Package org.apache.commons.numbers.core

Class ArithmeticUtils

  • public final class ArithmeticUtils
extends Object
    Some useful, arithmetics related, additions to the built-in functions in Math.

    • Method Summary

      All Methods Static Methods Concrete Methods 
      Modifier and Type Method Description
      static int divideUnsigned​(int dividend, int divisor)
      Returns the unsigned quotient of dividing the first argument by the second where each argument and the result is interpreted as an unsigned value.
      static long divideUnsigned​(long dividend, long divisor)
      Returns the unsigned quotient of dividing the first argument by the second where each argument and the result is interpreted as an unsigned value.
      static int gcd​(int p, int q)
      Computes the greatest common divisor of the absolute value of two numbers, using a modified version of the "binary gcd" method.
      static long gcd​(long p, long q)
      Gets the greatest common divisor of the absolute value of two numbers, using the "binary gcd" method which avoids division and modulo operations.
      static boolean isPowerOfTwo​(long n)
      Returns true if the argument is a power of two.
      static int lcm​(int a, int b)
      Returns the least common multiple of the absolute value of two numbers, using the formula lcm(a,b) = (a / gcd(a,b)) * b.
      static long lcm​(long a, long b)
      Returns the least common multiple of the absolute value of two numbers, using the formula lcm(a,b) = (a / gcd(a,b)) * b.
      static int pow​(int k, int e)
      Raise an int to an int power.
      static long pow​(long k, int e)
      Raise a long to an int power.
      static BigInteger pow​(BigInteger k, int e)
      Raise a BigInteger to an int power.
      static BigInteger pow​(BigInteger k, long e)
      Raise a BigInteger to a long power.
      static BigInteger pow​(BigInteger k, BigInteger e)
      Raise a BigInteger to a BigInteger power.
      static int remainderUnsigned​(int dividend, int divisor)
      Returns the unsigned remainder from dividing the first argument by the second where each argument and the result is interpreted as an unsigned value.
      static long remainderUnsigned​(long dividend, long divisor)
      Returns the unsigned remainder from dividing the first argument by the second where each argument and the result is interpreted as an unsigned value.

    • Method Detail

      • gcd

        public static int gcd​(int p,
                      int q)
        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 gcd(Integer.MIN_VALUE, Integer.MIN_VALUE), gcd(Integer.MIN_VALUE, 0) and gcd(0, Integer.MIN_VALUE) throw an ArithmeticException, because the result would be 2^31, which is too large for an int value.
        • The result of gcd(x, x), gcd(0, x) and gcd(x, 0) is the absolute value of x, except for the special cases above.
        • The invocation gcd(0, 0) is the only one which returns 0.

        Two numbers are relatively prime, or coprime, if their gcd is 1.

        Parameters:
        p - Number.
        q - Number.
        Returns:
        the greatest common divisor (never negative).
        Throws:
        ArithmeticException - if the result cannot be represented as a non-negative int value.

      • gcd

        public static long gcd​(long p,
                       long q)

        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 gcd(Long.MIN_VALUE, Long.MIN_VALUE), gcd(Long.MIN_VALUE, 0L) and gcd(0L, Long.MIN_VALUE) throw an ArithmeticException, because the result would be 2^63, which is too large for a long value.
        • The result of gcd(x, x), gcd(0L, x) and gcd(x, 0L) is the absolute value of x, except for the special cases above.
        • The invocation gcd(0L, 0L) is the only one which returns 0L.

        Two numbers are relatively prime, or coprime, if their gcd is 1.

        Parameters:
        p - Number.
        q - Number.
        Returns:
        the greatest common divisor, never negative.
        Throws:
        ArithmeticException - if the result cannot be represented as a non-negative long value.

      • lcm

        public static int lcm​(int a,
                      int b)

        Returns the least common multiple of the absolute value of two numbers, using the formula lcm(a,b) = (a / gcd(a,b)) * b.

        Special cases:
        • The invocations lcm(Integer.MIN_VALUE, n) and lcm(n, Integer.MIN_VALUE), where abs(n) is a power of 2, throw an ArithmeticException, because the result would be 2^31, which is too large for an int value.
        • The result of lcm(0, x) and lcm(x, 0) is 0 for any x.
        Parameters:
        a - Number.
        b - Number.
        Returns:
        the least common multiple, never negative.
        Throws:
        ArithmeticException - if the result cannot be represented as a non-negative int value.

      • lcm

        public static long lcm​(long a,
                       long b)

        Returns the least common multiple of the absolute value of two numbers, using the formula lcm(a,b) = (a / gcd(a,b)) * b.

        Special cases:
        • The invocations lcm(Long.MIN_VALUE, n) and lcm(n, Long.MIN_VALUE), where abs(n) is a power of 2, throw an ArithmeticException, because the result would be 2^63, which is too large for an int value.
        • The result of lcm(0L, x) and lcm(x, 0L) is 0L for any x.
        Parameters:
        a - Number.
        b - Number.
        Returns:
        the least common multiple, never negative.
        Throws:
        ArithmeticException - if the result cannot be represented as a non-negative long value.

      • pow

        public static int pow​(int k,
                      int e)
        Raise an int to an int power.

        Special cases:

        • k^0 returns 1 (including k=0)
        • k^1 returns k (including k=0)
        • 0^0 returns 1
        • 0^e returns 0
        • 1^e returns 1
        • (-1)^e returns -1 or 1 if e is odd or even
        Parameters:
        k - Number to raise.
        e - Exponent (must be positive or zero).
        Returns:
        \( k^e \)
        Throws:
        IllegalArgumentException - if e < 0.
        ArithmeticException - if the result would overflow.

      • pow

        public static long pow​(long k,
                       int e)
        Raise a long to an int power.

        Special cases:

        • k^0 returns 1 (including k=0)
        • k^1 returns k (including k=0)
        • 0^0 returns 1
        • 0^e returns 0
        • 1^e returns 1
        • (-1)^e returns -1 or 1 if e is odd or even
        Parameters:
        k - Number to raise.
        e - Exponent (must be positive or zero).
        Returns:
        \( k^e \)
        Throws:
        IllegalArgumentException - if e < 0.
        ArithmeticException - if the result would overflow.

      • isPowerOfTwo

        public static boolean isPowerOfTwo​(long n)
        Returns true if the argument is a power of two.
        Parameters:
        n - the number to test
        Returns:
        true if the argument is a power of two

      • remainderUnsigned

        public static int remainderUnsigned​(int dividend,
                                    int divisor)
        Returns the unsigned remainder from dividing the first argument by the second where each argument and the result is interpreted as an unsigned value.

        Implementation note

        In v1.0 this method did not use the long datatype. Modern 64-bit processors make use of the long datatype faster than an algorithm using the int datatype. This method now delegates to Integer.remainderUnsigned(int, int) which uses long arithmetic; or from JDK 19 an intrinsic method.

        Parameters:
        dividend - the value to be divided
        divisor - the value doing the dividing
        Returns:
        the unsigned remainder of the first argument divided by the second argument.
        See Also:
        Integer.remainderUnsigned(int, int)

      • remainderUnsigned

        public static long remainderUnsigned​(long dividend,
                                     long divisor)
        Returns the unsigned remainder from dividing the first argument by the second where each argument and the result is interpreted as an unsigned value.

        Implementation note

        This method does not use the BigInteger datatype. The JDK implementation of Long.remainderUnsigned(long, long) uses BigInteger prior to JDK 17 and this method is 15-25x faster. From JDK 17 onwards the JDK implementation is as fast; or from JDK 19 even faster due to use of an intrinsic method.

        Parameters:
        dividend - the value to be divided
        divisor - the value doing the dividing
        Returns:
        the unsigned remainder of the first argument divided by the second argument.
        See Also:
        Long.remainderUnsigned(long, long)

      • divideUnsigned

        public static int divideUnsigned​(int dividend,
                                 int divisor)
        Returns the unsigned quotient of dividing the first argument by the second where each argument and the result is interpreted as an unsigned value.

        Note that in two's complement arithmetic, the three other basic arithmetic operations of add, subtract, and multiply are bit-wise identical if the two operands are regarded as both being signed or both being unsigned. Therefore separate addUnsigned, etc. methods are not provided.

        Implementation note

        In v1.0 this method did not use the long datatype. Modern 64-bit processors make use of the long datatype faster than an algorithm using the int datatype. This method now delegates to Integer.divideUnsigned(int, int) which uses long arithmetic; or from JDK 19 an intrinsic method.

        Parameters:
        dividend - the value to be divided
        divisor - the value doing the dividing
        Returns:
        the unsigned quotient of the first argument divided by the second argument
        See Also:
        Integer.divideUnsigned(int, int)

      • divideUnsigned

        public static long divideUnsigned​(long dividend,
                                  long divisor)
        Returns the unsigned quotient of dividing the first argument by the second where each argument and the result is interpreted as an unsigned value.

        Note that in two's complement arithmetic, the three other basic arithmetic operations of add, subtract, and multiply are bit-wise identical if the two operands are regarded as both being signed or both being unsigned. Therefore separate addUnsigned, etc. methods are not provided.

        Implementation note

        This method does not use the BigInteger datatype. The JDK implementation of Long.divideUnsigned(long, long) uses BigInteger prior to JDK 17 and this method is 15-25x faster. From JDK 17 onwards the JDK implementation is as fast; or from JDK 19 even faster due to use of an intrinsic method.

        Parameters:
        dividend - the value to be divided
        divisor - the value doing the dividing
        Returns:
        the unsigned quotient of the first argument divided by the second argument.
        See Also:
        Long.divideUnsigned(long, long)