cern.jet.math

Class Arithmetic

java.lang.Object

cern.jet.math.Constants

cern.jet.math.Arithmetic

public class Arithmetic
extends Constants

Arithmetic functions.

Field Summary

Fields

Modifier and Type	Field and Description
protected static double[]	doubleFactorials
protected static double[]	logFactorials
protected static long[]	longFactorials

Fields inherited from class cern.jet.math.Constants

big, biginv, LOGPI, MACHEP, MAXGAM, MAXLOG, MINLOG, SQRTH, SQTPI

Constructor Summary

Constructors

Modifier	Constructor and Description
protected	Arithmetic() Makes this class non instantiable, but still let's others inherit from it.

Method Summary

Methods

Modifier and Type	Method and Description
static double	binomial(double n, long k) Efficiently returns the binomial coefficient, often also referred to as "n over k" or "n choose k".
static double	binomial(long n, long k) Efficiently returns the binomial coefficient, often also referred to as "n over k" or "n choose k".
static long	ceil(double value) Returns the smallest long >= value.
static double	chbevl(double x, double[] coef, int N) Evaluates the series of Chebyshev polynomials Ti at argument x/2.
static double	factorial(int k) Instantly returns the factorial k!.
static long	floor(double value) Returns the largest long <= value.
static double	log(double base, double value) Returns logbasevalue.
static double	log10(double value) Returns log10value.
static double	log2(double value) Returns log2value.
static double	logFactorial(int k) Returns log(k!).
static long	longFactorial(int k) Instantly returns the factorial k!.
static double	stirlingCorrection(int k) Returns the StirlingCorrection.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

logFactorials

protected static final double[] logFactorials

longFactorials

protected static final long[] longFactorials

doubleFactorials

protected static final double[] doubleFactorials

Constructor Detail

Arithmetic

protected Arithmetic()

Makes this class non instantiable, but still let's others inherit from it.

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 log_basevalue.

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.

longFactorial

public static long longFactorial(int k)
                          throws IllegalArgumentException

Instantly returns the factorial k!.

Parameters:

k - must hold k >= 0 && k < 21.

Throws:

IllegalArgumentException

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)