added documentation for FastMath in user guide.
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<li><a href="utilities.html#a6.2_Double_array_utilities">6.2 Double array utilities</a></li>
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<li><a href="utilities.html#a6.3_intdouble_hash_map">6.3 int/double hash map</a></li>
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<li><a href="utilities.html#a6.4_Continued_Fractions">6.4 Continued Fractions</a></li>
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<li><a href="utilities.html#a6.5_binomial_coefficients_factorials_and_other_common_math_functions">6.5 binomial coefficients, factorials and other common math functions</a></li>
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<li><a href="utilities.html#a6.5_binomial_coefficients_factorials_and_other_common_math_functions">6.5 Binomial coefficients, factorials and other common math functions</a></li>
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<li><a href="utilities.html#a6.6_fast_math">6.6 Fast mathematical functions</a></li>
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<li><a href="utilities.html#a6.7_miscellaneous">6.7 Miscellaneous</a></li>
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</ul></li>
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<li><a href="complex.html">7. Complex Numbers</a>
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<ul>
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</p>
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</subsection>
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<subsection name="6.5 binomial coefficients, factorials and other common math functions" href="math_utils">
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<subsection name="6.5 Binomial coefficients, factorials and other common math functions" href="binomial_coefficients_factorials_and_other_common_math_functions">
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<p>
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A collection of reusable math functions is provided in the
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<a href="../apidocs/org/apache/commons/math/util/MathUtils.html">MathUtils</a>
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</p>
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</subsection>
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<subsection name="6.7 Miscellaneous" href="math_utils">
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<subsection name="6.6 Fast mathematical functions" href="fast_math">
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<p>
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Apache Commons Math provides a faster, more accurate, portable alternative
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to the regular <code>Math</code> and <code>StrictMath</code>classes for large
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scale computation.
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</p>
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<p>
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FastMath is a drop-in replacement for both Math and StrictMath. This
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means that for any method in Math (say <code>Math.sin(x)</code> or
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<code>Math.cbrt(y)</code>), user can directly change the class and use the
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methods as is (using <code>FastMath.sin(x)</code> or <code>FastMath.cbrt(y)</code>
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in the previous example).
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</p>
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<p>
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FastMath speed is achieved by relying heavily on optimizing compilers to
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native code present in many JVM todays and use of large tables. Precomputed
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literal arrays are provided in this class to speed up load time. These
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precomputed tables are used in the default configuration, to improve speed
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even at first use of the class. If users prefer to compute the tables
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automatically at load time, they can change a compile-time constant. This will
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increase class load time at first use, but this overhead will occur only once
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per run, regardless of the number of subsequent calls to computation methods.
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Note that FastMath is extensively used inside Apache Commons Math, so by
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calling some algorithms, the one-shot overhead when the constant is set to
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false will occur regardless of the end-user calling FastMath methods directly
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or not. Performance figures for a specific JVM and hardware can be evaluated by
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running the FastMathTestPerformance tests in the test directory of the source
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distribution.
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</p>
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<p>
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FastMath accuracy should be mostly independent of the JVM as it relies only
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on IEEE-754 basic operations and on embedded tables. Almost all operations
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are accurate to about 0.5 ulp throughout the domain range. This statement, of
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course is only a rough global observed behavior, it is <em>not</em> a guarantee
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for <em>every</em> double numbers input (see William Kahan's <a
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href="http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma">Table
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Maker's Dilemma</a>).
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</p>
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<p>
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FastMath additionally implements the following methods not found in Math/StrictMath:
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<ul>
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<li>asinh(double)</li>
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<li>acosh(double)</li>
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<li>atanh(double)</li>
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</ul>
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The following methods are found in Math/StrictMath since 1.6 only, they are
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provided by FastMath even in 1.5 Java virtual machines
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<ul>
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<li>copySign(double, double)</li>
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<li>getExponent(double)</li>
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<li>nextAfter(double,double)</li>
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<li>nextUp(double)</li>
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<li>scalb(double, int)</li>
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<li>copySign(float, float)</li>
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<li>getExponent(float)</li>
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<li>nextAfter(float,double)</li>
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<li>nextUp(float)</li>
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<li>scalb(float, int)</li>
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</ul>
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</p>
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</subsection>
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<subsection name="6.7 Miscellaneous" href="miscellaneous">
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The <a href="../apidocs/org/apache/commons/math/util/MultidimensionalCounter.html">
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MultidimensionalCounter</a> is a utility class that converts a set of indices
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(identifying points in a multidimensional space) to a single index (e.g. identifying
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