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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* 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.math.BigInteger;
import java.util.concurrent.atomic.AtomicReference;
import org.apache.commons.math3.exception.MathArithmeticException;
import org.apache.commons.math3.exception.NotPositiveException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.util.Localizable;
import org.apache.commons.math3.exception.util.LocalizedFormats;
/**
* Some useful, arithmetics related, additions to the built-in functions in
* {@link Math}.
*
* @version $Id: ArithmeticUtils.java 1422313 2012-12-15 18:53:41Z psteitz $
*/
public final class ArithmeticUtils {
/** All long-representable factorials */
static final long[] FACTORIALS = new long[] {
1l, 1l, 2l,
6l, 24l, 120l,
720l, 5040l, 40320l,
362880l, 3628800l, 39916800l,
479001600l, 6227020800l, 87178291200l,
1307674368000l, 20922789888000l, 355687428096000l,
6402373705728000l, 121645100408832000l, 2432902008176640000l };
/** Stirling numbers of the second kind. */
static final AtomicReference STIRLING_S2 = new AtomicReference (null);
/** Private constructor. */
private ArithmeticUtils() {
super();
}
/**
* Add two integers, checking for overflow.
*
* @param x an addend
* @param y an addend
* @return the sum {@code x+y}
* @throws MathArithmeticException if the result can not be represented
* as an {@code int}.
* @since 1.1
*/
public static int addAndCheck(int x, int y)
throws MathArithmeticException {
long s = (long)x + (long)y;
if (s < Integer.MIN_VALUE || s > Integer.MAX_VALUE) {
throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_ADDITION, x, y);
}
return (int)s;
}
/**
* Add two long integers, checking for overflow.
*
* @param a an addend
* @param b an addend
* @return the sum {@code a+b}
* @throws MathArithmeticException if the result can not be represented as an
* long
* @since 1.2
*/
public static long addAndCheck(long a, long b) throws MathArithmeticException {
return ArithmeticUtils.addAndCheck(a, b, LocalizedFormats.OVERFLOW_IN_ADDITION);
}
/**
* Returns an exact representation of the Binomial
* Coefficient, "{@code n choose k}", the number of
* {@code k}-element subsets that can be selected from an
* {@code n}-element set.
*
* Preconditions:
*
* - {@code 0 <= k <= n } (otherwise
* {@code IllegalArgumentException} is thrown)
* - The result is small enough to fit into a {@code long}. The
* largest value of {@code n} for which all coefficients are
* {@code < Long.MAX_VALUE} is 66. If the computed value exceeds
* {@code Long.MAX_VALUE} an {@code ArithMeticException} is
* thrown.
*
*
* @param n the size of the set
* @param k the size of the subsets to be counted
* @return {@code n choose k}
* @throws NotPositiveException if {@code n < 0}.
* @throws NumberIsTooLargeException if {@code k > n}.
* @throws MathArithmeticException if the result is too large to be
* represented by a long integer.
*/
public static long binomialCoefficient(final int n, final int k)
throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
ArithmeticUtils.checkBinomial(n, k);
if ((n == k) || (k == 0)) {
return 1;
}
if ((k == 1) || (k == n - 1)) {
return n;
}
// Use symmetry for large k
if (k > n / 2) {
return binomialCoefficient(n, n - k);
}
// We use the formula
// (n choose k) = n! / (n-k)! / k!
// (n choose k) == ((n-k+1)*...*n) / (1*...*k)
// which could be written
// (n choose k) == (n-1 choose k-1) * n / k
long result = 1;
if (n <= 61) {
// For n <= 61, the naive implementation cannot overflow.
int i = n - k + 1;
for (int j = 1; j <= k; j++) {
result = result * i / j;
i++;
}
} else if (n <= 66) {
// For n > 61 but n <= 66, the result cannot overflow,
// but we must take care not to overflow intermediate values.
int i = n - k + 1;
for (int j = 1; j <= k; j++) {
// We know that (result * i) is divisible by j,
// but (result * i) may overflow, so we split j:
// Filter out the gcd, d, so j/d and i/d are integer.
// result is divisible by (j/d) because (j/d)
// is relative prime to (i/d) and is a divisor of
// result * (i/d).
final long d = gcd(i, j);
result = (result / (j / d)) * (i / d);
i++;
}
} else {
// For n > 66, a result overflow might occur, so we check
// the multiplication, taking care to not overflow
// unnecessary.
int i = n - k + 1;
for (int j = 1; j <= k; j++) {
final long d = gcd(i, j);
result = mulAndCheck(result / (j / d), i / d);
i++;
}
}
return result;
}
/**
* Returns a {@code double} representation of the Binomial
* Coefficient, "{@code n choose k}", the number of
* {@code k}-element subsets that can be selected from an
* {@code n}-element set.
*
* Preconditions:
*
* - {@code 0 <= k <= n } (otherwise
* {@code IllegalArgumentException} is thrown)
* - The result is small enough to fit into a {@code double}. The
* largest value of {@code n} for which all coefficients are <
* Double.MAX_VALUE is 1029. If the computed value exceeds Double.MAX_VALUE,
* Double.POSITIVE_INFINITY is returned
*
*
* @param n the size of the set
* @param k the size of the subsets to be counted
* @return {@code n choose k}
* @throws NotPositiveException if {@code n < 0}.
* @throws NumberIsTooLargeException if {@code k > n}.
* @throws MathArithmeticException if the result is too large to be
* represented by a long integer.
*/
public static double binomialCoefficientDouble(final int n, final int k)
throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
ArithmeticUtils.checkBinomial(n, k);
if ((n == k) || (k == 0)) {
return 1d;
}
if ((k == 1) || (k == n - 1)) {
return n;
}
if (k > n/2) {
return binomialCoefficientDouble(n, n - k);
}
if (n < 67) {
return binomialCoefficient(n,k);
}
double result = 1d;
for (int i = 1; i <= k; i++) {
result *= (double)(n - k + i) / (double)i;
}
return FastMath.floor(result + 0.5);
}
/**
* Returns the natural {@code log} of the Binomial
* Coefficient, "{@code n choose k}", the number of
* {@code k}-element subsets that can be selected from an
* {@code n}-element set.
*
* Preconditions:
*
* - {@code 0 <= k <= n } (otherwise
* {@code IllegalArgumentException} is thrown)
*
*
* @param n the size of the set
* @param k the size of the subsets to be counted
* @return {@code n choose k}
* @throws NotPositiveException if {@code n < 0}.
* @throws NumberIsTooLargeException if {@code k > n}.
* @throws MathArithmeticException if the result is too large to be
* represented by a long integer.
*/
public static double binomialCoefficientLog(final int n, final int k)
throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
ArithmeticUtils.checkBinomial(n, k);
if ((n == k) || (k == 0)) {
return 0;
}
if ((k == 1) || (k == n - 1)) {
return FastMath.log(n);
}
/*
* For values small enough to do exact integer computation,
* return the log of the exact value
*/
if (n < 67) {
return FastMath.log(binomialCoefficient(n,k));
}
/*
* Return the log of binomialCoefficientDouble for values that will not
* overflow binomialCoefficientDouble
*/
if (n < 1030) {
return FastMath.log(binomialCoefficientDouble(n, k));
}
if (k > n / 2) {
return binomialCoefficientLog(n, n - k);
}
/*
* Sum logs for values that could overflow
*/
double logSum = 0;
// n!/(n-k)!
for (int i = n - k + 1; i <= n; i++) {
logSum += FastMath.log(i);
}
// divide by k!
for (int i = 2; i <= k; i++) {
logSum -= FastMath.log(i);
}
return logSum;
}
/**
* Returns n!. Shorthand for {@code n} Factorial, the
* product of the numbers {@code 1,...,n}.
*
* Preconditions:
*
* - {@code n >= 0} (otherwise
* {@code IllegalArgumentException} is thrown)
* - The result is small enough to fit into a {@code long}. The
* largest value of {@code n} for which {@code n!} <
* Long.MAX_VALUE} is 20. If the computed value exceeds {@code Long.MAX_VALUE}
* an {@code ArithMeticException } is thrown.
*
*
*
* @param n argument
* @return {@code n!}
* @throws MathArithmeticException if the result is too large to be represented
* by a {@code long}.
* @throws NotPositiveException if {@code n < 0}.
* @throws MathArithmeticException if {@code n > 20}: The factorial value is too
* large to fit in a {@code long}.
*/
public static long factorial(final int n) throws NotPositiveException, MathArithmeticException {
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,
n);
}
if (n > 20) {
throw new MathArithmeticException();
}
return FACTORIALS[n];
}
/**
* Compute n!, the
* factorial of {@code n} (the product of the numbers 1 to n), as a
* {@code double}.
* The result should be small enough to fit into a {@code double}: The
* largest {@code n} for which {@code n! < Double.MAX_VALUE} is 170.
* If the computed value exceeds {@code Double.MAX_VALUE},
* {@code Double.POSITIVE_INFINITY} is returned.
*
* @param n Argument.
* @return {@code n!}
* @throws NotPositiveException if {@code n < 0}.
*/
public static double factorialDouble(final int n) throws NotPositiveException {
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,
n);
}
if (n < 21) {
return FACTORIALS[n];
}
return FastMath.floor(FastMath.exp(ArithmeticUtils.factorialLog(n)) + 0.5);
}
/**
* Compute the natural logarithm of the factorial of {@code n}.
*
* @param n Argument.
* @return {@code n!}
* @throws NotPositiveException if {@code n < 0}.
*/
public static double factorialLog(final int n) throws NotPositiveException {
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,
n);
}
if (n < 21) {
return FastMath.log(FACTORIALS[n]);
}
double logSum = 0;
for (int i = 2; i <= n; i++) {
logSum += FastMath.log(i);
}
return logSum;
}
/**
* Computes the greatest common divisor of the absolute value of two
* numbers, using a modified version of the "binary gcd" method.
* See Knuth 4.5.2 algorithm B.
* The algorithm is due to Josef Stein (1961).
*
* Special cases:
*
* - The invocations
* {@code gcd(Integer.MIN_VALUE, Integer.MIN_VALUE)},
* {@code gcd(Integer.MIN_VALUE, 0)} and
* {@code gcd(0, Integer.MIN_VALUE)} throw an
* {@code ArithmeticException}, because the result would be 2^31, which
* is too large for an int value.
* - The result of {@code gcd(x, x)}, {@code gcd(0, x)} and
* {@code gcd(x, 0)} is the absolute value of {@code x}, except
* for the special cases above.
* - The invocation {@code gcd(0, 0)} is the only one which returns
* {@code 0}.
*
*
* @param p Number.
* @param q Number.
* @return the greatest common divisor (never negative).
* @throws MathArithmeticException if the result cannot be represented as
* a non-negative {@code int} value.
* @since 1.1
*/
public static int gcd(int p,
int q)
throws MathArithmeticException {
int a = p;
int b = q;
if (a == 0 ||
b == 0) {
if (a == Integer.MIN_VALUE ||
b == Integer.MIN_VALUE) {
throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_32_BITS,
p, q);
}
return FastMath.abs(a + b);
}
long al = a;
long bl = b;
boolean useLong = false;
if (a < 0) {
if(Integer.MIN_VALUE == a) {
useLong = true;
} else {
a = -a;
}
al = -al;
}
if (b < 0) {
if (Integer.MIN_VALUE == b) {
useLong = true;
} else {
b = -b;
}
bl = -bl;
}
if (useLong) {
if(al == bl) {
throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_32_BITS,
p, q);
}
long blbu = bl;
bl = al;
al = blbu % al;
if (al == 0) {
if (bl > Integer.MAX_VALUE) {
throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_32_BITS,
p, q);
}
return (int) bl;
}
blbu = bl;
// Now "al" and "bl" fit in an "int".
b = (int) al;
a = (int) (blbu % al);
}
return gcdPositive(a, b);
}
/**
* Computes the greatest common divisor of two positive numbers
* (this precondition is not checked and the result is undefined
* if not fulfilled) using the "binary gcd" method which avoids division
* and modulo operations.
* See Knuth 4.5.2 algorithm B.
* The algorithm is due to Josef Stein (1961).
*
* Special cases:
*
* - The result of {@code gcd(x, x)}, {@code gcd(0, x)} and
* {@code gcd(x, 0)} is the value of {@code x}.
* - The invocation {@code gcd(0, 0)} is the only one which returns
* {@code 0}.
*
*
* @param a Positive number.
* @param b Positive number.
* @return the greatest common divisor.
*/
private static int gcdPositive(int a,
int b) {
if (a == 0) {
return b;
}
else if (b == 0) {
return a;
}
// Make "a" and "b" odd, keeping track of common power of 2.
final int aTwos = Integer.numberOfTrailingZeros(a);
a >>= aTwos;
final int bTwos = Integer.numberOfTrailingZeros(b);
b >>= bTwos;
final int shift = Math.min(aTwos, bTwos);
// "a" and "b" are positive.
// If a > b then "gdc(a, b)" is equal to "gcd(a - b, b)".
// If a < b then "gcd(a, b)" is equal to "gcd(b - a, a)".
// Hence, in the successive iterations:
// "a" becomes the absolute difference of the current values,
// "b" becomes the minimum of the current values.
while (a != b) {
final int delta = a - b;
b = Math.min(a, b);
a = Math.abs(delta);
// Remove any power of 2 in "a" ("b" is guaranteed to be odd).
a >>= Integer.numberOfTrailingZeros(a);
}
// Recover the common power of 2.
return a << shift;
}
/**
*
* Gets the greatest common divisor of the absolute value of two numbers,
* using the "binary gcd" method which avoids division and modulo
* operations. See Knuth 4.5.2 algorithm B. This algorithm is due to Josef
* Stein (1961).
*
* Special cases:
*
* - The invocations
* {@code gcd(Long.MIN_VALUE, Long.MIN_VALUE)},
* {@code gcd(Long.MIN_VALUE, 0L)} and
* {@code gcd(0L, Long.MIN_VALUE)} throw an
* {@code ArithmeticException}, because the result would be 2^63, which
* is too large for a long value.
* - The result of {@code gcd(x, x)}, {@code gcd(0L, x)} and
* {@code gcd(x, 0L)} is the absolute value of {@code x}, except
* for the special cases above.
*
- The invocation {@code gcd(0L, 0L)} is the only one which returns
* {@code 0L}.
*
*
* @param p Number.
* @param q Number.
* @return the greatest common divisor, never negative.
* @throws MathArithmeticException if the result cannot be represented as
* a non-negative {@code long} value.
* @since 2.1
*/
public static long gcd(final long p, final long q) throws MathArithmeticException {
long u = p;
long v = q;
if ((u == 0) || (v == 0)) {
if ((u == Long.MIN_VALUE) || (v == Long.MIN_VALUE)){
throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_64_BITS,
p, q);
}
return FastMath.abs(u) + FastMath.abs(v);
}
// keep u and v negative, as negative integers range down to
// -2^63, while positive numbers can only be as large as 2^63-1
// (i.e. we can't necessarily negate a negative number without
// overflow)
/* assert u!=0 && v!=0; */
if (u > 0) {
u = -u;
} // make u negative
if (v > 0) {
v = -v;
} // make v negative
// B1. [Find power of 2]
int k = 0;
while ((u & 1) == 0 && (v & 1) == 0 && k < 63) { // while u and v are
// both even...
u /= 2;
v /= 2;
k++; // cast out twos.
}
if (k == 63) {
throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_64_BITS,
p, q);
}
// B2. Initialize: u and v have been divided by 2^k and at least
// one is odd.
long t = ((u & 1) == 1) ? v : -(u / 2)/* B3 */;
// t negative: u was odd, v may be even (t replaces v)
// t positive: u was even, v is odd (t replaces u)
do {
/* assert u<0 && v<0; */
// B4/B3: cast out twos from t.
while ((t & 1) == 0) { // while t is even..
t /= 2; // cast out twos
}
// B5 [reset max(u,v)]
if (t > 0) {
u = -t;
} else {
v = t;
}
// B6/B3. at this point both u and v should be odd.
t = (v - u) / 2;
// |u| larger: t positive (replace u)
// |v| larger: t negative (replace v)
} while (t != 0);
return -u * (1L << k); // gcd is u*2^k
}
/**
*
* Returns the least common multiple of the absolute value of two numbers,
* using the formula {@code lcm(a,b) = (a / gcd(a,b)) * b}.
*
* Special cases:
*
* - The invocations {@code lcm(Integer.MIN_VALUE, n)} and
* {@code lcm(n, Integer.MIN_VALUE)}, where {@code abs(n)} is a
* power of 2, throw an {@code ArithmeticException}, because the result
* would be 2^31, which is too large for an int value.
* - The result of {@code lcm(0, x)} and {@code lcm(x, 0)} is
* {@code 0} for any {@code x}.
*
*
* @param a Number.
* @param b Number.
* @return the least common multiple, never negative.
* @throws MathArithmeticException if the result cannot be represented as
* a non-negative {@code int} value.
* @since 1.1
*/
public static int lcm(int a, int b) throws MathArithmeticException {
if (a == 0 || b == 0){
return 0;
}
int lcm = FastMath.abs(ArithmeticUtils.mulAndCheck(a / gcd(a, b), b));
if (lcm == Integer.MIN_VALUE) {
throw new MathArithmeticException(LocalizedFormats.LCM_OVERFLOW_32_BITS,
a, b);
}
return lcm;
}
/**
*
* Returns the least common multiple of the absolute value of two numbers,
* using the formula {@code lcm(a,b) = (a / gcd(a,b)) * b}.
*
* Special cases:
*
* - The invocations {@code lcm(Long.MIN_VALUE, n)} and
* {@code lcm(n, Long.MIN_VALUE)}, where {@code abs(n)} is a
* power of 2, throw an {@code ArithmeticException}, because the result
* would be 2^63, which is too large for an int value.
* - The result of {@code lcm(0L, x)} and {@code lcm(x, 0L)} is
* {@code 0L} for any {@code x}.
*
*
* @param a Number.
* @param b Number.
* @return the least common multiple, never negative.
* @throws MathArithmeticException if the result cannot be represented
* as a non-negative {@code long} value.
* @since 2.1
*/
public static long lcm(long a, long b) throws MathArithmeticException {
if (a == 0 || b == 0){
return 0;
}
long lcm = FastMath.abs(ArithmeticUtils.mulAndCheck(a / gcd(a, b), b));
if (lcm == Long.MIN_VALUE){
throw new MathArithmeticException(LocalizedFormats.LCM_OVERFLOW_64_BITS,
a, b);
}
return lcm;
}
/**
* Multiply two integers, checking for overflow.
*
* @param x Factor.
* @param y Factor.
* @return the product {@code x * y}.
* @throws MathArithmeticException if the result can not be
* represented as an {@code int}.
* @since 1.1
*/
public static int mulAndCheck(int x, int y) throws MathArithmeticException {
long m = ((long)x) * ((long)y);
if (m < Integer.MIN_VALUE || m > Integer.MAX_VALUE) {
throw new MathArithmeticException();
}
return (int)m;
}
/**
* Multiply two long integers, checking for overflow.
*
* @param a Factor.
* @param b Factor.
* @return the product {@code a * b}.
* @throws MathArithmeticException if the result can not be represented
* as a {@code long}.
* @since 1.2
*/
public static long mulAndCheck(long a, long b) throws MathArithmeticException {
long ret;
if (a > b) {
// use symmetry to reduce boundary cases
ret = mulAndCheck(b, a);
} else {
if (a < 0) {
if (b < 0) {
// check for positive overflow with negative a, negative b
if (a >= Long.MAX_VALUE / b) {
ret = a * b;
} else {
throw new MathArithmeticException();
}
} else if (b > 0) {
// check for negative overflow with negative a, positive b
if (Long.MIN_VALUE / b <= a) {
ret = a * b;
} else {
throw new MathArithmeticException();
}
} else {
// assert b == 0
ret = 0;
}
} else if (a > 0) {
// assert a > 0
// assert b > 0
// check for positive overflow with positive a, positive b
if (a <= Long.MAX_VALUE / b) {
ret = a * b;
} else {
throw new MathArithmeticException();
}
} else {
// assert a == 0
ret = 0;
}
}
return ret;
}
/**
* Subtract two integers, checking for overflow.
*
* @param x Minuend.
* @param y Subtrahend.
* @return the difference {@code x - y}.
* @throws MathArithmeticException if the result can not be represented
* as an {@code int}.
* @since 1.1
*/
public static int subAndCheck(int x, int y) throws MathArithmeticException {
long s = (long)x - (long)y;
if (s < Integer.MIN_VALUE || s > Integer.MAX_VALUE) {
throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_SUBTRACTION, x, y);
}
return (int)s;
}
/**
* Subtract two long integers, checking for overflow.
*
* @param a Value.
* @param b Value.
* @return the difference {@code a - b}.
* @throws MathArithmeticException if the result can not be represented as a
* {@code long}.
* @since 1.2
*/
public static long subAndCheck(long a, long b) throws MathArithmeticException {
long ret;
if (b == Long.MIN_VALUE) {
if (a < 0) {
ret = a - b;
} else {
throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_ADDITION, a, -b);
}
} else {
// use additive inverse
ret = addAndCheck(a, -b, LocalizedFormats.OVERFLOW_IN_ADDITION);
}
return ret;
}
/**
* Raise an int to an int power.
*
* @param k Number to raise.
* @param e Exponent (must be positive or zero).
* @return ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static int pow(final int k, int e) throws NotPositiveException {
if (e < 0) {
throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
}
int result = 1;
int k2p = k;
while (e != 0) {
if ((e & 0x1) != 0) {
result *= k2p;
}
k2p *= k2p;
e = e >> 1;
}
return result;
}
/**
* Raise an int to a long power.
*
* @param k Number to raise.
* @param e Exponent (must be positive or zero).
* @return ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static int pow(final int k, long e) throws NotPositiveException {
if (e < 0) {
throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
}
int result = 1;
int k2p = k;
while (e != 0) {
if ((e & 0x1) != 0) {
result *= k2p;
}
k2p *= k2p;
e = e >> 1;
}
return result;
}
/**
* Raise a long to an int power.
*
* @param k Number to raise.
* @param e Exponent (must be positive or zero).
* @return ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static long pow(final long k, int e) throws NotPositiveException {
if (e < 0) {
throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
}
long result = 1l;
long k2p = k;
while (e != 0) {
if ((e & 0x1) != 0) {
result *= k2p;
}
k2p *= k2p;
e = e >> 1;
}
return result;
}
/**
* Raise a long to a long power.
*
* @param k Number to raise.
* @param e Exponent (must be positive or zero).
* @return ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static long pow(final long k, long e) throws NotPositiveException {
if (e < 0) {
throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
}
long result = 1l;
long k2p = k;
while (e != 0) {
if ((e & 0x1) != 0) {
result *= k2p;
}
k2p *= k2p;
e = e >> 1;
}
return result;
}
/**
* Raise a BigInteger to an int power.
*
* @param k Number to raise.
* @param e Exponent (must be positive or zero).
* @return ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static BigInteger pow(final BigInteger k, int e) throws NotPositiveException {
if (e < 0) {
throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
}
return k.pow(e);
}
/**
* Raise a BigInteger to a long power.
*
* @param k Number to raise.
* @param e Exponent (must be positive or zero).
* @return ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static BigInteger pow(final BigInteger k, long e) throws NotPositiveException {
if (e < 0) {
throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
}
BigInteger result = BigInteger.ONE;
BigInteger k2p = k;
while (e != 0) {
if ((e & 0x1) != 0) {
result = result.multiply(k2p);
}
k2p = k2p.multiply(k2p);
e = e >> 1;
}
return result;
}
/**
* Raise a BigInteger to a BigInteger power.
*
* @param k Number to raise.
* @param e Exponent (must be positive or zero).
* @return ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static BigInteger pow(final BigInteger k, BigInteger e) throws NotPositiveException {
if (e.compareTo(BigInteger.ZERO) < 0) {
throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
}
BigInteger result = BigInteger.ONE;
BigInteger k2p = k;
while (!BigInteger.ZERO.equals(e)) {
if (e.testBit(0)) {
result = result.multiply(k2p);
}
k2p = k2p.multiply(k2p);
e = e.shiftRight(1);
}
return result;
}
/**
* Returns the
* Stirling number of the second kind, "{@code S(n,k)}", the number of
* ways of partitioning an {@code n}-element set into {@code k} non-empty
* subsets.
*
* The preconditions are {@code 0 <= k <= n } (otherwise
* {@code NotPositiveException} is thrown)
*
* @param n the size of the set
* @param k the number of non-empty subsets
* @return {@code S(n,k)}
* @throws NotPositiveException if {@code k < 0}.
* @throws NumberIsTooLargeException if {@code k > n}.
* @throws MathArithmeticException if some overflow happens, typically for n exceeding 25 and
* k between 20 and n-2 (S(n,n-1) is handled specifically and does not overflow)
* @since 3.1
*/
public static long stirlingS2(final int n, final int k)
throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
if (k < 0) {
throw new NotPositiveException(k);
}
if (k > n) {
throw new NumberIsTooLargeException(k, n, true);
}
long[][] stirlingS2 = STIRLING_S2.get();
if (stirlingS2 == null) {
// the cache has never been initialized, compute the first numbers
// by direct recurrence relation
// as S(26,9) = 11201516780955125625 is larger than Long.MAX_VALUE
// we must stop computation at row 26
final int maxIndex = 26;
stirlingS2 = new long[maxIndex][];
stirlingS2[0] = new long[] { 1l };
for (int i = 1; i < stirlingS2.length; ++i) {
stirlingS2[i] = new long[i + 1];
stirlingS2[i][0] = 0;
stirlingS2[i][1] = 1;
stirlingS2[i][i] = 1;
for (int j = 2; j < i; ++j) {
stirlingS2[i][j] = j * stirlingS2[i - 1][j] + stirlingS2[i - 1][j - 1];
}
}
// atomically save the cache
STIRLING_S2.compareAndSet(null, stirlingS2);
}
if (n < stirlingS2.length) {
// the number is in the small cache
return stirlingS2[n][k];
} else {
// use explicit formula to compute the number without caching it
if (k == 0) {
return 0;
} else if (k == 1 || k == n) {
return 1;
} else if (k == 2) {
return (1l << (n - 1)) - 1l;
} else if (k == n - 1) {
return binomialCoefficient(n, 2);
} else {
// definition formula: note that this may trigger some overflow
long sum = 0;
long sign = ((k & 0x1) == 0) ? 1 : -1;
for (int j = 1; j <= k; ++j) {
sign = -sign;
sum += sign * binomialCoefficient(k, j) * pow(j, n);
if (sum < 0) {
// there was an overflow somewhere
throw new MathArithmeticException(LocalizedFormats.ARGUMENT_OUTSIDE_DOMAIN,
n, 0, stirlingS2.length - 1);
}
}
return sum / factorial(k);
}
}
}
/**
* Add two long integers, checking for overflow.
*
* @param a Addend.
* @param b Addend.
* @param pattern Pattern to use for any thrown exception.
* @return the sum {@code a + b}.
* @throws MathArithmeticException if the result cannot be represented
* as a {@code long}.
* @since 1.2
*/
private static long addAndCheck(long a, long b, Localizable pattern) throws MathArithmeticException {
long ret;
if (a > b) {
// use symmetry to reduce boundary cases
ret = addAndCheck(b, a, pattern);
} else {
// assert a <= b
if (a < 0) {
if (b < 0) {
// check for negative overflow
if (Long.MIN_VALUE - b <= a) {
ret = a + b;
} else {
throw new MathArithmeticException(pattern, a, b);
}
} else {
// opposite sign addition is always safe
ret = a + b;
}
} else {
// assert a >= 0
// assert b >= 0
// check for positive overflow
if (a <= Long.MAX_VALUE - b) {
ret = a + b;
} else {
throw new MathArithmeticException(pattern, a, b);
}
}
}
return ret;
}
/**
* Check binomial preconditions.
*
* @param n Size of the set.
* @param k Size of the subsets to be counted.
* @throws NotPositiveException if {@code n < 0}.
* @throws NumberIsTooLargeException if {@code k > n}.
*/
private static void checkBinomial(final int n, final int k) throws NumberIsTooLargeException, NotPositiveException {
if (n < k) {
throw new NumberIsTooLargeException(LocalizedFormats.BINOMIAL_INVALID_PARAMETERS_ORDER,
k, n, true);
}
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.BINOMIAL_NEGATIVE_PARAMETER, n);
}
}
/**
* Returns true if the argument is a power of two.
*
* @param n the number to test
* @return true if the argument is a power of two
*/
public static boolean isPowerOfTwo(long n) {
return (n > 0) && ((n & (n - 1)) == 0);
}
}