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High performance scientific and technical computing data structures and methods,
mostly based on CERN's
Colt Java API
/**
* 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.
*/
/*
Copyright 1999 CERN - European Organization for Nuclear Research.
Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose
is hereby granted without fee, provided that the above copyright notice appear in all copies and
that both that copyright notice and this permission notice appear in supporting documentation.
CERN makes no representations about the suitability of this software for any purpose.
It is provided "as is" without expressed or implied warranty.
*/
package org.apache.mahout.collections;
/**
* Arithmetic functions.
*/
public final class Arithmetic extends Constants {
// for method STIRLING_CORRECTION(...)
private static final double[] STIRLING_CORRECTION = {
0.0,
8.106146679532726e-02, 4.134069595540929e-02,
2.767792568499834e-02, 2.079067210376509e-02,
1.664469118982119e-02, 1.387612882307075e-02,
1.189670994589177e-02, 1.041126526197209e-02,
9.255462182712733e-03, 8.330563433362871e-03,
7.573675487951841e-03, 6.942840107209530e-03,
6.408994188004207e-03, 5.951370112758848e-03,
5.554733551962801e-03, 5.207655919609640e-03,
4.901395948434738e-03, 4.629153749334029e-03,
4.385560249232324e-03, 4.166319691996922e-03,
3.967954218640860e-03, 3.787618068444430e-03,
3.622960224683090e-03, 3.472021382978770e-03,
3.333155636728090e-03, 3.204970228055040e-03,
3.086278682608780e-03, 2.976063983550410e-03,
2.873449362352470e-03, 2.777674929752690e-03,
};
// for method logFactorial(...)
// log(k!) for k = 0, ..., 29
private static final double[] LOG_FACTORIALS = {
0.00000000000000000, 0.00000000000000000, 0.69314718055994531,
1.79175946922805500, 3.17805383034794562, 4.78749174278204599,
6.57925121201010100, 8.52516136106541430, 10.60460290274525023,
12.80182748008146961, 15.10441257307551530, 17.50230784587388584,
19.98721449566188615, 22.55216385312342289, 25.19122118273868150,
27.89927138384089157, 30.67186010608067280, 33.50507345013688888,
36.39544520803305358, 39.33988418719949404, 42.33561646075348503,
45.38013889847690803, 48.47118135183522388, 51.60667556776437357,
54.78472939811231919, 58.00360522298051994, 61.26170176100200198,
64.55753862700633106, 67.88974313718153498, 71.25703896716800901
};
// k! for k = 0, ..., 20
private static final long[] LONG_FACTORIALS = {
1L,
1L,
2L,
6L,
24L,
120L,
720L,
5040L,
40320L,
362880L,
3628800L,
39916800L,
479001600L,
6227020800L,
87178291200L,
1307674368000L,
20922789888000L,
355687428096000L,
6402373705728000L,
121645100408832000L,
2432902008176640000L
};
// k! for k = 21, ..., 170
private static final double[] DOUBLE_FACTORIALS = {
5.109094217170944E19,
1.1240007277776077E21,
2.585201673888498E22,
6.204484017332394E23,
1.5511210043330984E25,
4.032914611266057E26,
1.0888869450418352E28,
3.048883446117138E29,
8.841761993739701E30,
2.652528598121911E32,
8.222838654177924E33,
2.6313083693369355E35,
8.68331761881189E36,
2.952327990396041E38,
1.0333147966386144E40,
3.719933267899013E41,
1.3763753091226346E43,
5.23022617466601E44,
2.0397882081197447E46,
8.15915283247898E47,
3.34525266131638E49,
1.4050061177528801E51,
6.041526306337384E52,
2.6582715747884495E54,
1.196222208654802E56,
5.502622159812089E57,
2.5862324151116827E59,
1.2413915592536068E61,
6.082818640342679E62,
3.0414093201713376E64,
1.5511187532873816E66,
8.06581751709439E67,
4.274883284060024E69,
2.308436973392413E71,
1.2696403353658264E73,
7.109985878048632E74,
4.052691950487723E76,
2.350561331282879E78,
1.386831185456898E80,
8.32098711274139E81,
5.075802138772246E83,
3.146997326038794E85,
1.9826083154044396E87,
1.2688693218588414E89,
8.247650592082472E90,
5.443449390774432E92,
3.6471110918188705E94,
2.48003554243683E96,
1.7112245242814127E98,
1.1978571669969892E100,
8.504785885678624E101,
6.123445837688612E103,
4.470115461512686E105,
3.307885441519387E107,
2.4809140811395404E109,
1.8854947016660506E111,
1.451830920282859E113,
1.1324281178206295E115,
8.94618213078298E116,
7.15694570462638E118,
5.797126020747369E120,
4.7536433370128435E122,
3.94552396972066E124,
3.314240134565354E126,
2.8171041143805494E128,
2.4227095383672744E130,
2.107757298379527E132,
1.854826422573984E134,
1.6507955160908465E136,
1.4857159644817605E138,
1.3520015276784033E140,
1.2438414054641305E142,
1.156772507081641E144,
1.0873661566567426E146,
1.0329978488239061E148,
9.916779348709491E149,
9.619275968248216E151,
9.426890448883248E153,
9.332621544394415E155,
9.332621544394418E157,
9.42594775983836E159,
9.614466715035125E161,
9.902900716486178E163,
1.0299016745145631E166,
1.0813967582402912E168,
1.1462805637347086E170,
1.2265202031961373E172,
1.324641819451829E174,
1.4438595832024942E176,
1.5882455415227423E178,
1.7629525510902457E180,
1.974506857221075E182,
2.2311927486598138E184,
2.543559733472186E186,
2.925093693493014E188,
3.393108684451899E190,
3.96993716080872E192,
4.6845258497542896E194,
5.574585761207606E196,
6.689502913449135E198,
8.094298525273444E200,
9.875044200833601E202,
1.2146304367025332E205,
1.506141741511141E207,
1.882677176888926E209,
2.3721732428800483E211,
3.0126600184576624E213,
3.856204823625808E215,
4.974504222477287E217,
6.466855489220473E219,
8.471580690878813E221,
1.1182486511960037E224,
1.4872707060906847E226,
1.99294274616152E228,
2.690472707318049E230,
3.6590428819525483E232,
5.0128887482749884E234,
6.917786472619482E236,
9.615723196941089E238,
1.3462012475717523E241,
1.8981437590761713E243,
2.6953641378881633E245,
3.8543707171800694E247,
5.550293832739308E249,
8.047926057471989E251,
1.1749972043909107E254,
1.72724589045464E256,
2.5563239178728637E258,
3.8089226376305687E260,
5.7133839564458575E262,
8.627209774233244E264,
1.3113358856834527E267,
2.0063439050956838E269,
3.0897696138473515E271,
4.789142901463393E273,
7.471062926282892E275,
1.1729568794264134E278,
1.8532718694937346E280,
2.946702272495036E282,
4.714723635992061E284,
7.590705053947223E286,
1.2296942187394494E289,
2.0044015765453032E291,
3.287218585534299E293,
5.423910666131583E295,
9.003691705778434E297,
1.5036165148649983E300,
2.5260757449731988E302,
4.2690680090047056E304,
7.257415615308004E306
};
/** Makes this class non instantiable, but still let's others inherit from it. */
Arithmetic() {
}
/**
* Efficiently returns the binomial coefficient, often also referred to as "n over k" or "n choose k". The binomial
* coefficient is defined as (n * n-1 * ... * n-k+1 ) / ( 1 * 2 * ... * k ). - k<0: 0.
*
- k==0: 1.
- k==1: n.
- else: (n * n-1 * ... * n-k+1 ) / ( 1 * 2 * ... * k
* ).
*
* @return the binomial coefficient.
*/
public static double binomial(double n, long k) {
if (k < 0) {
return 0;
}
if (k == 0) {
return 1;
}
if (k == 1) {
return n;
}
// binomial(n,k) = (n * n-1 * ... * n-k+1 ) / ( 1 * 2 * ... * k )
double a = n - k + 1;
double b = 1;
double binomial = 1;
for (long i = k; i-- > 0;) {
binomial *= (a++) / (b++);
}
return binomial;
}
/**
* Efficiently returns the binomial coefficient, often also referred to as "n over k" or "n choose k". The binomial
* coefficient is defined as - k<0: 0.
- k==0 || k==n: 1.
- k==1 || k==n-1:
* n.
- else: (n * n-1 * ... * n-k+1 ) / ( 1 * 2 * ... * k ).
*
* @return the binomial coefficient.
*/
public static double binomial(long n, long k) {
if (k < 0) {
return 0;
}
if (k == 0 || k == n) {
return 1;
}
if (k == 1 || k == n - 1) {
return n;
}
// try quick version and see whether we get numeric overflows.
// factorial(..) is O(1); requires no loop; only a table lookup.
if (n > k) {
int max = LONG_FACTORIALS.length + DOUBLE_FACTORIALS.length;
if (n < max) { // if (n! < inf && k! < inf)
double n_fac = factorial((int) n);
double k_fac = factorial((int) k);
double n_minus_k_fac = factorial((int) (n - k));
double nk = n_minus_k_fac * k_fac;
if (nk != Double.POSITIVE_INFINITY) { // no numeric overflow?
// now this is completely safe and accurate
return n_fac / nk;
}
}
if (k > n / 2) {
k = n - k;
} // quicker
}
// binomial(n,k) = (n * n-1 * ... * n-k+1 ) / ( 1 * 2 * ... * k )
long a = n - k + 1;
long b = 1;
double binomial = 1;
for (long i = k; i-- > 0;) {
binomial *= (double) a++ / (b++);
}
return binomial;
}
/**
* Returns the smallest long >= value. Examples: {@code 1.0 -> 1, 1.2 -> 2, 1.9 -> 2}. This
* method is safer than using (long) Math.ceil(value), because of possible rounding error.
*/
public static long ceil(double value) {
return Math.round(Math.ceil(value));
}
/**
* Evaluates the series of Chebyshev polynomials Ti at argument x/2. The series is given by
*
* N-1
* - '
* y = > coef[i] T (x/2)
* - i
* i=0
*
* Coefficients are stored in reverse order, i.e. the zero order term is last in the array. Note N is the number of
* coefficients, not the order. If coefficients are for the interval a to b, x must have been transformed to x ->
* 2(2x - b - a)/(b-a) before entering the routine. This maps x from (a, b) to (-1, 1), over which the Chebyshev
* polynomials are defined.
If the coefficients are for the inverted interval, in which (a, b) is mapped to (1/b,
* 1/a), the transformation required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity, this becomes x -> 4a/x - 1.
*
SPEED:
Taking advantage of the recurrence properties of the Chebyshev polynomials, the routine requires one
* more addition per loop than evaluating a nested polynomial of the same degree.
*
* @param x argument to the polynomial.
* @param coef the coefficients of the polynomial.
* @param N the number of coefficients.
*/
public static double chbevl(double x, double[] coef, int N) {
int p = 0;
double b0 = coef[p++];
double b1 = 0.0;
int i = N - 1;
double b2;
do {
b2 = b1;
b1 = b0;
b0 = x * b1 - b2 + coef[p++];
} while (--i > 0);
return 0.5 * (b0 - b2);
}
/**
* Instantly returns the factorial k!.
*
* @param k must hold k >= 0.
*/
private static double factorial(int k) {
if (k < 0) {
throw new IllegalArgumentException();
}
int length1 = LONG_FACTORIALS.length;
if (k < length1) {
return LONG_FACTORIALS[k];
}
int length2 = DOUBLE_FACTORIALS.length;
if (k < length1 + length2) {
return DOUBLE_FACTORIALS[k - length1];
} else {
return Double.POSITIVE_INFINITY;
}
}
/**
* Returns the largest long <= value.
*
Examples: {@code 1.0 -> 1, 1.2 -> 1, 1.9 -> 1 2.0 -> 2,} 2.2 -> 2, 2.9 -> 2 Correction term of the Stirling approximation for log(k!) (series in * 1/k, or table values for small k) with int parameter k.
log k! = (k + 1/2)log(k + 1) - (k + 1) + * (1/2)log(2Pi) + STIRLING_CORRECTION(k + 1)
log k! = (k + 1/2)log(k) - k + (1/2)log(2Pi) +
* STIRLING_CORRECTION(k)
*/
public static double stirlingCorrection(int k) {
if (k > 30) {
double r = 1.0 / k;
double rr = r * r;
double C7 = -5.95238095238095238e-04; // -1/1680
double C5 = 7.93650793650793651e-04; // +1/1260
double C3 = -2.77777777777777778e-03; // -1/360
double C1 = 8.33333333333333333e-02; // +1/12
return r * (C1 + rr * (C3 + rr * (C5 + rr * C7)));
} else {
return STIRLING_CORRECTION[k];
}
}
}