diff --git a/commons-statistics-distribution/pom.xml b/commons-statistics-distribution/pom.xml index 578d34ff..e11328b4 100644 --- a/commons-statistics-distribution/pom.xml +++ b/commons-statistics-distribution/pom.xml @@ -40,6 +40,11 @@ ${statistics.build.outputTimestamp} distribution + + + + 0.999 + 0.999 diff --git a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/HurwitzZeta.java b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/HurwitzZeta.java new file mode 100644 index 00000000..d07635d5 --- /dev/null +++ b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/HurwitzZeta.java @@ -0,0 +1,188 @@ +/* + * 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 + * + * https://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.statistics.distribution; + +/** + * Utility class used to compute the + * Hurwitz zeta function. + * + * 
+ *                 oo    1
+ * zeta(s, a) = sum    ------
+ *                 k=0      s
+ *                     (k+a)
+ *
+ * + *

The function is formally defined for complex variable {@code s} with {@code Re(s) > 1} + * and real {@code a != 0, -1, -2, ...}. This series is absolutely convergent for the given + * values of {@code s} and {@code a}. Note the special case zeta(s, 1) is the Riemann zeta function. + * + *

This implementation uses real-valued {@code s > 1} and {@code a >= 1}. + * Specialisation to a smaller domain than any finite {@code a} allows optimisation for + * a {@code double} precision result. + * + *

The implementation is performed by spitting the integral into two parts and using + * the Euler-Maclaurin formula to approximate the second integral {@code I + T + R} + * with a continuous integral {@code I}, a tail {@code T}, and a residual error term + * {@code R} (not computed). + * + * 

+ *                 N-1           oo
+ * zeta(s, a) = sum    f(k) + sum    f(k) = S + I + T + R
+ *                 k=0           k=N
+ *
+ *          1
+ * f(k) = ------
+ *             s
+ *        (a+k)
+ *
+ *                            1-s
+ *      ,-oo   1         (a+N)
+ * I =  |    ------ dt = --------
+ *     -' N       s        s-1
+ *           (a+t)
+ *
+ *            /             B     (s)      \
+ *       1    | 1      M     2k      2k-1  |
+ * T = ------ | - + sum    ----- --------- |
+ *          s | 2      k=1 (2k)!      2k-1 |
+ *     (a+N)  \                  (a+N)     /
+ *
+ * B   = Bernoulli number
+ *  2k
+ *
+ *           ___n-1
+ * (s)     = | |    (x+i)    (rising factorial Pochhammer function)
+ *    n      | |i=0
+ *
+ * + *

These formulas for the real-valued {@code s} are provided in Johansson (2015) as + * equations 5-9. The implementation omits the residual term {@code R}. + * + *

References + *

    + *
  1. Johansson (2015) + * Rigorous high-precision computation of the Hurwitz zeta function and its derivatives + * Numerical Algorithms (69) 253–270
    2. + *
  2. Hurwitz zeta function (Wikipedia)
    3. + *
  3. Riemann zeta function (Wikipedia)
    4. + *
  4. Euler–Maclaurin formula (Wikipedia)
    5. + *
  5. Bernoulli number (Wikipedia)
    6. + *
  6. Rising and falling factorials (Wikipedia)
    7. + *
+ * + * @since 1.4 + */ +final class HurwitzZeta { + /** Number of terms of the series summation S. */ + private static final int N = 8; + /** Convergence epsilon for the sum of the tail function. This prevents summation + * of terms that do not affect the final result. */ + private static final double EPS = 0x1.0p-53; + + /** + * Precomputed factors for {@code k}-th element of the tail function {@code T}. + * Uses {@code 2k!} divided by Bernoulli number {@code B_2k}. + * Provides M=14 terms. The test suite uses max 9 before convergence. + */ + private static final double[] F = { + 12.0, // 2! / (1 / 6) + -720.0, // 4! / (-1 / 30) + 30240.0, // 6! / (1 / 42) + -1209600.0, // 8! / (-1 / 30) + 4.790016E7, // 10! / (5 / 66) + -1.8924375803183792E9, // 12! / (-691 / 2730) + 7.47242496E10, // 14! / (7 / 6) + -2.950130727918164E12, // 16! / (-3617 / 510) + 1.1646782814350067E14, // 18! / (43867 / 798) + -4.597978722407473E15, // 20! / (-174611 / 330) + 1.81521054019435456E17, // 22! / (854513 / 138) + -7.1661652561756672E18, // 24! / (-236364091 / 2730) + 2.82908877253043E20, // 26! / (8553103 / 6) + -1.1168794925000445E22, // 28! / (-23749461029 / 870) + }; + + /** No instances. */ + private HurwitzZeta() {} + + /** + * Compute the value of the Hurwitz zeta function {@code zeta(s, a)}. + * + * 
+     *                 oo    1
+     * zeta(s, a) = sum    ------
+     *                 k=0      s
+     *                     (k+a)
+     *
+ * + *

Warning: No parameter validation is performed. + * The domain of {@code a} is expected to be a positive integer {@code [1, 2^31)}. + * + * @param s Argument {@code s > 1} + * @param a Argument {@code a >= 1} + * @return zeta(s, a) + */ + static double value(double s, double a) { + final double apn = a + N; + double p = Math.pow(apn, -s); + + // Initialise sum with the first tail term + double sum = 0.5 * p; + // S : k in [0, n-1] + for (int k = N - 1; k >= 0; k--) { + // Descending k sums in order of magnitude for increased precision. + // Prevents early exit for large s when the term (a+k)^-s is below + // machine epsilon of the ascending series sum. + sum += Math.pow(a + k, -s); + } + + // I + sum += Math.pow(apn, 1 - s) / (s - 1); + + // T + // The following recycles the power term p: (a+n)^-(2k-1+s). + // This incorporates the factor for T, (a+n)^-s, into the sum terms. + // The first power is (a+n)^-(1+s) not (a+n)^-1. + // When s is large the loop exits before the rising factorial overflows. + + // Rising factorial term : (s)_{2k-1} + double f = s; + // 2k - 1 + double k2 = 1; + // Sum of an alternating series as each F changes sign. + // Sum until terms will not impact the result. + double tsum = 0; + final double stop = sum * EPS; + int i; + for (i = 0; i < F.length; i++) { + // p = (a+n)^-(2k-1+s) + p /= apn; + final double t = f * p / F[i]; + tsum += t; + if (Math.abs(t) <= stop) { + break; + } + p /= apn; + // f = s * (s+1) * (s+2) * ... * (s+2k-2) + f *= s + k2; + k2 += 1.0; + f *= s + k2; + k2 += 1.0; + } + return sum + tsum; + } +} diff --git a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ZipfDistribution.java b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ZipfDistribution.java index 713bba40..d026939a 100644 --- a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ZipfDistribution.java +++ b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ZipfDistribution.java @@ -34,13 +34,26 @@ * * generalized harmonic number of order N of s. * - *

Note: The generalized harmonic number \( H_{N,s} \) is computed - * by direct summation of \( N \) terms. Construction of the distribution, and the first + *

\[ \sum_{k=1}^N \frac{1}{k^s} \] + * + *

Implementation note + * + *

The sum of the power series in harmonic numbers or cumulative + * probability functions may be computed + * using the Hurwitz zeta function + * when \( s \gt 1 \): + * + *

\[ \begin{aligned} + * \sum_{k=a}^b \frac{1}{k^s} &= \sum_{n=0}^\infty \frac{1}{(n+a)^s} - \sum_{n=0}^\infty \frac{1}{(n+b+1)^s} \\ + * &= \zeta(s, a) - \zeta(s, b+1) \end{aligned} \] + * + *

This is performed unless there is significant cancellation in the two zeta terms. + * In all other cases the sum is computed by direct summation of terms with performance + * implications for large \( N \). Construction of the distribution, and the first * call to {@link #getMean()} or {@link #getVariance()}, is \( O(N) \); each call to * {@link #cumulativeProbability(int) cumulativeProbability(x)} or - * {@link #survivalProbability(int) survivalProbability(x)} is \( O(x) \) (the - * partial harmonic sum is not cached between calls); the inverse probability - * functions perform a search using \( O(\log N) \) cumulative probability + * {@link #survivalProbability(int) survivalProbability(x)} is \( O(x) \); the inverse + * probability functions perform a search using \( O(\log N) \) cumulative probability * evaluations. A number of elements of order 231 requires billions of * {@code Math.pow} evaluations for construction alone. Take this run-time cost into * account when the parameters are derived from untrusted input, and bound the number @@ -49,30 +62,167 @@ * on the number of elements. * * @see Zipf distribution (Wikipedia) + * @see Hurwitz zeta function (Wikipedia) */ public final class ZipfDistribution extends AbstractDiscreteDistribution { + /** Minimum number of terms required to use the Hurwitz zeta function for cumulative + * probability functions. Below this level a regular sum of the terms is used. + * Note the evaluation of the Hurwitz zeta function requires multiple calls to + * {@link Math#pow(double, double)}. If the number of probability terms is low it is + * more efficient to sum the terms directly. */ + private static final int MIN_TERMS = 10; + /** Maximum ratio between the small term and large term to avoid significant cancellation + * in the sum {@code large - small}, i.e. {@code small / large <= ratio}. + * Note that the ratio {@code 1 - 2^-b} will lose {@code b - 1} bits in the result. + * This value allows a loss of 2-bits. */ + private static final double MAX_RATIO = 0.875; + /** Number of elements. */ private final int numberOfElements; /** Exponent parameter of the distribution. */ private final double exponent; - /** Cached value of the nth generalized harmonic. */ + /** Cached value of the N-th generalized harmonic. */ private final double nthHarmonic; - /** Cached value of the log of the nth generalized harmonic. */ + /** Cached value of the log of the N-th generalized harmonic. */ private final double logNthHarmonic; - /** Cached value of the nth generalized harmonic using (exponent - 1). */ + /** Cached value of the N-th generalized harmonic using (exponent - 1). */ private double nthHarmonicM1 = Double.NaN; - /** Cached value of the nth generalized harmonic using (exponent - 2). */ + /** Cached value of the N-th generalized harmonic using (exponent - 2). */ private double nthHarmonicM2 = Double.NaN; + /** Function to compute the generalised harmonic series. */ + private final HarmonicSeries genHarmonic; + + /** + * Compute the value of the generalised harmonic series. + * 

+     *    b      1
+     * sum      ---
+     *    k=a     s
+     *           k
+     *
+ */ + @FunctionalInterface + private interface HarmonicSeries { + /** + * Compute the value. + * 
+         *    b      1
+         * sum      ---
+         *    k=a     s
+         *           k
+         *
+ *

The exponent is assumed to be known. + * @param a Lower bound. + * @param b Upper bound. + * @return value + */ + double value(int a, int b); + } + + /** + * Compute the value of the generalised harmonic series using a difference of Hurwitz + * zeta functions. + * 

+     *    b      1
+     * sum      ---   = zeta(s, a) - zeta(s, b + 1)
+     *    k=a     s
+     *           k
+     *
+     *                 oo    1
+     * zeta(s, a) = sum    ------
+     *                 k=0      s
+     *                     (k+a)
+     *
+ * + *

If subtraction of terms results in significant loss of bits then the summation + * uses (b - a + 1) terms of the power series. + * + *

Note: The Hurwitz zeta function is defined for {@code s > 1} where it is absolutely + * convergent. + */ + private static class ZetaHarmonicSeries implements HarmonicSeries { + /** Number of elements parameter. */ + private final int n; + /** Exponent parameter. */ + private final double s; + /** zeta(s, 1) where s is the exponent of the distribution. */ + private final double zeta1; + /** zeta(s, 1 + n) where s is the exponent of the distribution; n is the number of elements. */ + private final double zeta1pN; + /** Cached value of the N-th generalized harmonic. */ + private final double nthHarmonic; + + /** Create an instance. + * @param n Maximum number of elements (n). + * @param exponent Exponent (s). + * @param zeta1 zeta(s, 1) + * @param zeta1pN zeta(s, 1 + n) + */ + ZetaHarmonicSeries(int n, double exponent, + double zeta1, double zeta1pN) { + this.n = n; + this.s = exponent; + this.zeta1 = zeta1; + this.zeta1pN = zeta1pN; + this.nthHarmonic = zeta1 - zeta1pN; + } + + @Override + public double value(int a, int b) { + if (b - a >= MIN_TERMS) { + final double z1 = a == 1 ? zeta1 : HurwitzZeta.value(s, a); + final double z2 = b == n ? zeta1pN : HurwitzZeta.value(s, 1 + b); + if (allowedDifference(z1, z2)) { + return applyBounds(z1 - z2); + } + } + return applyBounds(generalizedHarmonic(a, b, s)); + } + + /** + * Ensure the value is within the bound {@code [0, N-th harmonic]}. Ensures + * probability normalisation by N-th harmonic is in the range [0, 1]. In practice + * this may not be required. + * + *

Note: It should not be possible for the series summation to exceed the N-th + * harmonic. This has been computed using a difference of zeta terms: + * + * 

+         *   zeta(s, a) - zeta(s, b+1) <= zeta(s, 1) - zeta(s, N+1)  where a >= 1 and b <= N
+         *
+ * + *

The zeta difference will only exceed the normalizing constant due to + * floating-point error in the zeta function. + * + *

The series sum may exceed the N-th harmonic if there is an error in the zeta + * function. This may occur as the power terms approach zero (sub-normal + * summation) when s or n are both large. Since n is bounded to an integer the + * value is always suitable for evaluation of k^-s with k in integer [1, n], i.e. + * we never see 1 + n == n as n < 2^53. The only errors are expected when s is + * very large. + * + * @param x Value + * @return bounded value + */ + private double applyBounds(double x) { + return x < nthHarmonic ? x : nthHarmonic; + } + } /** Create an instance. * @param numberOfElements Number of elements. * @param exponent Exponent. + * @param nthHarmonic N-th generalized harmonic number + * @param genHarmonic Function to compute the generalised harmonic series. */ private ZipfDistribution(int numberOfElements, - double exponent) { + double exponent, + double nthHarmonic, + HarmonicSeries genHarmonic) { this.numberOfElements = numberOfElements; this.exponent = exponent; - this.nthHarmonic = generalizedHarmonic(numberOfElements, exponent); + this.nthHarmonic = nthHarmonic; + this.genHarmonic = genHarmonic; logNthHarmonic = Math.log(nthHarmonic); } @@ -101,7 +251,23 @@ public static ZipfDistribution of(int numberOfElements, throw new DistributionException(DistributionException.NEGATIVE, exponent); } - return new ZipfDistribution(numberOfElements, exponent); + + // If s > 1 and the size is non-trivial then use the Hurwitz zeta function + // to compute the harmonic series. Note if size is close to MIN_TERMS then + // the implementation may repeatedly call the zeta function and reject using + // it so the threshold is 4 * MIN_TERMS. + if (exponent > 1 && (numberOfElements >>> 2) > MIN_TERMS) { + final double zeta1 = HurwitzZeta.value(exponent, 1); + final double zeta1pN = HurwitzZeta.value(exponent, 1 + numberOfElements); + if (allowedDifference(zeta1, zeta1pN)) { + return new ZipfDistribution(numberOfElements, exponent, zeta1 - zeta1pN, + new ZetaHarmonicSeries(numberOfElements, exponent, zeta1, zeta1pN)); + } + } + + final double nthHarmonic = generalizedHarmonic(1, numberOfElements, exponent); + return new ZipfDistribution(numberOfElements, exponent, nthHarmonic, + (a, b) -> generalizedHarmonic(a, b, exponent)); } /** @@ -152,7 +318,7 @@ public double probability(int x0, } // Here: 1 <= x0 < x1 < n: // sum(pdf(x)) for x in (x0, x1] - return generalizedHarmonic(x0 + 1, x1, exponent) / nthHarmonic; + return genHarmonic.value(x0 + 1, x1) / nthHarmonic; } /** {@inheritDoc} */ @@ -167,14 +333,14 @@ public double logProbability(int x) { /** {@inheritDoc} */ @Override - public double cumulativeProbability(final int x) { + public double cumulativeProbability(int x) { if (x <= 0) { return 0; } else if (x >= numberOfElements) { return 1; } - return generalizedHarmonic(x, exponent) / nthHarmonic; + return genHarmonic.value(1, x) / nthHarmonic; } /** {@inheritDoc} */ @@ -188,7 +354,7 @@ public double survivalProbability(int x) { // Compute summation of terms omitted in the CDF. // The raw sums in CDF(x) + SF(x) = N-th harmonic - return generalizedHarmonic(x + 1, numberOfElements, exponent) / nthHarmonic; + return genHarmonic.value(x + 1, numberOfElements) / nthHarmonic; } /** @@ -262,31 +428,6 @@ private double nthHarmonicExpMinus2() { return h; } - /** - * Calculates the {@code n}-th generalized harmonic number. - * - * 

-     *          1
-     *   sum  -----  for k in [1, n]
-     *         k^m
-     *
- * - *

Assumes {@code exponent > 0} to arrange the terms to sum from small to large. - * - * @param n Last term in the series to calculate. - * @param m Exponent (special case {@code m = 1} is the harmonic series). - * @return the sum - */ - private static double generalizedHarmonic(final int n, final double m) { - double value = 0; - // Sum small to large - for (int k = n; k >= 2; k--) { - value += Math.pow(k, -m); - } - // 1^-m = 1 - return value + 1.0; - } - /** * Calculates the sum of terms of the * Harmonic @@ -307,7 +448,7 @@ private static double generalizedHarmonic(final int n, final double m) { * @param m Exponent (special case {@code m = 1} is the harmonic series). * @return the sum */ - private static double generalizedHarmonic(final int from, final int to, final double m) { + static double generalizedHarmonic(int from, int to, double m) { double value = 0; // Sum small to large for (int k = to; k >= from; k--) { @@ -325,24 +466,44 @@ private static double generalizedHarmonic(final int from, final int to, final do * @param m Exponent (special case {@code m = 1} is the harmonic series). * @return the nth generalized harmonic number. */ - private static double generalizedHarmonicAscendingSum(final int n, final double m) { - double value; + private static double generalizedHarmonicAscendingSum(int n, double m) { // Sum small to large. // If m < 0 then sum ascending, otherwise descending. // Note: 1^-m = 1 if (m < 0) { - value = 1.0; + double value = 1.0; for (int k = 2; k <= n; k++) { value += Math.pow(k, -m); } - } else { - value = 0; - for (int k = n; k >= 2; k--) { - value += Math.pow(k, -m); + return value; + } + + // If s > 1 and the size is non-trivial then use the Hurwitz zeta function + // to compute the harmonic series. + if (m > 1 && (n >>> 2) > MIN_TERMS) { + final double z1 = HurwitzZeta.value(m, 1); + final double z2 = HurwitzZeta.value(m, 1 + n); + if (allowedDifference(z1, z2)) { + return z1 - z2; } - value += 1.0; } - return value; + + return generalizedHarmonic(1, n, m); + } + + /** + * Check the difference {@code large - small} does not result in cancellation and + * potential loss of significant bits in the result. + * + *

This method is used to test if the difference of zeta functions is suitable + * to avoid the summation of the power series. + * + * @param large the large value + * @param small the small value + * @return true if {@code large - small} is allowed + */ + static boolean allowedDifference(double large, double small) { + return small <= large * MAX_RATIO; } /** diff --git a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java index 5a6c48c1..bc747faf 100644 --- a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java +++ b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java @@ -53,7 +53,7 @@ class ExtendedPrecisionTest { * Class to compute the root mean squared error (RMS). * @see Wikipedia: RMS */ - private static final class RMS { + static final class RMS { private double ss; private double max; private int n; @@ -249,12 +249,28 @@ void testExpmhxxStandardPrecision() { assertPrecision(EXPMHXX_RMS2, 400, 60); } - private static void assertPrecision(RMS rms, double maxError, double rmsError) { + /** + * Assert the precision of the RMS statistics. + * + * @param rms RMS statistics. + * @param maxError Allowed maximum error. + * @param rmsError Allowed RMS error. + */ + static void assertPrecision(RMS rms, double maxError, double rmsError) { Assertions.assertTrue(rms.getMax() < maxError, () -> "max error: " + rms.getMax()); Assertions.assertTrue(rms.getRMS() < rmsError, () -> "rms error: " + rms.getRMS()); } - private static void addError(double z, BigDecimal expected, double e, RMS rms) { + /** + * Adds the error to the RMS statistics. + * + * @param z Value. + * @param expected Expected value. + * @param e Expected value as a double. + * @param rms RMS statistics. + * @return the error in units of least precision (ULPs) + */ + static double addError(double z, BigDecimal expected, double e, RMS rms) { double error; if (z == e) { error = 0; @@ -264,5 +280,6 @@ private static void addError(double z, BigDecimal expected, double e, RMS rms) { .divide(new BigDecimal(Math.ulp(e)), MathContext.DECIMAL64).doubleValue(); } rms.add(error); + return error; } } diff --git a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/HurwitzZetaTest.java b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/HurwitzZetaTest.java new file mode 100644 index 00000000..47c18a37 --- /dev/null +++ b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/HurwitzZetaTest.java @@ -0,0 +1,695 @@ +/* + * 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 + * + * https://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.statistics.distribution; + +import java.math.BigDecimal; +import java.math.BigInteger; +import java.math.RoundingMode; +import java.util.function.DoubleBinaryOperator; +import java.util.stream.IntStream; +import java.util.stream.Stream; +import org.apache.commons.numbers.fraction.BigFraction; +import org.apache.commons.statistics.distribution.ExtendedPrecisionTest.RMS; +import org.junit.jupiter.api.Assertions; +import org.junit.jupiter.api.MethodOrderer; +import org.junit.jupiter.api.Order; +import org.junit.jupiter.api.Test; +import org.junit.jupiter.api.TestMethodOrder; +import org.junit.jupiter.params.ParameterizedTest; +import org.junit.jupiter.params.provider.Arguments; +import org.junit.jupiter.params.provider.CsvFileSource; +import org.junit.jupiter.params.provider.MethodSource; + +/** + * Test the {@link HurwitzZeta} function. + */ +@TestMethodOrder(MethodOrderer.OrderAnnotation.class) +class HurwitzZetaTest { + /** Table used to create a histogram of the number of steps to converge the tail series. + * Used in the {@link #zeta(double, double, int, int)} implementation. */ + private static final int[] M = new int[53]; + /** Optimal N used to test the zeta function. */ + private static final int N = 8; + /** Minimum N used to test the zeta function. + * Used for reporting RMS errors with varying N. When MIN_N == MAX_N no report is printed. */ + private static final int MIN_N = N; // e.g. 5 + /** Maximum N used to test the zeta function. Used for reporting RMS errors with varying N. */ + private static final int MAX_N = N; // e.g. 12 + /** Flag set when the JVM version is printed. Used for testing. */ + private static boolean jvm = false; + + /** + * Numerators of the even Bernoulli numbers {@code B_{2k}}. + * Taken from: + * 

+     * A000367 Numerators of Bernoulli numbers B_2n.
+     * https://oeis.org/A164020/b164020.txt
+     *
+ * + *

Contains the sequence up to 2k = 106 required for M=53 in the zeta implementation. + * Johansson (2015) suggests N ~ M ~ P for P-bits of precision. + */ + private static final String[] NUM = { + "0 1", + "1 1", + "2 -1", + "3 1", + "4 -1", + "5 5", + "6 -691", + "7 7", + "8 -3617", + "9 43867", + "10 -174611", + "11 854513", + "12 -236364091", + "13 8553103", + "14 -23749461029", + "15 8615841276005", + "16 -7709321041217", + "17 2577687858367", + "18 -26315271553053477373", + "19 2929993913841559", + "20 -261082718496449122051", + "21 1520097643918070802691", + "22 -27833269579301024235023", + "23 596451111593912163277961", + "24 -5609403368997817686249127547", + "25 495057205241079648212477525", + "26 -801165718135489957347924991853", + "27 29149963634884862421418123812691", + "28 -2479392929313226753685415739663229", + "29 84483613348880041862046775994036021", + "30 -1215233140483755572040304994079820246041491", + "31 12300585434086858541953039857403386151", + "32 -106783830147866529886385444979142647942017", + "33 1472600022126335654051619428551932342241899101", + "34 -78773130858718728141909149208474606244347001", + "35 1505381347333367003803076567377857208511438160235", + "36 -5827954961669944110438277244641067365282488301844260429", + "37 34152417289221168014330073731472635186688307783087", + "38 -24655088825935372707687196040585199904365267828865801", + "39 414846365575400828295179035549542073492199375372400483487", + "40 -4603784299479457646935574969019046849794257872751288919656867", + "41 1677014149185145836823154509786269900207736027570253414881613", + "42 -2024576195935290360231131160111731009989917391198090877281083932477", + "43 660714619417678653573847847426261496277830686653388931761996983", + "44 -1311426488674017507995511424019311843345750275572028644296919890574047", + "45 1179057279021082799884123351249215083775254949669647116231545215727922535", + "46 -1295585948207537527989427828538576749659341483719435143023316326829946247", + "47 1220813806579744469607301679413201203958508415202696621436215105284649447", + "48 -211600449597266513097597728109824233673043954389060234150638733420050668349987259", + "49 67908260672905495624051117546403605607342195728504487509073961249992947058239", + "50 -94598037819122125295227433069493721872702841533066936133385696204311395415197247711", + "51 3204019410860907078243020782116241775491817197152717450679002501086861530836678158791", + "52 -319533631363830011287103352796174274671189606078272738327103470162849568365549721224053", + "53 36373903172617414408151820151593427169231298640581690038930816378281879873386202346572901", + }; + + /** + * Denominators of the even Bernoulli numbers {@code B_{2k}}. + * Taken from: + * 

+     * A002445 Denominators of Bernoulli numbers B_{2n}.
+     * https://oeis.org/A002445/b002445.txt
+     *
+ */ + private static final String[] DENOM = { + "0 1", + "1 6", + "2 30", + "3 42", + "4 30", + "5 66", + "6 2730", + "7 6", + "8 510", + "9 798", + "10 330", + "11 138", + "12 2730", + "13 6", + "14 870", + "15 14322", + "16 510", + "17 6", + "18 1919190", + "19 6", + "20 13530", + "21 1806", + "22 690", + "23 282", + "24 46410", + "25 66", + "26 1590", + "27 798", + "28 870", + "29 354", + "30 56786730", + "31 6", + "32 510", + "33 64722", + "34 30", + "35 4686", + "36 140100870", + "37 6", + "38 30", + "39 3318", + "40 230010", + "41 498", + "42 3404310", + "43 6", + "44 61410", + "45 272118", + "46 1410", + "47 6", + "48 4501770", + "49 6", + "50 33330", + "51 4326", + "52 1590", + "53 642", + }; + + /** + * Precomputed factors for {@code k}-th element of the tail function {@code T}. + * Uses {@code 2k!} divided by Bernoulli number {@code B_2k}. + * The table size is suitable for N ~ M ~ P for P-bits of precision (53 entries) + * as stated in Johansson (2015) section 3.1. In practice the result in double + * precision requires lower N & M values. + */ + private static final double[] F = { + 12.0, // 2! / (1 / 6) + -720.0, // 4! / (-1 / 30) + 30240.0, // 6! / (1 / 42) + -1209600.0, // 8! / (-1 / 30) + 4.790016E7, // 10! / (5 / 66) + -1.8924375803183792E9, // 12! / (-691 / 2730) + 7.47242496E10, // 14! / (7 / 6) + -2.950130727918164E12, // 16! / (-3617 / 510) + 1.1646782814350067E14, // 18! / (43867 / 798) + -4.597978722407473E15, // 20! / (-174611 / 330) + 1.81521054019435456E17, // 22! / (854513 / 138) + -7.1661652561756672E18, // 24! / (-236364091 / 2730) + 2.82908877253043E20, // 26! / (8553103 / 6) + -1.1168794925000445E22, // 28! / (-23749461029 / 870) + 4.4092635141854666E23, // 30! / (8615841276005 / 14322) + -1.7407074646225822E25, // 32! / (-7709321041217 / 510) + 6.872037622739274E26, // 34! / (2577687858367 / 6) + -2.7129717107520044E28, // 36! / (-26315271553053477373 / 1919190) + 1.0710383014704457E30, // 38! / (2929993913841559 / 6) + -4.228289733582729E31, // 40! / (-261082718496449122051 / 13530) + 1.669261878547101E33, // 42! / (1520097643918070802691 / 1806) + -6.589981753232787E34, // 44! / (-27833269579301024235023 / 690) + 2.6016205165923062E36, // 46! / (596451111593912163277961 / 282) + -1.0270786120209628E38, // 48! / (-5609403368997817686249127547 / 46410) + 4.054743835786749E39, // 50! / (495057205241079648212477525 / 66) + -1.6007487042788347E41, // 52! / (-801165718135489957347924991853 / 1590) + 6.319502582715391E42, // 54! / (29149963634884862421418123812691 / 798) + -2.4948396201225357E44, // 56! / (-2479392929313226753685415739663229 / 870) + 9.849232037909393E45, // 58! / (84483613348880041862046775994036021 / 354) + -3.888320954748035E47, // 60! / (-1215233140483755572040304994079820246041491 / 56786730) + 1.535047584313167E49, // 62! / (12300585434086858541953039857403386151 / 6) + -6.060124957607529E50, // 64! / (-106783830147866529886385444979142647942017 / 510) + 2.3924414381101893E52, // 66! / (1472600022126335654051619428551932342241899101 / 64722) + -9.444980218768351E53, // 68! / (-78773130858718728141909149208474606244347001 / 30) + 3.728728733414322E55, // 70! / (1505381347333367003803076567377857208511438160235 / 4686) + -1.4720431007109737E57, // 72! / (-5827954961669944110438277244641067365282488301844260429 / 140100870) + 5.811393226148101E58, // 74! / (34152417289221168014330073731472635186688307783087 / 6) + -2.2942460864500874E60, // 76! / (-24655088825935372707687196040585199904365267828865801 / 30) + 9.057320508803927E61, // 78! / (414846365575400828295179035549542073492199375372400483487 / 3318) + -3.575686814230726E63, // 80! / (-4603784299479457646935574969019046849794257872751288919656867 / 230010) + 1.4116245727459505E65, // 82! / (1677014149185145836823154509786269900207736027570253414881613 / 498) + -5.5728704383437275E66, // 84! / (-2024576195935290360231131160111731009989917391198090877281083932477 / 3404310) + 2.2000810641991213E68, // 86! / (660714619417678653573847847426261496277830686653388931761996983 / 6) + -8.685571901589203E69, // 88! / (-1311426488674017507995511424019311843345750275572028644296919890574047 / 61410) + 3.428926346636115E71, // 90! / (1179057279021082799884123351249215083775254949669647116231545215727922535 / 272118) + -1.3536858624708422E73, // 92! / (-1295585948207537527989427828538576749659341483719435143023316326829946247 / 1410) + 5.344137578373867E74, // 94! / (1220813806579744469607301679413201203958508415202696621436215105284649447 / 6) + -2.10978095054183E76, // 96! / (-211600449597266513097597728109824233673043954389060234150638733420050668349987259 / 4501770) + 8.329081341920855E77, // 98! / (67908260672905495624051117546403605607342195728504487509073961249992947058239 / 6) + -3.288189514770133E79, // 100! / (-94598037819122125295227433069493721872702841533066936133385696204311395415197247711 / 33330) + 1.2981251882636475E81, // 102! / (3204019410860907078243020782116241775491817197152717450679002501086861530836678158791 / 4326) + -5.124792828500739E82, // 104! / (-319533631363830011287103352796174274671189606078272738327103470162849568365549721224053 / 1590) + 2.023187114193683E84, // 106! / (36373903172617414408151820151593427169231298640581690038930816378281879873386202346572901 / 642) + }; + + + /** + * Precomputed factors for {@code k}-th element of the tail function {@code T}. + * Uses Bernoulli number {@code B_2k} divided by {@code 2k!}. + * This is the inverse of table {@link #M}. + */ + private static final double[] FM = { + 0.08333333333333333, + -0.001388888888888889, + 3.306878306878307E-5, + -8.267195767195768E-7, + 2.08767569878681E-8, + -5.284190138687493E-10, + 1.3382536530684679E-11, + -3.3896802963225827E-13, + 8.586062056277845E-15, + -2.174868698558062E-16, + 5.5090028283602295E-18, + -1.3954464685812522E-19, + 3.534707039629467E-21, + -8.953517427037546E-23, + 2.267952452337683E-24, + -5.744790668872202E-26, + 1.455172475614865E-27, + -3.6859949406653103E-29, + 9.336734257095045E-31, + -2.36502241570063E-32, + 5.990671762482134E-34, + -1.5174548844682903E-35, + 3.843758125454189E-37, + -9.736353072646691E-39, + 2.466247044200681E-40, + -6.247076741820743E-42, + 1.5824030244644914E-43, + -4.008273685948936E-45, + 1.0153075855569557E-46, + -2.5718041582418717E-48, + 6.514456035233815E-50, + -1.6501309906896525E-51, + 4.179830628539476E-53, + -1.058763466770291E-54, + 2.6818791912607708E-56, + -6.793279351107421E-58, + 1.7207577616681404E-59, + -4.358730329348894E-61, + 1.1040792903684666E-62, + -2.7966655133781345E-64, + 7.084036501679471E-66, + -1.794407408289224E-67, + 4.545287063611096E-69, + -1.1513346631982051E-70, + 2.9163647710923614E-72, + -7.387238263497337E-74, + 1.8712093117637953E-75, + -4.739828557761799E-77, + 1.2006125993354507E-78, + -3.0411872415142924E-80, + 7.703417274705106E-82, + -1.951298390909883E-83, + 4.942696565159462E-85, + }; + + /** RMS error of the test zeta function using different N. */ + private static final RMS[] RMS_ZETA = IntStream.range(0, 54).mapToObj(x -> new RMS()).toArray(RMS[]::new); + /** RMS error of the final zeta function. */ + private static final RMS RMS_ZETA_FINAL = new RMS(); + + // Test zeta implementation + // + // This class contains a parameterized version of the final implementation. + // See STATISTICS-100 for variations tested during development. + // + // Note: The method is sensitive to the initial loop over N to create S. + // Under certain conditions the N cannot be too high if using an ascending + // sum of k as the sum does not converge and later terms are added with + // low precision. + // + // Better results are obtained using descending k. However this prevents + // an early exit if the series is rapidly converging and the term (a+k)^-s + // drops below machine epsilon of the sum. + // + // Summing in extended precision requires a double-double (DD) sum to be used + // throughout. Use in S and then not in the tail T does not lower the RMS. + + /** + * Compute the value of the Hurwitz zeta function {@code zeta(s, a)}. + * See {@link HurwitzZeta} for the formula details. + * + *

Warning: No parameter validation is performed. + * + * @param s Argument {@code s > 1} + * @param a Argument {@code a >= 1} + * @param n Argument {@code N} + * @param m Argument {@code M} + * @return zeta(s, a) + */ + static double zeta(double s, double a, int n, int m) { + // Asymptotic Behavior as a -> inf + // https://dlmf.nist.gov/25.11#E43 + // When a is large the series cannot use a+k. + // This reduces to N=0, the I term and the first term of T. + if (a > 1e15) { + return Math.pow(a, 1 - s) / (s - 1) + Math.pow(a, -s) * 0.5; + } + + final double apn = a + n; + double p = Math.pow(apn, -s); + + // Initialise sum with the first tail term + double sum = 0.5 * p; + // S : k in [0, n-1] + for (int k = n; --k >= 0;) { + // Descending k sums in order of magnitude for increased precision. + // Prevents early exit for large s when the term (a+k)^-s is below + // machine epsilon of the ascending series sum. + sum += Math.pow(a + k, -s); + } + + // I + // Use of (a+n)^(1-s) = (a+n)^-1 * apn to recycle the power lowers precision. + sum += Math.pow(apn, 1 - s) / (s - 1); + + // T + // The following recycles the power term p: (a+n)^-(2k-1+s). + // This incorporates the factor for T into the sum terms. + // This sets the first power as (a+n)^-(1+s) not (a+n)^-1. + // When s is large the loop exits before the rising factorial overflows. + + // Rising factorial term : (s)_{2k-1} + double f = s; + // 2k - 1 + double k2 = 1; + // Sum of an alternating series as each F changes sign. + // Sum until terms will not impact the result. + // Note: if the factor is too small (e.g. 0x1p-63) then the series continues + // further and terms may be less accurate (i.e. add noise to the T sum). + double tsum = 0; + final double stop = sum * 0x1p-53; + int i; + for (i = 0; i < m; i++) { + // p = (a+n)^-(2k-1+s) + p /= apn; + final double t = f * p / F[i]; + tsum += t; + if (Math.abs(t) <= stop) { + break; + } + p /= apn; + // f = s * (s+1) * (s+2) * ... * (s+2k-2) + f *= s + k2; + k2 += 1.0; + f *= s + k2; + k2 += 1.0; + } + // Used to histogram convergence when testing + if (n == N && MIN_N != MAX_N) { + M[i]++; + } + return sum + tsum; + } + + /** + * Test the factors required for the tail sum. These are computed from the numerator + * and denominator of the Bernoulli numbers, and the factorial of 2k. The test asserts + * that using 2k! / B_2k is more accurate than B_2k / 2k! when limited to double precision. + */ + @Test + void testFactors() { + // Factorial of 2k. Initialise at k=0. + BigInteger factorial = BigInteger.ONE; + double sum1 = 0; + double sum2 = 0; + // If this is too small the BigDecimal created by BigFraction is truncated + final int scale = 200; + // Check factors + for (int k = 1; k < NUM.length; k++) { + factorial = factorial.multiply(BigInteger.valueOf(2 * k - 1)).multiply(BigInteger.valueOf(2 * k)); + final BigInteger num = new BigInteger(NUM[k].substring(NUM[k].indexOf(' ') + 1)); + final BigInteger denom = new BigInteger(DENOM[k].substring(DENOM[k].indexOf(' ') + 1)); + + // 2k! / B_2k + final BigFraction factor1 = BigFraction.of(factorial.multiply(denom), num); + final double d1 = factor1.doubleValue(); + // Cross verify BigFraction vs BigDecimal + BigDecimal v = factor1.bigDecimalValue(scale, RoundingMode.HALF_EVEN); + Assertions.assertEquals(v.doubleValue(), d1); + // Find ULP precision + final double e1 = new BigDecimal(d1).subtract(v) + .divide(new BigDecimal(Math.ulp(d1)), scale, RoundingMode.HALF_EVEN).doubleValue(); + sum1 += Math.abs(e1); + + Assertions.assertEquals(d1, F[k - 1]); + + // Format to print the table: + // "%s, // %s! / (%s / %s)%n", d2, 2 * k, num, denom + + // B_2k / 2k! + final BigFraction factor2 = BigFraction.of(num, factorial.multiply(denom)); + final double d2 = factor2.doubleValue(); + // Cross verify BigFraction vs BigDecimal + v = factor2.bigDecimalValue(scale, RoundingMode.HALF_EVEN); + Assertions.assertEquals(v.doubleValue(), d2); + // Find ULP precision + final double e2 = new BigDecimal(d2).subtract(v) + .divide(new BigDecimal(Math.ulp(d2)), scale, RoundingMode.HALF_EVEN).doubleValue(); + sum2 += Math.abs(e2); + + Assertions.assertEquals(d2, FM[k - 1]); + + // Format to print the table: + // "%s, // (%s / %s) / %s!%n", d2, num, denom, 2 * k + + // For any M, the cumulative error is lower using 2k! / B_2k + // as the first 6/7 factors are exact and errors in the later factors + // are comparable. + Assertions.assertTrue(sum1 < sum2, "2k! / B_2k does not have lower combined error"); + } + } + + @ParameterizedTest + @Order(1) + @CsvFileSource(resources = "hurwitzzeta.csv") + void testZeta(double s, double a, BigDecimal expected) { + // Test with varying N. + // Changes to M can be made but since the implementation quickly converges in the + // tail sum M only has to be a small double digit number. + for (int i = MIN_N; i <= MAX_N; i++) { + final int n = i; + assertZeta(s, a, expected, (x, p) -> zeta(x, p, n, 15), n < 6 ? 25 : 4, RMS_ZETA[i]); + } + } + + @Test + void testZetaPrecision() { + // Output the RMS with varying N. + for (int i = 0; i < RMS_ZETA.length; i++) { + reportPrecision(String.format("zeta %-2d", i), RMS_ZETA[i]); + } + ExtendedPrecisionTest.assertPrecision(RMS_ZETA[N], 3.5, 0.60); + } + + @ParameterizedTest + @Order(1) + @CsvFileSource(resources = "hurwitzzeta.csv") + void testZetaFinal(double s, double a, BigDecimal expected) { + assertZeta(s, a, expected, HurwitzZeta::value, 3, RMS_ZETA_FINAL); + } + + @Test + void testZetaFinalPrecision() { + reportPrecision("zeta final", RMS_ZETA_FINAL); + ExtendedPrecisionTest.assertPrecision(RMS_ZETA_FINAL, 3.5, 0.60); + + // Test the formal implementation matches the test implementation + if (!Double.isNaN(RMS_ZETA[N].getRMS())) { + Assertions.assertEquals(RMS_ZETA[N].getMax(), RMS_ZETA_FINAL.getMax(), "Test max != final max"); + Assertions.assertEquals(RMS_ZETA[N].getRMS(), RMS_ZETA_FINAL.getRMS(), "Test rms != final rms"); + } + } + + private static double assertZeta(double s, double a, BigDecimal expected, + DoubleBinaryOperator f, int ulp, RMS rms) { + final double e = expected.doubleValue(); + final double x = f.applyAsDouble(s, a); + TestUtils.assertEquals(e, x, DoubleTolerances.ulps(ulp)); + return ExtendedPrecisionTest.addError(x, expected, e, rms); + } + + private static void reportPrecision(String name, RMS rms) { + // Note: The RMS changes with the JDK due to the use of Math.pow. + + // Eclipse Adoptium 17.0.6+10 + // zeta final max 3.463160912553975 rms 0.581703219550096 + // zeta 5 max 20.5732122922785 rms 2.1416222789789 + // zeta 6 max 3.739011012156354 rms 0.605560612860319 + // zeta 7 max 3.698235736226918 rms 0.5978604595147342 + // zeta 8 max 3.463160912553975 rms 0.581703219550096 + // zeta 9 max 3.739011012156354 rms 0.6009469175550126 + // zeta 10 max 3.463160912553975 rms 0.5855515390508779 + // zeta 11 max 3.78356780754505 rms 0.5916447716325028 + // zeta 12 max 3.78356780754505 rms 0.5910072153542633 + + // Only report implementation precision when testing different N. + if (MAX_N <= MIN_N) { + return; + } + final double v = rms.getRMS(); + if (!Double.isNaN(v)) { + // CHECKSTYLE: stop regex + if (!jvm) { + jvm = true; + System.out.printf("// %s %s%n", + System.getProperty("java.vm.vendor"), + System.getProperty("java.vm.version") + ); + } + System.out.printf("// %-10s max %-25s rms %s%n", name, rms.getMax(), v); + // CHECKSTYLE: resume regex + } + } + + @Test + @Order(2) + void testZetaM() { + // Check the M used to converge the tail series in the chosen implementation. + // This depends on the epsilon used to stop the sum. + int m = 0; + for (int i = 0; i < M.length; i++) { + if (M[i] != 0) { + m = i + 1; + // This is used for testing. + // CHECKSTYLE: stop regex + System.out.printf("// zeta N=%-2d M=%-2d %d%n", N, m, M[i]); + // CHECKSTYLE: resume regex + } + } + Assertions.assertTrue(m < 15); + } + + /** + * Spot tests for the zeta function to check various points in the domain and extreme values. + */ + @ParameterizedTest + @MethodSource(value = "testZetaSpot") + void testZetaSpot(double s, double a, double z, int ulp) { + final double v = HurwitzZeta.value(s, a); + TestUtils.assertEquals(z, v, DoubleTolerances.ulps(ulp)); + } + + static Stream testZetaSpot() { + return Stream.of( + // Reference values from mpmath version 1.4.1. + // from mpmath import mp, zeta + // mp.dps = 30; mp.pretty = True + // def f(s, a): + // print(f'Arguments.of({s}, {a}, {zeta(s, a)}, 0),') + // f(1.001, 1) etc. + Arguments.of(1.001, 1, 1000.57728847601162684806668989, 0), + Arguments.of(1.001, 3, 999.077634929516400548962369402, 0), + Arguments.of(1.001, 156.78, 994.961087152463264230812967825, 0), + Arguments.of(1.001, 345600, 987.327939500118316838908716646, 1), + Arguments.of(1.001, 100000000.0, 981.747943025003254769065166246, 1), + Arguments.of(1.001, 1000000000.0, 979.48998540929872849188230117, 1), + Arguments.of(1.001, 10000000000.0, 977.237220955969649930501114761, 0), + Arguments.of(1.001, 8.374e+19, 955.1620677642774549247109353, 0), + Arguments.of(1.001, 7.2834e+238, 576.949320020341985622075858479, 0), + Arguments.of(1.1678, 1, 6.54877176355186372373480299218, 1), + Arguments.of(1.1678, 2.5, 5.29477174574315816738362779184, 1), + Arguments.of(1.1678, 5.765, 4.50848224904053531500311915417, 1), + Arguments.of(1.1678, 87698, 0.882622082916184877125466190512, 1), + Arguments.of(1.1678, 1098765, 0.577489707637839655591145476126, 1), + Arguments.of(1.1678, 100000000.0, 0.27089940045667952429754209707, 0), + Arguments.of(1.1678, 1000000000.0, 0.184080609461973448176589687532, 1), + Arguments.of(1.1678, 10000000000.0, 0.125085809513774573322512635067, 1), + Arguments.of(1.1678, 6.786e+70, 7.75645430994177324455600419939e-12, 0), + Arguments.of(1.3567, 1, 3.40603490577277492861753138946, 1), + Arguments.of(1.3567, 12, 1.17295098623816176759602458034, 1), + Arguments.of(1.3567, 267, 0.382340735582659556038655214163, 0), + Arguments.of(1.3567, 100000000.0, 0.00392732544562110257556960028487, 0), + Arguments.of(1.3567, 1000000000.0, 0.0017274158124759523689829065186, 0), + Arguments.of(1.3567, 6780000000.0, 0.000872764873809386121835245990354, 1), + Arguments.of(1.3567, 10000000000.0, 0.000759795803739606906900648768122, 1), + Arguments.of(1.3567, 2.394279e+25, 2.48282011395930739748191376465e-09, 0), + Arguments.of(1.3567, 1.37e+201, 5.03765781298315620677652110278e-72, 0), + Arguments.of(1.9183, 1.234, 1.31039727875809754102393633608, 0), + Arguments.of(1.9183, 2.234, 0.642314727500847848717986207886, 1), + Arguments.of(1.9183, 32, 0.0458231265360237920443268300232, 1), + Arguments.of(1.9183, 189, 0.00886331331419119681212176796288, 0), + Arguments.of(1.9183, 26378, 9.48410976427270589834458588242e-05, 0), + Arguments.of(1.9183, 1484793, 2.34193449523082289840265727697e-06, 1), + Arguments.of(1.9183, 100000000.0, 4.90473353191884545392736492566e-08, 1), + Arguments.of(1.9183, 1000000000.0, 5.91991424826323516752379877396e-09, 1), + Arguments.of(1.9183, 10000000000.0, 7.14521688264220831940065076935e-10, 1), + Arguments.of(1.9183, 7.18923e+79, 5.06551945929778382002806813887e-74, 1), + Arguments.of(1.9183, 2.3423e+159, 4.87385597076733868223619309795e-147, 0), + Arguments.of(1.9183, 3.4535e+303, 1.98532564832533048061338098412e-279, 1), + + // Large s + Arguments.of(1001, 1, 1.0, 0), + Arguments.of(1001, 1.3, 8.76403979411431591130552997055e-115, 0), + Arguments.of(1001, 2, 4.66631809251609439495044772362e-302, 0), + Arguments.of(1001, 3, 0, 0), + + Arguments.of(26783400000000.0, 1.0000000000000002, 0.994070539579705196633883729806, 0), + Arguments.of(26783400000000.0, 1.0000000000002, 0.00470868956401873280652247209542, 0), + Arguments.of(26783400000000.0, 1.0000000002, 0, 0), + + Arguments.of(1e+18, 1, 1.0, 0), + Arguments.of(1e+18, 1.0000000000000002, 3.69192903582901397705695784205e-97, 0), + Arguments.of(1e+18, 1.0000000000000004, 1.36303400055980247979692825583e-193, 1), + + Arguments.of(1e+19, 1, 1.0, 0), + Arguments.of(1e+19, 1.0000000000000002, 0, 0), + + // Requires M=12 when N=9. + // This is largest M noted during development when the RMS error is close to optimal. + Arguments.of(31.76, 8.23, 8.6811830191090714059294777379e-30, 1), + Arguments.of(31.76, 11.23, 4.68180272588528321856530594913e-34, 0), + Arguments.of(31.76, 12.23, 3.17258072050987981477830566397e-35, 0), + Arguments.of(31.76, 13.23, 2.66312289027731475813712555339e-36, 1), + Arguments.of(31.76, 14.23, 2.68377485329827353841985109724e-37, 2), + Arguments.of(31.76, 15.23, 3.1669792006042583105605159352e-38, 1), + Arguments.of(31.76, 17.23, 6.55526711653597399706064463689e-40, 0), + Arguments.of(31.76, 19.23, 2.08909657211945256234757218604e-41, 0), + + Arguments.of(61.76, 30.23, 4.27875492887441652081612955782e-92, 2), + + // s -> large, a == 1 + // Asymptote of Reimann zeta function: 1 + 2^-s + Arguments.of(25.67, 1, 1.00000001873152459253171691257, 0), + Arguments.of(29.67, 1, 1.00000000117069191296622655835, 0), + Arguments.of(39.67, 1, 1.00000000000114324712238181431, 0), + Arguments.of(59.67, 1, 1.00000000000000000109028530523, 0), + Arguments.of(69.67, 1, 1.00000000000000000000106473174, 0), + Arguments.of(89.67, 1, 1.00000000000000000000000000102, 0), + + // ------- + + // Reference values using Matlab R2026a Symbolic Math Toolbox + // vpa(hurwitzZeta(sym(s, 'f'), sym(a, 'f'))) + // Note: The use of 'f' uses the floating-point conversion as N * 2^e + // where N is the mantissa and e is the exponent. + + Arguments.of(1.5, 4789, 0.0289021565574206125831622859402, 0), + Arguments.of(2.345, 12.789, 0.0254390524135780630410689989495, 1), + Arguments.of(1.345, 12.789, 1.2196695365743183226009917322, 1), + Arguments.of(1.345, 1278.9, 0.245686258677206272154248922088, 1), + Arguments.of(4.345, 28697.9, 0.000000000000000366539298049937530342298075842, 1), + Arguments.of(1.345, 1278562927.9, 0.00209103729002248575358360674936, 0), + // large s before underflow + Arguments.of(23.45, 12.789, 0.0000000000000000000000000134520611728481431909082787144, 0), + Arguments.of(23.45, 1278.9, 8.01879331040682555147782226582e-72, 1), + Arguments.of(23.45, 1278562927.9, 1.59541010013538002585127908428e-206, 1), + Arguments.of(234.5, 12.789, 2.7978232305220904641253847499e-260, 0), + Arguments.of(234.5, 17.89, 1.83179298527375942363255194085e-294, 0), + // small s -> 1 + Arguments.of(1.00001, 1.789, 99999.7231466483928005870213366, 1), + Arguments.of(1.00001, 17.89, 99997.1440070978817356297446315, 1), + Arguments.of(1.00001, 1278562927.89, 99979.0331951153772621341875866, 1), + Arguments.of(1.0000000000000002, 1.789, 4503599627370495.72314713725233, 0), + Arguments.of(1.0000000000000002, 1278562927.89, 4503599627370475.03099742881301, 0) + ); + } +} diff --git a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ZipfDistributionTest.java b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ZipfDistributionTest.java index fcb349d5..18a70834 100644 --- a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ZipfDistributionTest.java +++ b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ZipfDistributionTest.java @@ -68,12 +68,20 @@ protected double getRelativeTolerance() { @ParameterizedTest @CsvSource({ // Generated using scipy 1.16.3 using scipy.stats.zipfian.stats(exp, n) - "150, 0.512, 52.707637767916495, 1966.9356468021338", - "73, 1.67, 4.937625767687036, 87.76033876340095", - "999, 2.1, 3.5725516349635846, 343.7153292773371", + "150, 0.512, 52.707637767916495, 1966.9356468021338, 2e-15", + "73, 1.67, 4.937625767687036, 87.76033876340095, 1e-15", + "999, 2.1, 3.5725516349635846, 343.7153292773371, 1e-15", + // Attempt to use the zeta function but the two values are close: zeta(s-1, 1) - zeta(s-1, 1+n) + "100, 2.02, 3.080144872398558, 48.00225370319864, 1e-15", + "1000, 2.001, 4.5415154015097565, 584.4277334272527, 1e-15", + // Large n is not practical without the zeta function. + // Ensure n - 2 > 1 to use the zeta function for the variance. + "999999, 3.1, 1.3184365884771752, 5.083318190566237, 1e-15", + "987654321, 3.4, 1.2148826443135665, 1.2508670058966394, 1e-15", + "987654321, 5.4, 1.0312467279214397, 0.045058034902836094, 2e-14", }) - void testAdditionalMoments(int n, double exp, double mean, double variance) { - final DoubleTolerance tolerance = createRelTolerance(1e-14); + void testAdditionalMoments(int n, double exp, double mean, double variance, double eps) { + final DoubleTolerance tolerance = createRelTolerance(eps); final ZipfDistribution dist = ZipfDistribution.of(n, exp); testMoments(dist, mean, variance, tolerance); // Run twice to check the cached N-th harmonic numbers diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/hurwitzzeta.csv b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/hurwitzzeta.csv new file mode 100644 index 00000000..21bb51ac --- /dev/null +++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/hurwitzzeta.csv @@ -0,0 +1,2447 @@ +# 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. + +# High-precision test data for the Hurwitz zeta function. +# +# Test (s, a) created using (limited to 50 by Collector.joining): +# static SplittableRandom g = new SplittableRandom(42); +# static Stream range(int a, int b, int n) { +# return g.ints(n, a, b).mapToDouble(x -> x + g.nextDouble()).boxed(); +# } +# String s = Stream.of( +# rng.doubles(8).map(x -> 1 + x * 0.01).boxed(), +# range(1, 2, 8), range(2, 3, 8), range(3, 5, 8), range(5, 25, 16) +# ).flatMap(Function.identity()).sorted() +# .map(String::valueOf).collect(Collectors.joining(",")); +# String a = Stream.of( +# range(1, 5, 10), range(5, 10, 8), range(10, 100, 8), range(100, 10000, 8), +# range(10000, 1000000, 8), range(1000000, 1000000000, 8) +# ).flatMap(Function.identity()).sorted() +# .map(String::valueOf).collect(Collectors.joining(",")); +# +# Matlab R2026a Hurwitz zeta evaluation (requires Symbolic Math Toolbox) +# +# ss = [1.0006167373880737,1.000837831969073,1.0013335008901918,1.0016028781357047,1.007918781538738,1.0092859696644174,1.00932136215088,1.0098210648580728,1.15991039287692,1.2034351093002307,1.3441907165236375,1.4929891857946924,1.52001329960324,1.6184820663561348,1.8006318767135032,1.8682280765465324,2.0730537691034647,2.277567379985672,2.4954986581492435,2.6198190348990975,2.6889463724014133,2.694177574321061,2.7854994594961,2.840516482087527,3.0919669672136787,3.1425018997680585,3.321391289337802,3.3800228685822176,3.774783122268348,3.7821562891438543,4.159054910020286,4.2660528284133425,5.924392435714811,8.308535429890805,8.816805086255101,9.964802685014341,10.032260281541108,12.188497456354446,13.969371237967552,15.623165142423973,16.822502311331153,18.835195189585164,19.672902052767473,20.496411460832384,21.810311386916062,22.32015186103379,23.555964997504226,24.04488824777366]; +# aa = [1.2041330017152543,1.2265460891928393,1.3496515186903868,1.4561891368135123,1.505408944679143,2.7166669767929994,2.726054404097307,2.730289055066077,4.05172768030195,4.212481787507753,6.16172420394647,6.809209225225226,6.823627236462992,7.028332179450887,7.41802758378929,7.604703847746029,8.383906908241919,8.953759909753053,22.445567410957914,34.50882161224855,36.835964339959254,41.993572536077544,50.876045582169034,58.390227406731306,67.10660283843828,72.04866994353,155.018653878391,4114.635962637943,4782.866406950918,5832.707445185928,5833.772069174886,7530.585189204646,9437.17981778131,9714.481644420117,91324.59743878205,204107.607053725,256840.68070942213,350404.8197437515,691957.3706941172,791516.8138645723,882912.7638920178,911565.6220658533,1.2333347534928627E8,1.2402498806286132E8,4.633754213481752E8,7.482264661031605E8,7.511855225150903E8,8.039668683292435E8,9.683359524993379E8,9.77667584000786E8]; +# fileID = fopen('hurwitzzeta.csv', 'w'); +# for i = 1:numel(ss) +# for j = 1:numel(aa) +# s = ss(i); +# a = aa(j); +# fprintf(fileID, '%.17g, %.17g, %s\n', s, a, vpa(hurwitzZeta(sym(s, 'f'), sym(a, 'f')))); +# end +# end +# fclose(fileID); + +1.0006167373880737, 1.2041330017152543, 1621.719473252515312763352702405 +1.0006167373880737, 1.2265460891928393, 1621.691562703148010863568687139 +1.0006167373880737, 1.3496515186903868, 1621.5499505025638698461701440922 +1.0006167373880737, 1.4561891368135123, 1621.4409278977707971125280885662 +1.0006167373880737, 1.505408944679143, 1621.3941158029178515622565953908 +1.0006167373880737, 2.7166669767929994, 1620.6316547775399172030588465003 +1.0006167373880737, 2.7260544040973071, 1620.6274978488730291195233049531 +1.0006167373880737, 2.730289055066077, 1620.6256282304933911994791858949 +1.0006167373880737, 4.0517276803019504, 1620.1654726058177756670543350325 +1.0006167373880737, 4.2124817875077527, 1620.1215142111329951919773122083 +1.0006167373880737, 6.1617242039464699, 1619.7015757475924884667097049307 +1.0006167373880737, 6.8092092252252261, 1619.5936638241372450999487334012 +1.0006167373880737, 6.8236272364629924, 1619.5913885175881447503378483592 +1.0006167373880737, 7.0283321794508868, 1619.5596302892179534832660465785 +1.0006167373880737, 7.4180275837892902, 1619.5018253743599021116444130048 +1.0006167373880737, 7.6047038477460287, 1619.4752757749118150385608027882 +1.0006167373880737, 8.3839069082419186, 1619.3714924711347788725764551927 +1.0006167373880737, 8.9537599097530531, 1619.3018819077904931870577826636 +1.0006167373880737, 22.445567410957914, 1618.3499581093010906550543708198 +1.0006167373880737, 34.508821612248553, 1617.9128464301917888452008917322 +1.0006167373880737, 36.835964339959254, 1617.8468079865745927222611653255 +1.0006167373880737, 41.993572536077544, 1617.7143840019387190331048768859 +1.0006167373880737, 50.876045582169034, 1617.5208712936731467789944026415 +1.0006167373880737, 58.390227406731306, 1617.382184088501627776884936482 +1.0006167373880737, 67.106602838438278, 1617.2422886749851942499557469229 +1.0006167373880737, 72.048669943530001, 1617.1709024046196714862246019045 +1.0006167373880737, 155.01865387839101, 1616.403179285347322066637548796 +1.0006167373880737, 4114.6359626379426, 1613.1348065429878491971520887792 +1.0006167373880737, 4782.8664069509177, 1612.9850773558180361809203945695 +1.0006167373880737, 5832.707445185928, 1612.7876636956393157501122091846 +1.0006167373880737, 5833.7720691748864, 1612.7874821436157706842573640505 +1.0006167373880737, 7530.5851892046458, 1612.5335357082317430952830980016 +1.0006167373880737, 9437.1798177813107, 1612.3090926248273625650353172096 +1.0006167373880737, 9714.4816444201169, 1612.2802938505421970974689413031 +1.0006167373880737, 91324.597438782046, 1610.0536372736031158845933012716 +1.0006167373880737, 204107.60705372499, 1609.2552505633676293281946018657 +1.0006167373880737, 256840.68070942213, 1609.0271839438428337821696504693 +1.0006167373880737, 350404.81974375149, 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3.5930158304838043816986181835202e-209 +24.04488824777366, 977667584.00078595, 2.8805373148768169587520773382915e-209 diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.zipf.3.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.zipf.3.properties index 47679093..2b76e263 100644 --- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.zipf.3.properties +++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.zipf.3.properties @@ -19,7 +19,7 @@ variance = 3.5719600967876493 lower = 1 upper = 15 cdf.points = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 -# Reference values from scipy.stats (1.7.1) zipf(2.23, 15) +# Reference values from scipy.stats (1.7.1) zipfian(2.23, 15) cdf.values = \ 0. , 0.6924674979902014, 0.8400729855406271,\ 0.8998341423916065, 0.931297539617085 , 0.9504267165563961,\ diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.zipf.5.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.zipf.5.properties new file mode 100644 index 00000000..8069f7bd --- /dev/null +++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.zipf.5.properties @@ -0,0 +1,44 @@ +# 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. + +parameters = 1500 2.23 +mean = 2.8078406400341707 +variance = 237.80763667804035 +lower = 1 +upper = 1500 +cdf.points = 0, 1, 2, 10, 20, 100, 1000, 1400, 1480, 1490, 1498, 1499, 1500 +# Reference values from scipy.stats (1.18.1) zipfian(2.23, 1500) +cdf.values = \ + 0. , 0.6793839804663769, 0.8242006021298353,\ + 0.9694700732079443, 0.9866211655284569, 0.9981650124548695,\ + 0.999955755489256 , 0.9999939389122287, 0.9999988606992284,\ + 0.9999994346109918, 0.9999998875952036, 0.9999999438393927,\ + 1. +pmf.values = \ + 0.0000000000000000e+00, 6.7938398046637694e-01,\ + 1.4481662166345818e-01, 4.0005094644785166e-03,\ + 8.5274348856558132e-04, 2.3556746163481584e-05,\ + 1.3871240519188244e-07, 6.5501279360686049e-08,\ + 5.7867094478904714e-08, 5.7004602764855378e-08,\ + 5.6327951634616137e-08, 5.6244189259847323e-08,\ + 5.6160607179436723e-08 +sf.values = \ + 1.0000000000000000e+00, 3.2061601953362301e-01,\ + 1.7579939787016483e-01, 3.0529926792055753e-02,\ + 1.3378834471543130e-02, 1.8349875451304459e-03,\ + 4.4244510744057607e-05, 6.0610877713223825e-06,\ + 1.1393007716559116e-06, 5.6538900821481440e-07,\ + 1.1240479643928405e-07, 5.6160607179436723e-08,\ + 0.0000000000000000e+00 diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.zipf.6.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.zipf.6.properties new file mode 100644 index 00000000..2a25ad60 --- /dev/null +++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.zipf.6.properties @@ -0,0 +1,48 @@ +# 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. + +# Chosen to trigger mixed use of the zeta function in the CDF +# zeta(1.03, 1) = 33.913 +# zeta(1.03, 11) = 31.063 : 31.063 / 33.913 = 0.916 +# zeta(1.03, 15000) = 24.980 : 24.980 / 33.913 = 0.736 +parameters = 15000 1.03 +mean = 1297.3454661861774 +variance = 7899568.145569441 +lower = 1 +upper = 15000 +cdf.points = 0, 1, 2, 10, 100, 1000, 5000, 10000, 14000, 14900, 14990, 14998, 14999, 15000 +# Reference values from scipy.stats (1.18.1) zipfian(1.03, 15000) +cdf.values = \ + 0. , 0.11195013330273845, 0.16677324973987057,\ + 0.319070742935133 , 0.5468715627914645 , 0.7633638796885669 ,\ + 0.9063008653138818 , 0.9657767683975667 , 0.9942059773775886 ,\ + 0.9994387828696124 , 0.9999440519631826 , 0.9999888134665544 ,\ + 0.9999944069253194 , 1. +pmf.values = \ + 0.0000000000000000e+00, 1.1195013330273845e-01,\ + 5.4823116437132113e-02, 1.0447794337957712e-02,\ + 9.7504489997414848e-04, 9.0996484637104598e-05,\ + 1.7341454093654414e-05, 8.4922860644985181e-06,\ + 6.0049962190942215e-06, 5.6317421009313979e-06,\ + 5.5969178591540315e-06, 5.5938429002196576e-06,\ + 5.5934587644489349e-06, 5.5930746806645358e-06 +sf.values = \ + 1.0000000000000000e+00, 8.8804986669726238e-01,\ + 8.3322675026013027e-01, 6.8092925706486807e-01,\ + 4.5312843720853546e-01, 2.3663612031143266e-01,\ + 9.3699134686117999e-02, 3.4223231602433762e-02,\ + 5.7940226224110197e-03, 5.6121713038718817e-04,\ + 5.5948036817508440e-05, 1.1186533445113472e-05,\ + 5.5930746806645358e-06, 0.0000000000000000e+0 diff --git a/src/changes/changes.xml b/src/changes/changes.xml index fc14f6d5..d74f096f 100644 --- a/src/changes/changes.xml +++ b/src/changes/changes.xml @@ -53,6 +53,10 @@ If the output is not quite correct, check for invisible trailing spaces! + + "ZipfDistribution": Use the Hurwitz zeta function when parameters are suitable + for a significant performance improvment when the number of elements is large. + "MannWhitneyUTest": Limit the allocation size for the exact p-value computation. The maximum byte allocation can be configured using a system