li>
     * <li>throws <code>MathIllegalArgumentException</code> if any of the following are true:
     * <ul><li>the values array is null</li>
     *     <li>the weights array is null</li>
     *     <li>the weights array does not have the same length as the values array</li>
     *     <li>the weights array contains one or more infinite values</li>
     *     <li>the weights array contains one or more NaN values</li>
     *     <li>the weights array contains negative values</li>
     *     <li>the weights array does not contain at least one non-zero value (applies when length is non zero)</li>
     *     <li>the start and length arguments do not determine a valid array</li></ul>
     * </li>
     * <li>returns <code>false</code> if the array is non-null, but
     * <code>length</code> is 0 unless <code>allowEmpty</code> is <code>true</code>.</li>
     * </ul>
     *
     * @param values the input array.
     * @param weights the weights array.
     * @param begin index of the first array element to include.
     * @param length the number of elements to include.
     * @param allowEmpty if {@code true} than allow zero length arrays to pass.
     * @return {@code true} if the parameters are valid.
     * @throws NullArgumentException if either of the arrays are null
     * @throws MathIllegalArgumentException if the array indices are not valid,
     * the weights array contains NaN, infinite or negative elements, or there
     * are no positive weights.
     * @since 3.3
     */
    public static boolean verifyValues(final double[] values, final double[] weights,
                                       final int begin, final int length, final boolean allowEmpty) {

        if (weights == null || values == null) {
            throw new NullArgumentException(LocalizedFormats.INPUT_ARRAY);
        }

        checkEqualLength(weights, values);

        if (length != 0) {
            boolean containsPositiveWeight = false;
            for (int i = begin; i < begin + length; i++) {
                final double weight = weights[i];
                if (Double.isNaN(weight)) {
                    throw new MathIllegalArgumentException(LocalizedFormats.NAN_ELEMENT_AT_INDEX, Integer.valueOf(i));
                }
                if (Double.isInfinite(weight)) {
                    throw new MathIllegalArgumentException(LocalizedFormats.INFINITE_ARRAY_ELEMENT,
                        Double.valueOf(weight), Integer.valueOf(i));
                }
                if (weight < 0) {
                    throw new MathIllegalArgumentException(LocalizedFormats.NEGATIVE_ELEMENT_AT_INDEX,
                        Integer.valueOf(i), Double.valueOf(weight));
                }
                if (!containsPositiveWeight && weight > 0.0) {
                    containsPositiveWeight = true;
                }
            }

            if (!containsPositiveWeight) {
                throw new MathIllegalArgumentException(LocalizedFormats.WEIGHT_AT_LEAST_ONE_NON_ZERO);
            }
        }

        return verifyValues(values, begin, length, allowEmpty);
    }

    /**
     * Concatenates a sequence of arrays. The return array consists of the
     * entries of the input arrays concatenated in the order they appear in
     * the argument list.  Null arrays cause NullPointerExceptions; zero
     * length arrays are allowed (contributing nothing to the output array).
     *
     * @param x list of double[] arrays to concatenate
     * @return a new array consisting of the entries of the argument arrays
     * @throws NullPointerException if any of the arrays are null
     * @since 3.6
     */
    public static double[] concatenate(double[]... x) {
        int combinedLength = 0;
        for (double[] a : x) {
            combinedLength += a.length;
        }
        int offset = 0;
        int curLength = 0;
        final double[] combined = new double[combinedLength];
        for (int i = 0; i < x.length; i++) {
            curLength = x[i].length;
            System.arraycopy(x[i], 0, combined, offset, curLength);
            offset += curLength;
        }
        return combined;
    }

    /**
     * Returns an array consisting of the unique values in {@code data}.
     * The return array is sorted in descending order.  Empty arrays
     * are allowed, but null arrays result in NullPointerException.
     * Infinities are allowed.  NaN values are allowed with maximum
     * sort order - i.e., if there are NaN values in {@code data},
     * {@code Double.NaN} will be the first element of the output array,
     * even if the array also contains {@code Double.POSITIVE_INFINITY}.
     *
     * @param data array to scan
     * @return descending list of values included in the input array
     * @throws NullPointerException if data is null
     * @since 3.6
     */
    public static double[] unique(double[] data) {
        TreeSet<Double> values = new TreeSet<>();
        for (int i = 0; i < data.length; i++) {
            values.add(data[i]);
        }
        final int count = values.size();
        final double[] out = new double[count];
        Iterator<Double> iterator = values.descendingIterator();
        int i = 0;
        while (iterator.hasNext()) {
            out[i++] = iterator.next();
        }
        return out;
    }
}