6.1 Overview

The org.apache.commons.math4.util package collects a group of array utilities, value transformers, and numerical routines used by implementation classes in commons-math.

6.2 Double array utilities

To maintain statistics based on a "rolling" window of values, a resizable array implementation was developed and is provided for reuse in the util package. The core functionality provided is described in the documentation for the interface, DoubleArray. This interface adds one method, addElementRolling(double) to basic list accessors. The addElementRolling method adds an element (the actual parameter) to the end of the list and removes the first element in the list.

The ResizableDoubleArray class provides a configurable, array-backed implementation of the DoubleArray interface. When addElementRolling is invoked, the underlying array is expanded if necessary, the new element is added to the end of the array and the "usable window" of the array is moved forward, so that the first element is effectively discarded, what was the second becomes the first, and so on. To efficiently manage storage, two maintenance operations need to be periodically performed -- orphaned elements at the beginning of the array need to be reclaimed and space for new elements at the end needs to be created. Both of these operations are handled automatically, with frequency / effect driven by the configuration properties expansionMode, expansionFactor and contractionCriteria. See ResizableDoubleArray for details.

6.3 int/double hash map

The OpenIntToDoubleHashMap class provides a specialized hash map implementation for int/double. This implementation has a much smaller memory overhead than standard java.util.HashMap class. It uses open addressing and primitive arrays, which greatly reduces the number of intermediate objects and improve data locality.

6.4 Continued Fractions

The ContinuedFraction class provides a generic way to create and evaluate continued fractions. The easiest way to create a continued fraction is to subclass ContinuedFraction and override the getA and getB methods which return the continued fraction terms. The precise definition of these terms is explained in Continued Fraction, equation (1) from MathWorld.

As an example, the constant π can be computed using a continued fraction. The following anonymous class provides the implementation:

ContinuedFraction c = new ContinuedFraction() {
    public double getA(int n, double x) {
        switch(n) {
            case 0: return 3.0;
            default: return 6.0;
        }
    }

    public double getB(int n, double x) {
        double y = (2.0 * n) - 1.0;
        return y * y;
    }
}

Then, to evalute π, simply call any of the evalute methods (Note, the point of evaluation in this example is meaningless since π is a constant).

For a more practical use of continued fractions, consider the exponential function. The following anonymous class provides its implementation:

ContinuedFraction c = new ContinuedFraction() {
    public double getA(int n, double x) {
        if (n % 2 == 0) {
            switch(n) {
                case 0: return 1.0;
                default: return 2.0;
            }
        } else {
            return n;
        }
    }

    public double getB(int n, double x) {
        if (n % 2 == 0) {
            return -x;
        } else {
            return x;
        }
    }
}

Then, to evalute e^x for any value x, simply call any of the evalute methods.

6.5 Binomial coefficients, factorials, Stirling numbers and other common math functions

A collection of reusable math functions is provided in the ArithmeticUtils utility class. ArithmeticUtils currently includes methods to compute the following:

• Binomial coefficients -- "n choose k" available as an (exact) long value, binomialCoefficient(int, int) for small n, k; as a double, binomialCoefficientDouble(int, int) for larger values; and in a "super-sized" version, binomialCoefficientLog(int, int) that returns the natural logarithm of the value.
• Stirling numbers of the second kind -- S(n,k) as an exact long value stirlingS2(int, int) for small n, k.
• Factorials -- like binomial coefficients, these are available as exact long values, factorial(int); doubles, factorialDouble(int); or logs, factorialLog(int).
• Least common multiple and greatest common denominator functions.

6.6 Fast mathematical functions

Apache Commons Math provides a faster, more accurate, portable alternative to the regular Math and StrictMathclasses for large scale computation.

FastMath is a drop-in replacement for both Math and StrictMath. This means that for any method in Math (say Math.sin(x) or Math.cbrt(y)), user can directly change the class and use the methods as is (using FastMath.sin(x) or FastMath.cbrt(y) in the previous example).

FastMath speed is achieved by relying heavily on optimizing compilers to native code present in many JVM todays and use of large tables. Precomputed literal arrays are provided in this class to speed up load time. These precomputed tables are used in the default configuration, to improve speed even at first use of the class. If users prefer to compute the tables automatically at load time, they can change a compile-time constant. This will increase class load time at first use, but this overhead will occur only once per run, regardless of the number of subsequent calls to computation methods. Note that FastMath is extensively used inside Apache Commons Math, so by calling some algorithms, the one-shot overhead when the constant is set to false will occur regardless of the end-user calling FastMath methods directly or not. Performance figures for a specific JVM and hardware can be evaluated by running the FastMathTestPerformance tests in the test directory of the source distribution.

FastMath accuracy should be mostly independent of the JVM as it relies only on IEEE-754 basic operations and on embedded tables. Almost all operations are accurate to about 0.5 ulp throughout the domain range. This statement, of course is only a rough global observed behavior, it is not a guarantee for every double numbers input (see William Kahan's Table Maker's Dilemma).

FastMath additionally implements the following methods not found in Math/StrictMath:

• asinh(double)
• acosh(double)
• atanh(double)
• pow(double,int)

• copySign(double, double)
• getExponent(double)
• nextAfter(double,double)
• nextUp(double)
• scalb(double, int)
• copySign(float, float)
• getExponent(float)
• nextAfter(float,double)
• nextUp(float)
• scalb(float, int)

6.7 Miscellaneous