Java tutorial
/* * 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.math3.util; import java.util.List; import java.util.ArrayList; import java.util.Comparator; import java.util.Collections; import org.apache.commons.math3.exception.DimensionMismatchException; import org.apache.commons.math3.exception.MathInternalError; import org.apache.commons.math3.exception.NonMonotonicSequenceException; import org.apache.commons.math3.exception.NotPositiveException; import org.apache.commons.math3.exception.NotStrictlyPositiveException; import org.apache.commons.math3.exception.NullArgumentException; import org.apache.commons.math3.exception.MathIllegalArgumentException; import org.apache.commons.math3.exception.util.LocalizedFormats; import org.apache.commons.math3.exception.MathArithmeticException; /** * Arrays utilities. * * @since 3.0 * @version $Id: MathArrays.java 1422313 2012-12-15 18:53:41Z psteitz $ */ public class MathArrays { /** Factor used for splitting double numbers: n = 2^27 + 1 (i.e. {@value}). */ private static final int SPLIT_FACTOR = 0x8000001; /** * 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); } /** * 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) { if (a.length != b.length) { throw new DimensionMismatchException(a.length, b.length); } 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) { if (a.length != b.length) { throw new DimensionMismatchException(a.length, b.length); } 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) { if (a.length != b.length) { throw new DimensionMismatchException(a.length, b.length); } 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) { if (a.length != b.length) { throw new DimensionMismatchException(a.length, b.length); } 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 */ public static double distance1(double[] p1, double[] p2) { double sum = 0; for (int i = 0; i < p1.length; i++) { sum += FastMath.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 */ public static int distance1(int[] p1, int[] p2) { int sum = 0; for (int i = 0; i < p1.length; i++) { sum += FastMath.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 */ public static double distance(double[] p1, double[] p2) { double sum = 0; for (int i = 0; i < p1.length; i++) { final double dp = p1[i] - p2[i]; sum += dp * dp; } return FastMath.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 */ public static double distance(int[] p1, int[] p2) { double sum = 0; for (int i = 0; i < p1.length; i++) { final double dp = p1[i] - p2[i]; sum += dp * dp; } return FastMath.sqrt(sum); } /** * Calculates the L<sub>∞</sub> (max of abs) distance between two points. * * @param p1 the first point * @param p2 the second point * @return the L<sub>∞</sub> distance between the two points */ public static double distanceInf(double[] p1, double[] p2) { double max = 0; for (int i = 0; i < p1.length; i++) { max = FastMath.max(max, FastMath.abs(p1[i] - p2[i])); } return max; } /** * Calculates the L<sub>∞</sub> (max of abs) distance between two points. * * @param p1 the first point * @param p2 the second point * @return the L<sub>∞</sub> distance between the two points */ public static int distanceInf(int[] p1, int[] p2) { int max = 0; for (int i = 0; i < p1.length; i++) { max = FastMath.max(max, FastMath.abs(p1[i] - p2[i])); } return max; } /** * Specification of ordering direction. */ public static 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 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) throws NonMonotonicSequenceException { 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, 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) throws NonMonotonicSequenceException { 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) throws NonMonotonicSequenceException { 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) throws NullArgumentException, DimensionMismatchException { MathUtils.checkNotNull(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) throws NotStrictlyPositiveException { for (int i = 0; i < in.length; i++) { if (in[i] <= 0) { throw new NotStrictlyPositiveException(in[i]); } } } /** * Check that all entries of the input array are >= 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) throws NotPositiveException { for (int i = 0; i < in.length; i++) { if (in[i] < 0) { throw new NotPositiveException(in[i]); } } } /** * Check all entries of the input array are >= 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) throws NotPositiveException { 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 the Cartesian norm (2-norm), handling both overflow and underflow. * Translation of the minpack enorm subroutine. * * The redistribution policy for MINPACK is available * <a href="http://www.netlib.org/minpack/disclaimer">here</a>, for * convenience, it is reproduced below.</p> * * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0"> * <tr><td> * Minpack Copyright Notice (1999) University of Chicago. * All rights reserved * </td></tr> * <tr><td> * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * <ol> * <li>Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer.</li> * <li>Redistributions in binary form must reproduce the above * copyright notice, this list of conditions and the following * disclaimer in the documentation and/or other materials provided * with the distribution.</li> * <li>The end-user documentation included with the redistribution, if any, * must include the following acknowledgment: * {@code This product includes software developed by the University of * Chicago, as Operator of Argonne National Laboratory.} * Alternately, this acknowledgment may appear in the software itself, * if and wherever such third-party acknowledgments normally appear.</li> * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS" * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE * UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND * THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR * IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE * OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY * OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR * USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF * THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4) * DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION * UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL * BE CORRECTED.</strong></li> * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT, * INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF * ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE, * EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE * POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li> * <ol></td></tr> * </table> * * @param v Vector of doubles. * @return the 2-norm of the vector. * @since 2.2 */ public static double safeNorm(double[] v) { double rdwarf = 3.834e-20; double rgiant = 1.304e+19; double s1 = 0; double s2 = 0; double s3 = 0; double x1max = 0; double x3max = 0; double floatn = v.length; double agiant = rgiant / floatn; for (int i = 0; i < v.length; i++) { double xabs = Math.abs(v[i]); if (xabs < rdwarf || xabs > agiant) { if (xabs > rdwarf) { if (xabs > x1max) { double r = x1max / xabs; s1 = 1 + s1 * r * r; x1max = xabs; } else { double r = xabs / x1max; s1 += r * r; } } else { if (xabs > x3max) { double r = x3max / xabs; s3 = 1 + s3 * r * r; x3max = xabs; } else { if (xabs != 0) { double r = xabs / x3max; s3 += r * r; } } } } else { s2 += xabs * xabs; } } double norm; if (s1 != 0) { norm = x1max * Math.sqrt(s1 + (s2 / x1max) / x1max); } else { if (s2 == 0) { norm = x3max * Math.sqrt(s3); } else { if (s2 >= x3max) { norm = Math.sqrt(s2 * (1 + (x3max / s2) * (x3max * s3))); } else { norm = Math.sqrt(x3max * ((s2 / x3max) + (x3max * s3))); } } } return norm; } /** * Sort an array in ascending order in place and perform the same reordering * of entries on other arrays. For example, if * {@code x = [3, 1, 2], y = [1, 2, 3]} and {@code z = [0, 5, 7]}, then * {@code sortInPlace(x, y, z)} will update {@code x} to {@code [1, 2, 3]}, * {@code y} to {@code [2, 3, 1]} and {@code z} to {@code [5, 7, 0]}. * * @param x Array to be sorted and used as a pattern for permutation * of the other arrays. * @param yList Set of arrays whose permutations of entries will follow * those performed on {@code x}. * @throws DimensionMismatchException if any {@code y} is not the same * size as {@code x}. * @throws NullArgumentException if {@code x} or any {@code y} is null. * @since 3.0 */ public static void sortInPlace(double[] x, double[]... yList) throws DimensionMismatchException, NullArgumentException { sortInPlace(x, OrderDirection.INCREASING, yList); } /** * Sort an array in place and perform the same reordering of entries on * other arrays. This method works the same as the other * {@link #sortInPlace(double[], double[][]) sortInPlace} method, but * allows the order of the sort to be provided in the {@code dir} * parameter. * * @param x Array to be sorted and used as a pattern for permutation * of the other arrays. * @param dir Order direction. * @param yList Set of arrays whose permutations of entries will follow * those performed on {@code x}. * @throws DimensionMismatchException if any {@code y} is not the same * size as {@code x}. * @throws NullArgumentException if {@code x} or any {@code y} is null * @since 3.0 */ public static void sortInPlace(double[] x, final OrderDirection dir, double[]... yList) throws NullArgumentException, DimensionMismatchException { if (x == null) { throw new NullArgumentException(); } final int len = x.length; final List<Pair<Double, double[]>> list = new ArrayList<Pair<Double, double[]>>(len); final int yListLen = yList.length; for (int i = 0; i < len; i++) { final double[] yValues = new double[yListLen]; for (int j = 0; j < yListLen; j++) { double[] y = yList[j]; if (y == null) { throw new NullArgumentException(); } if (y.length != len) { throw new DimensionMismatchException(y.length, len); } yValues[j] = y[i]; } list.add(new Pair<Double, double[]>(x[i], yValues)); } final Comparator<Pair<Double, double[]>> comp = new Comparator<Pair<Double, double[]>>() { public int compare(Pair<Double, double[]> o1, Pair<Double, double[]> o2) { int val; switch (dir) { case INCREASING: val = o1.getKey().compareTo(o2.getKey()); break; case DECREASING: val = o2.getKey().compareTo(o1.getKey()); break; default: // Should never happen. throw new MathInternalError(); } return val; } }; Collections.sort(list, comp); for (int i = 0; i < len; i++) { final Pair<Double, double[]> e = list.get(i); x[i] = e.getKey(); final double[] yValues = e.getValue(); for (int j = 0; j < yListLen; j++) { yList[j][i] = yValues[j]; } } } /** * Creates a copy of the {@code source} array. * * @param source Array to be copied. * @return the copied array. */ public static int[] copyOf(int[] source) { return copyOf(source, source.length); } /** * Creates a copy of the {@code source} array. * * @param source Array to be copied. * @return the copied array. */ public static double[] copyOf(double[] source) { return copyOf(source, source.length); } /** * Creates a copy of the {@code source} array. * * @param source Array to be copied. * @param len Number of entries to copy. If smaller then the source * length, the copy will be truncated, if larger it will padded with * zeroes. * @return the copied array. */ public static int[] copyOf(int[] source, int len) { final int[] output = new int[len]; System.arraycopy(source, 0, output, 0, FastMath.min(len, source.length)); return output; } /** * Creates a copy of the {@code source} array. * * @param source Array to be copied. * @param len Number of entries to copy. If smaller then the source * length, the copy will be truncated, if larger it will padded with * zeroes. * @return the copied array. */ public static double[] copyOf(double[] source, int len) { final double[] output = new double[len]; System.arraycopy(source, 0, output, 0, FastMath.min(len, source.length)); return output; } /** * Compute a linear combination accurately. * This method computes the sum of the products * <code>a<sub>i</sub> b<sub>i</sub></code> to high accuracy. * It does so by using specific multiplication and addition algorithms to * preserve accuracy and reduce cancellation effects. * <br/> * It is based on the 2005 paper * <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.2.1547"> * Accurate Sum and Dot Product</a> by Takeshi Ogita, Siegfried M. Rump, * and Shin'ichi Oishi published in SIAM J. Sci. Comput. * * @param a Factors. * @param b Factors. * @return <code>Σ<sub>i</sub> a<sub>i</sub> b<sub>i</sub></code>. * @throws DimensionMismatchException if arrays dimensions don't match */ public static double linearCombination(final double[] a, final double[] b) throws DimensionMismatchException { final int len = a.length; if (len != b.length) { throw new DimensionMismatchException(len, b.length); } final double[] prodHigh = new double[len]; double prodLowSum = 0; for (int i = 0; i < len; i++) { final double ai = a[i]; final double ca = SPLIT_FACTOR * ai; final double aHigh = ca - (ca - ai); final double aLow = ai - aHigh; final double bi = b[i]; final double cb = SPLIT_FACTOR * bi; final double bHigh = cb - (cb - bi); final double bLow = bi - bHigh; prodHigh[i] = ai * bi; final double prodLow = aLow * bLow - (((prodHigh[i] - aHigh * bHigh) - aLow * bHigh) - aHigh * bLow); prodLowSum += prodLow; } final double prodHighCur = prodHigh[0]; double prodHighNext = prodHigh[1]; double sHighPrev = prodHighCur + prodHighNext; double sPrime = sHighPrev - prodHighNext; double sLowSum = (prodHighNext - (sHighPrev - sPrime)) + (prodHighCur - sPrime); final int lenMinusOne = len - 1; for (int i = 1; i < lenMinusOne; i++) { prodHighNext = prodHigh[i + 1]; final double sHighCur = sHighPrev + prodHighNext; sPrime = sHighCur - prodHighNext; sLowSum += (prodHighNext - (sHighCur - sPrime)) + (sHighPrev - sPrime); sHighPrev = sHighCur; } double result = sHighPrev + (prodLowSum + sLowSum); if (Double.isNaN(result)) { // either we have split infinite numbers or some coefficients were NaNs, // just rely on the naive implementation and let IEEE754 handle this result = 0; for (int i = 0; i < len; ++i) { result += a[i] * b[i]; } } return result; } /** * Compute a linear combination accurately. * <p> * This method computes a<sub>1</sub>×b<sub>1</sub> + * a<sub>2</sub>×b<sub>2</sub> to high accuracy. It does * so by using specific multiplication and addition algorithms to * preserve accuracy and reduce cancellation effects. It is based * on the 2005 paper <a * href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.2.1547"> * Accurate Sum and Dot Product</a> by Takeshi Ogita, * Siegfried M. Rump, and Shin'ichi Oishi published in SIAM J. Sci. Comput. * </p> * @param a1 first factor of the first term * @param b1 second factor of the first term * @param a2 first factor of the second term * @param b2 second factor of the second term * @return a<sub>1</sub>×b<sub>1</sub> + * a<sub>2</sub>×b<sub>2</sub> * @see #linearCombination(double, double, double, double, double, double) * @see #linearCombination(double, double, double, double, double, double, double, double) */ public static double linearCombination(final double a1, final double b1, final double a2, final double b2) { // the code below is split in many additions/subtractions that may // appear redundant. However, they should NOT be simplified, as they // use IEEE754 floating point arithmetic rounding properties. // as an example, the expression "ca1 - (ca1 - a1)" is NOT the same as "a1" // The variable naming conventions are that xyzHigh contains the most significant // bits of xyz and xyzLow contains its least significant bits. So theoretically // xyz is the sum xyzHigh + xyzLow, but in many cases below, this sum cannot // be represented in only one double precision number so we preserve two numbers // to hold it as long as we can, combining the high and low order bits together // only at the end, after cancellation may have occurred on high order bits // split a1 and b1 as two 26 bits numbers final double ca1 = SPLIT_FACTOR * a1; final double a1High = ca1 - (ca1 - a1); final double a1Low = a1 - a1High; final double cb1 = SPLIT_FACTOR * b1; final double b1High = cb1 - (cb1 - b1); final double b1Low = b1 - b1High; // accurate multiplication a1 * b1 final double prod1High = a1 * b1; final double prod1Low = a1Low * b1Low - (((prod1High - a1High * b1High) - a1Low * b1High) - a1High * b1Low); // split a2 and b2 as two 26 bits numbers final double ca2 = SPLIT_FACTOR * a2; final double a2High = ca2 - (ca2 - a2); final double a2Low = a2 - a2High; final double cb2 = SPLIT_FACTOR * b2; final double b2High = cb2 - (cb2 - b2); final double b2Low = b2 - b2High; // accurate multiplication a2 * b2 final double prod2High = a2 * b2; final double prod2Low = a2Low * b2Low - (((prod2High - a2High * b2High) - a2Low * b2High) - a2High * b2Low); // accurate addition a1 * b1 + a2 * b2 final double s12High = prod1High + prod2High; final double s12Prime = s12High - prod2High; final double s12Low = (prod2High - (s12High - s12Prime)) + (prod1High - s12Prime); // final rounding, s12 may have suffered many cancellations, we try // to recover some bits from the extra words we have saved up to now double result = s12High + (prod1Low + prod2Low + s12Low); if (Double.isNaN(result)) { // either we have split infinite numbers or some coefficients were NaNs, // just rely on the naive implementation and let IEEE754 handle this result = a1 * b1 + a2 * b2; } return result; } /** * Compute a linear combination accurately. * <p> * This method computes a<sub>1</sub>×b<sub>1</sub> + * a<sub>2</sub>×b<sub>2</sub> + a<sub>3</sub>×b<sub>3</sub> * to high accuracy. It does so by using specific multiplication and * addition algorithms to preserve accuracy and reduce cancellation effects. * It is based on the 2005 paper <a * href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.2.1547"> * Accurate Sum and Dot Product</a> by Takeshi Ogita, * Siegfried M. Rump, and Shin'ichi Oishi published in SIAM J. Sci. Comput. * </p> * @param a1 first factor of the first term * @param b1 second factor of the first term * @param a2 first factor of the second term * @param b2 second factor of the second term * @param a3 first factor of the third term * @param b3 second factor of the third term * @return a<sub>1</sub>×b<sub>1</sub> + * a<sub>2</sub>×b<sub>2</sub> + a<sub>3</sub>×b<sub>3</sub> * @see #linearCombination(double, double, double, double) * @see #linearCombination(double, double, double, double, double, double, double, double) */ public static double linearCombination(final double a1, final double b1, final double a2, final double b2, final double a3, final double b3) { // the code below is split in many additions/subtractions that may // appear redundant. However, they should NOT be simplified, as they // do use IEEE754 floating point arithmetic rounding properties. // as an example, the expression "ca1 - (ca1 - a1)" is NOT the same as "a1" // The variables naming conventions are that xyzHigh contains the most significant // bits of xyz and xyzLow contains its least significant bits. So theoretically // xyz is the sum xyzHigh + xyzLow, but in many cases below, this sum cannot // be represented in only one double precision number so we preserve two numbers // to hold it as long as we can, combining the high and low order bits together // only at the end, after cancellation may have occurred on high order bits // split a1 and b1 as two 26 bits numbers final double ca1 = SPLIT_FACTOR * a1; final double a1High = ca1 - (ca1 - a1); final double a1Low = a1 - a1High; final double cb1 = SPLIT_FACTOR * b1; final double b1High = cb1 - (cb1 - b1); final double b1Low = b1 - b1High; // accurate multiplication a1 * b1 final double prod1High = a1 * b1; final double prod1Low = a1Low * b1Low - (((prod1High - a1High * b1High) - a1Low * b1High) - a1High * b1Low); // split a2 and b2 as two 26 bits numbers final double ca2 = SPLIT_FACTOR * a2; final double a2High = ca2 - (ca2 - a2); final double a2Low = a2 - a2High; final double cb2 = SPLIT_FACTOR * b2; final double b2High = cb2 - (cb2 - b2); final double b2Low = b2 - b2High; // accurate multiplication a2 * b2 final double prod2High = a2 * b2; final double prod2Low = a2Low * b2Low - (((prod2High - a2High * b2High) - a2Low * b2High) - a2High * b2Low); // split a3 and b3 as two 26 bits numbers final double ca3 = SPLIT_FACTOR * a3; final double a3High = ca3 - (ca3 - a3); final double a3Low = a3 - a3High; final double cb3 = SPLIT_FACTOR * b3; final double b3High = cb3 - (cb3 - b3); final double b3Low = b3 - b3High; // accurate multiplication a3 * b3 final double prod3High = a3 * b3; final double prod3Low = a3Low * b3Low - (((prod3High - a3High * b3High) - a3Low * b3High) - a3High * b3Low); // accurate addition a1 * b1 + a2 * b2 final double s12High = prod1High + prod2High; final double s12Prime = s12High - prod2High; final double s12Low = (prod2High - (s12High - s12Prime)) + (prod1High - s12Prime); // accurate addition a1 * b1 + a2 * b2 + a3 * b3 final double s123High = s12High + prod3High; final double s123Prime = s123High - prod3High; final double s123Low = (prod3High - (s123High - s123Prime)) + (s12High - s123Prime); // final rounding, s123 may have suffered many cancellations, we try // to recover some bits from the extra words we have saved up to now double result = s123High + (prod1Low + prod2Low + prod3Low + s12Low + s123Low); if (Double.isNaN(result)) { // either we have split infinite numbers or some coefficients were NaNs, // just rely on the naive implementation and let IEEE754 handle this result = a1 * b1 + a2 * b2 + a3 * b3; } return result; } /** * Compute a linear combination accurately. * <p> * This method computes a<sub>1</sub>×b<sub>1</sub> + * a<sub>2</sub>×b<sub>2</sub> + a<sub>3</sub>×b<sub>3</sub> + * a<sub>4</sub>×b<sub>4</sub> * to high accuracy. It does so by using specific multiplication and * addition algorithms to preserve accuracy and reduce cancellation effects. * It is based on the 2005 paper <a * href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.2.1547"> * Accurate Sum and Dot Product</a> by Takeshi Ogita, * Siegfried M. Rump, and Shin'ichi Oishi published in SIAM J. Sci. Comput. * </p> * @param a1 first factor of the first term * @param b1 second factor of the first term * @param a2 first factor of the second term * @param b2 second factor of the second term * @param a3 first factor of the third term * @param b3 second factor of the third term * @param a4 first factor of the third term * @param b4 second factor of the third term * @return a<sub>1</sub>×b<sub>1</sub> + * a<sub>2</sub>×b<sub>2</sub> + a<sub>3</sub>×b<sub>3</sub> + * a<sub>4</sub>×b<sub>4</sub> * @see #linearCombination(double, double, double, double) * @see #linearCombination(double, double, double, double, double, double) */ public static double linearCombination(final double a1, final double b1, final double a2, final double b2, final double a3, final double b3, final double a4, final double b4) { // the code below is split in many additions/subtractions that may // appear redundant. However, they should NOT be simplified, as they // do use IEEE754 floating point arithmetic rounding properties. // as an example, the expression "ca1 - (ca1 - a1)" is NOT the same as "a1" // The variables naming conventions are that xyzHigh contains the most significant // bits of xyz and xyzLow contains its least significant bits. So theoretically // xyz is the sum xyzHigh + xyzLow, but in many cases below, this sum cannot // be represented in only one double precision number so we preserve two numbers // to hold it as long as we can, combining the high and low order bits together // only at the end, after cancellation may have occurred on high order bits // split a1 and b1 as two 26 bits numbers final double ca1 = SPLIT_FACTOR * a1; final double a1High = ca1 - (ca1 - a1); final double a1Low = a1 - a1High; final double cb1 = SPLIT_FACTOR * b1; final double b1High = cb1 - (cb1 - b1); final double b1Low = b1 - b1High; // accurate multiplication a1 * b1 final double prod1High = a1 * b1; final double prod1Low = a1Low * b1Low - (((prod1High - a1High * b1High) - a1Low * b1High) - a1High * b1Low); // split a2 and b2 as two 26 bits numbers final double ca2 = SPLIT_FACTOR * a2; final double a2High = ca2 - (ca2 - a2); final double a2Low = a2 - a2High; final double cb2 = SPLIT_FACTOR * b2; final double b2High = cb2 - (cb2 - b2); final double b2Low = b2 - b2High; // accurate multiplication a2 * b2 final double prod2High = a2 * b2; final double prod2Low = a2Low * b2Low - (((prod2High - a2High * b2High) - a2Low * b2High) - a2High * b2Low); // split a3 and b3 as two 26 bits numbers final double ca3 = SPLIT_FACTOR * a3; final double a3High = ca3 - (ca3 - a3); final double a3Low = a3 - a3High; final double cb3 = SPLIT_FACTOR * b3; final double b3High = cb3 - (cb3 - b3); final double b3Low = b3 - b3High; // accurate multiplication a3 * b3 final double prod3High = a3 * b3; final double prod3Low = a3Low * b3Low - (((prod3High - a3High * b3High) - a3Low * b3High) - a3High * b3Low); // split a4 and b4 as two 26 bits numbers final double ca4 = SPLIT_FACTOR * a4; final double a4High = ca4 - (ca4 - a4); final double a4Low = a4 - a4High; final double cb4 = SPLIT_FACTOR * b4; final double b4High = cb4 - (cb4 - b4); final double b4Low = b4 - b4High; // accurate multiplication a4 * b4 final double prod4High = a4 * b4; final double prod4Low = a4Low * b4Low - (((prod4High - a4High * b4High) - a4Low * b4High) - a4High * b4Low); // accurate addition a1 * b1 + a2 * b2 final double s12High = prod1High + prod2High; final double s12Prime = s12High - prod2High; final double s12Low = (prod2High - (s12High - s12Prime)) + (prod1High - s12Prime); // accurate addition a1 * b1 + a2 * b2 + a3 * b3 final double s123High = s12High + prod3High; final double s123Prime = s123High - prod3High; final double s123Low = (prod3High - (s123High - s123Prime)) + (s12High - s123Prime); // accurate addition a1 * b1 + a2 * b2 + a3 * b3 + a4 * b4 final double s1234High = s123High + prod4High; final double s1234Prime = s1234High - prod4High; final double s1234Low = (prod4High - (s1234High - s1234Prime)) + (s123High - s1234Prime); // final rounding, s1234 may have suffered many cancellations, we try // to recover some bits from the extra words we have saved up to now double result = s1234High + (prod1Low + prod2Low + prod3Low + prod4Low + s12Low + s123Low + s1234Low); if (Double.isNaN(result)) { // either we have split infinite numbers or some coefficients were NaNs, // just rely on the naive implementation and let IEEE754 handle this result = a1 * b1 + a2 * b2 + a3 * b3 + a4 * b4; } return result; } /** * 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 |-> 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> * * <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> * * <p>Ignores (i.e., copies unchanged to the output array) NaNs in the input array.</p> * * @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) throws MathIllegalArgumentException, MathArithmeticException { 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); } for (int i = 0; i < len; i++) { if (Double.isNaN(values[i])) { out[i] = Double.NaN; } else { out[i] = values[i] * normalizedSum / sum; } } return out; } }