Arithmetic
cern.jet.math

Class Arithmetic



  • public class Arithmeticextends Constants
    Arithmetic functions.
    • Method Summary

      Methods 
      Modifier and TypeMethod and Description
      static doublebinomial(double n, long k)
      Efficiently returns the binomial coefficient, often also referred to as "n over k" or "n choose k".
      static doublebinomial(long n, long k)
      Efficiently returns the binomial coefficient, often also referred to as "n over k" or "n choose k".
      static longceil(double value)
      Returns the smallest long >= value.
      static doublechbevl(double x, double[] coef, int N)
      Evaluates the series of Chebyshev polynomials Ti at argument x/2.
      static doublefactorial(int k)
      Instantly returns the factorial k!.
      static longfloor(double value)
      Returns the largest long <= value.
      static doublelog(double base, double value)
      Returns logbasevalue.
      static doublelog10(double value)
      Returns log10value.
      static doublelog2(double value)
      Returns log2value.
      static doublelogFactorial(int k)
      Returns log(k!).
      static longlongFactorial(int k)
      Instantly returns the factorial k!.
      static doublestirlingCorrection(int k)
      Returns the StirlingCorrection.
    • Method Detail

      • binomial

        public static double binomial(double n,              long k)
        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 ).
        Returns:
        the binomial coefficient.
      • binomial

        public static double binomial(long n,              long k)
        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 ).
        Returns:
        the binomial coefficient.
      • ceil

        public static long ceil(double value)
        Returns the smallest long >= value.
        Examples: 1.0 -> 1, 1.2 -> 2, 1.9 -> 2. This method is safer than using (long) Math.ceil(value), because of possible rounding error.
      • chbevl

        public static double chbevl(double x,            double[] coef,            int N)                     throws ArithmeticException
        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.

        Parameters:
        x - argument to the polynomial.
        coef - the coefficients of the polynomial.
        N - the number of coefficients.
        Throws:
        ArithmeticException
      • factorial

        public static double factorial(int k)
        Instantly returns the factorial k!.
        Parameters:
        k - must hold k >= 0.
      • floor

        public static long floor(double value)
        Returns the largest long <= value.
        Examples: 1.0 -> 1, 1.2 -> 1, 1.9 -> 1
        2.0 -> 2, 2.2 -> 2, 2.9 -> 2
        This method is safer than using (long) Math.floor(value), because of possible rounding error.
      • log

        public static double log(double base,         double value)
        Returns logbasevalue.
      • log10

        public static double log10(double value)
        Returns log10value.
      • log2

        public static double log2(double value)
        Returns log2value.
      • logFactorial

        public static double logFactorial(int k)
        Returns log(k!). Tries to avoid overflows. For k<30 simply looks up a table in O(1). For k>=30 uses stirlings approximation.
        Parameters:
        k - must hold k >= 0.
      • stirlingCorrection

        public static double stirlingCorrection(int k)
        Returns the StirlingCorrection.

        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) + stirlingCorrection(k + 1)

        log k! = (k + 1/2)log(k) - k + (1/2)log(2Pi) + stirlingCorrection(k)

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