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<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"/><title>Math Library</title><link rel="stylesheet" type="text/css" href="../scribble.css" title="default"/><link rel="stylesheet" type="text/css" href="../racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-style.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../doc-site.css" title="default"/><script type="text/javascript" src="../scribble-common.js"></script><script type="text/javascript" src="../manual-racket.js"></script><script type="text/javascript" src="../manual-racket.js"></script><script type="text/javascript" src="../doc-site.js"></script><script type="text/javascript" src="../local-redirect/local-redirect.js"></script><script type="text/javascript" src="../local-redirect/local-user-redirect.js"></script><!--[if IE 6]><style type="text/css">.SIEHidden { overflow: hidden; }</style><![endif]--></head><body id="doc-racket-lang-org"><div class="tocset"><div class="tocview"><div class="tocviewlist tocviewlisttopspace"><div class="tocviewtitle"><table cellspacing="0" cellpadding="0"><tr><td style="width: 1em;"><a href="javascript:void(0);" title="Expand/Collapse" class="tocviewtoggle" onclick="TocviewToggle(this,"tocview_0");">▼</a></td><td></td><td><a href="" class="tocviewselflink" data-pltdoc="x">Math Library</a></td></tr></table></div><div class="tocviewsublistonly" style="display: block;" id="tocview_0"><table cellspacing="0" cellpadding="0"><tr><td align="right">1 </td><td><a href="base.html" class="tocviewlink" data-pltdoc="x">Constants and Elementary Functions</a></td></tr><tr><td align="right">2 </td><td><a href="flonum.html" class="tocviewlink" data-pltdoc="x">Flonums</a></td></tr><tr><td align="right">3 </td><td><a href="special.html" class="tocviewlink" data-pltdoc="x">Special Functions</a></td></tr><tr><td align="right">4 </td><td><a href="number-theory.html" class="tocviewlink" data-pltdoc="x">Number Theory</a></td></tr><tr><td align="right">5 </td><td><a href="bigfloat.html" class="tocviewlink" data-pltdoc="x">Arbitrary-<wbr></wbr>Precision Floating-<wbr></wbr>Point Numbers (<a name="(tech._bigfloat)"></a><span style="font-style: italic">Bigfloats</span>)</a></td></tr><tr><td align="right">6 </td><td><a href="array.html" class="tocviewlink" data-pltdoc="x">Arrays</a></td></tr><tr><td align="right">7 </td><td><a href="matrices.html" class="tocviewlink" data-pltdoc="x">Matrices and Linear Algebra</a></td></tr><tr><td align="right">8 </td><td><a href="stats.html" class="tocviewlink" data-pltdoc="x">Statistics Functions</a></td></tr><tr><td align="right">9 </td><td><a href="dist.html" class="tocviewlink" data-pltdoc="x">Probability Distributions</a></td></tr><tr><td align="right">10 </td><td><a href="utils.html" class="tocviewlink" data-pltdoc="x">Stuff That Doesn’t Belong Anywhere Else</a></td></tr></table></div></div></div><div class="tocsub"><div class="tocsubtitle">On this page:</div><table class="tocsublist" cellspacing="0"><tr><td><span class="tocsublinknumber"></span><a href="#%28part._top%29" class="tocsubseclink" data-pltdoc="x">Math Library</a></td></tr></table></div></div><div class="maincolumn"><div class="main"><div class="versionbox"><span class="version">6.1</span></div><div class="navsettop"><span class="navleft"><form class="searchform"><input class="searchbox" style="color: #888;" type="text" value="...search manuals..." title="Enter a search string to search the manuals" onkeypress="return DoSearchKey(event, this, "6.1", "../");" onfocus="this.style.color="black"; this.style.textAlign="left"; if (this.value == "...search manuals...") this.value="";" onblur="if (this.value.match(/^ *$/)) { this.style.color="#888"; this.style.textAlign="center"; this.value="...search manuals..."; }"/></form> <a href="../index.html" title="up to the documentation top" data-pltdoc="x" onclick="return GotoPLTRoot("6.1");">top</a></span><span class="navright"> <span class="nonavigation">← prev</span> <a href="../index.html" title="up to the documentation top" data-pltdoc="x" onclick="return GotoPLTRoot("6.1");">up</a> <a href="base.html" title="forward to "1 Constants and Elementary Functions"" data-pltdoc="x">next →</a></span> </div><h2 x-source-module="(lib "math/scribblings/math.scrbl")" x-part-tag=""top""><a name="(part._top)"></a><a name="(mod-path._math)"></a>Math Library</h2><div class="SAuthorListBox"><span class="SAuthorList"><p class="author">Neil Toronto <<a href="mailto:ntoronto@racket-lang.org">ntoronto@racket-lang.org</a>></p><span class="SAuthorSep"><br/></span><p class="author">Jens Axel Søgaard <<a href="mailto:jensaxel@soegaard.net">jensaxel@soegaard.net</a>></p></span></div><p><div class="SIntrapara"></div><div class="SIntrapara"></div></p><p><table cellspacing="0" cellpadding="0" class="defmodule"><tr><td align="left"><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="http://download.racket-lang.org/docs/6.1/html/local-redirect/index.html?doc=reference&rel=require.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._require%2529%2529&version=6.1" class="RktStxLink Sq" data-pltdoc="x">require</a></span><span class="stt"> </span><a href="" class="RktModLink" data-pltdoc="x"><span class="RktSym">math</span></a><span class="RktPn">)</span></td><td align="right"><span class="RpackageSpec"><span class="Smaller"> package:</span> <span class="stt">math-lib</span></span></td></tr></table></p><p><div class="SIntrapara">The <a href="" class="RktModLink" data-pltdoc="x"><span class="RktSym">math</span></a> library provides functions and data structures useful
for working with numbers and collections of numbers. These include
</div><div class="SIntrapara"><ul><li><p><a href="base.html" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/base</span></a>: Constants and elementary functions</p></li><li><p><a href="flonum.html" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/flonum</span></a>: Flonum functions, including high-accuracy support</p></li><li><p><a href="special.html" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/special-functions</span></a>: Special (i.e. non-elementary) functions</p></li><li><p><a href="bigfloat.html" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/bigfloat</span></a>: Arbitrary-precision floating-point functions</p></li><li><p><a href="number-theory.html" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/number-theory</span></a>: Number-theoretic functions</p></li><li><p><a href="array.html" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/array</span></a>: Functional arrays for operating on large rectangular data sets</p></li><li><p><a href="matrices.html" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/matrix</span></a>: Linear algebra functions for arrays</p></li><li><p><a href="dist.html" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/distributions</span></a>: Probability distributions</p></li><li><p><a href="stats.html" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/statistics</span></a>: Statistical functions</p></li></ul></div></p><p><div class="SIntrapara">With this library, we hope to support a wide variety of applied mathematics in
Racket, including simulation, statistical inference, signal processing, and
combinatorics. If you find it lacking for your variety of mathematics, please
</div><div class="SIntrapara"><ul><li><p>Visit the <a href="https://github.com/plt/racket/wiki/Math-Library-Features">Math Library Features wiki page</a>
to see what is planned.</p></li><li><p>Contact us or post to one of the <a href="http://racket-lang.org/community.html">mailing lists</a> to
make suggestions or submit patches.</p></li></ul></div></p><p><span style="font-weight: bold">This is a Typed Racket library.</span> It is most efficient to use it in Typed
Racket, so that contracts are checked statically. However, almost all of it can
be used in untyped Racket. Exceptions and performance warnings are in <span style="font-weight: bold">bold text</span>.</p><table cellspacing="0" cellpadding="0"><tr><td><p><span class="hspace"> </span><a href="base.html" class="toptoclink" data-pltdoc="x">1<span class="hspace"> </span>Constants and Elementary Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="base.html#%28part._.Constants%29" class="toclink" data-pltdoc="x">1.1<span class="hspace"> </span>Constants</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="base.html#%28part._.Functions%29" class="toclink" data-pltdoc="x">1.2<span class="hspace"> </span>Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="base.html#%28part._.Random_.Number_.Generation%29" class="toclink" data-pltdoc="x">1.3<span class="hspace"> </span>Random Number Generation</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="base.html#%28part._.Measuring_.Error%29" class="toclink" data-pltdoc="x">1.4<span class="hspace"> </span>Measuring Error</a></p></td></tr><tr><td><p><span class="hspace"></span></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonum.html" class="toptoclink" data-pltdoc="x">2<span class="hspace"> </span>Flonums</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonum.html#%28part._.Additional_.Flonum_.Functions%29" class="toclink" data-pltdoc="x">2.1<span class="hspace"> </span>Additional Flonum Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonum.html#%28part._.Log-.Space_.Arithmetic%29" class="toclink" data-pltdoc="x">2.2<span class="hspace"> </span>Log-Space Arithmetic</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonum.html#%28part._.Debugging_.Flonum_.Functions%29" class="toclink" data-pltdoc="x">2.3<span class="hspace"> </span>Debugging Flonum Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonum.html#%28part._.Measuring_.Floating-.Point_.Error%29" class="toclink" data-pltdoc="x">2.3.1<span class="hspace"> </span>Measuring Floating-Point Error</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonum.html#%28part._.Flonum_.Constants%29" class="toclink" data-pltdoc="x">2.3.2<span class="hspace"> </span>Flonum Constants</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonum.html#%28part._.Low-.Level_.Flonum_.Operations%29" class="toclink" data-pltdoc="x">2.3.3<span class="hspace"> </span>Low-Level Flonum Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonum.html#%28part._.Double-.Double_.Operations%29" class="toclink" data-pltdoc="x">2.4<span class="hspace"> </span>Double-Double Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonum.html#%28part._.Debugging_.Double-.Double_.Functions%29" class="toclink" data-pltdoc="x">2.4.1<span class="hspace"> </span>Debugging Double-Double Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonum.html#%28part._.Low-.Level_.Double-.Double_.Operations%29" class="toclink" data-pltdoc="x">2.4.2<span class="hspace"> </span>Low-Level Double-Double Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="flonum.html#%28part._.Additional_.Flonum_.Vector_.Functions%29" class="toclink" data-pltdoc="x">2.5<span class="hspace"> </span>Additional Flonum Vector Functions</a></p></td></tr><tr><td><p><span class="hspace"></span></p></td></tr><tr><td><p><span class="hspace"> </span><a href="special.html" class="toptoclink" data-pltdoc="x">3<span class="hspace"> </span>Special Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="special.html#%28part._real-functions%29" class="toclink" data-pltdoc="x">3.1<span class="hspace"> </span>Real Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="special.html#%28part._flonum-functions%29" class="toclink" data-pltdoc="x">3.2<span class="hspace"> </span>Flonum Functions</a></p></td></tr><tr><td><p><span class="hspace"></span></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html" class="toptoclink" data-pltdoc="x">4<span class="hspace"> </span>Number Theory</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._congruences%29" class="toclink" data-pltdoc="x">4.1<span class="hspace"> </span>Congruences and Modular Arithmetic</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._modular%29" class="toclink" data-pltdoc="x">4.1.1<span class="hspace"> </span>Parameterized Modular Arithmetic</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._primes%29" class="toclink" data-pltdoc="x">4.2<span class="hspace"> </span>Primes</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._roots%29" class="toclink" data-pltdoc="x">4.3<span class="hspace"> </span>Roots</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._powers%29" class="toclink" data-pltdoc="x">4.4<span class="hspace"> </span>Powers</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._multiplicative%29" class="toclink" data-pltdoc="x">4.5<span class="hspace"> </span>Multiplicative Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._number-sequences%29" class="toclink" data-pltdoc="x">4.6<span class="hspace"> </span>Number Sequences</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._combinatorics%29" class="toclink" data-pltdoc="x">4.7<span class="hspace"> </span>Combinatorics</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._special_numbers%29" class="toclink" data-pltdoc="x">4.8<span class="hspace"> </span>Special Numbers</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._.Polygonal_.Numbers%29" class="toclink" data-pltdoc="x">4.8.1<span class="hspace"> </span>Polygonal Numbers</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._fractions%29" class="toclink" data-pltdoc="x">4.9<span class="hspace"> </span>Fractions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._quadratics%29" class="toclink" data-pltdoc="x">4.10<span class="hspace"> </span>The Quadratic Equation</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="number-theory.html#%28part._primitive_roots%29" class="toclink" data-pltdoc="x">4.11<span class="hspace"> </span>The group Zn and Primitive Roots</a></p></td></tr><tr><td><p><span class="hspace"></span></p></td></tr><tr><td><p><span class="hspace"> </span><a href="bigfloat.html" class="toptoclink" data-pltdoc="x">5<span class="hspace"> </span>Arbitrary-Precision Floating-Point Numbers (<a name="(tech._bigfloat)"></a><span style="font-style: italic">Bigfloats</span>)</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="bigfloat.html#%28part._quick%29" class="toclink" data-pltdoc="x">5.1<span class="hspace"> </span>Quick Start</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="bigfloat.html#%28part._.Fictionally_.Asked_.Questions%29" class="toclink" data-pltdoc="x">5.2<span class="hspace"> </span>Fictionally Asked Questions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="bigfloat.html#%28part._.Type_and_.Constructors%29" class="toclink" data-pltdoc="x">5.3<span class="hspace"> </span>Type and Constructors</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="bigfloat.html#%28part._.Accessors_and_.Conversion_.Functions%29" class="toclink" data-pltdoc="x">5.4<span class="hspace"> </span>Accessors and Conversion Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="bigfloat.html#%28part._.Parameters%29" class="toclink" data-pltdoc="x">5.5<span class="hspace"> </span>Parameters</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="bigfloat.html#%28part._constants%29" class="toclink" data-pltdoc="x">5.6<span class="hspace"> </span>Constants</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="bigfloat.html#%28part._predicates%29" class="toclink" data-pltdoc="x">5.7<span class="hspace"> </span>Predicates</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="bigfloat.html#%28part._rounding%29" class="toclink" data-pltdoc="x">5.8<span class="hspace"> </span>Rounding</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="bigfloat.html#%28part._ops%29" class="toclink" data-pltdoc="x">5.9<span class="hspace"> </span>Mathematical Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="bigfloat.html#%28part._misc%29" class="toclink" data-pltdoc="x">5.10<span class="hspace"> </span>Low-level Functions</a></p></td></tr><tr><td><p><span class="hspace"></span></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array.html" class="toptoclink" data-pltdoc="x">6<span class="hspace"> </span>Arrays</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_quick.html" class="toclink" data-pltdoc="x">6.1<span class="hspace"> </span>Quick Start</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_defs.html" class="toclink" data-pltdoc="x">6.2<span class="hspace"> </span>Definitions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_broadcasting.html" class="toclink" data-pltdoc="x">6.3<span class="hspace"> </span>Broadcasting</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_broadcasting.html#%28part._array~3abroadcasting~3arules%29" class="toclink" data-pltdoc="x">6.3.1<span class="hspace"> </span>Broadcasting Rules</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_broadcasting.html#%28part._array~3abroadcasting~3acontrol%29" class="toclink" data-pltdoc="x">6.3.2<span class="hspace"> </span>Broadcasting Control</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_slicing.html" class="toclink" data-pltdoc="x">6.4<span class="hspace"> </span>Slicing</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_slicing.html#%28part.__.Sequenceof_.Integer___pick_rows%29" class="toclink" data-pltdoc="x">6.4.1<span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><span class="RktStxLink">Sequenceof</span></span><span class="stt"> </span><span class="RktSym"><span class="RktStxLink">Integer</span></span><span class="RktPn">)</span>: pick rows</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_slicing.html#%28part._.Slice__pick_rows_in_a_length-aware_way%29" class="toclink" data-pltdoc="x">6.4.2<span class="hspace"> </span><span class="RktSym"><span class="RktStxLink">Slice</span></span>: pick rows in a length-aware way</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_slicing.html#%28part._.Slice-.Dots__preserve_remaining_axes%29" class="toclink" data-pltdoc="x">6.4.3<span class="hspace"> </span><span class="RktSym"><span class="RktStxLink">Slice-Dots</span></span>: preserve remaining axes</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_slicing.html#%28part._.Integer__remove_an_axis%29" class="toclink" data-pltdoc="x">6.4.4<span class="hspace"> </span><span class="RktSym"><span class="RktStxLink">Integer</span></span>: remove an axis</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_slicing.html#%28part._.Slice-.New-.Axis__add_an_axis%29" class="toclink" data-pltdoc="x">6.4.5<span class="hspace"> </span><span class="RktSym"><span class="RktStxLink">Slice-New-Axis</span></span>: add an axis</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_nonstrict.html" class="toclink" data-pltdoc="x">6.5<span class="hspace"> </span>Nonstrict Arrays</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_nonstrict.html#%28part._.Caching_.Nonstrict_.Elements%29" class="toclink" data-pltdoc="x">6.5.1<span class="hspace"> </span>Caching Nonstrict Elements</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_nonstrict.html#%28part._.Performance_.Considerations%29" class="toclink" data-pltdoc="x">6.5.2<span class="hspace"> </span>Performance Considerations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_types.html" class="toclink" data-pltdoc="x">6.6<span class="hspace"> </span>Types, Predicates and Accessors</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_construct.html" class="toclink" data-pltdoc="x">6.7<span class="hspace"> </span>Construction</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_convert.html" class="toclink" data-pltdoc="x">6.8<span class="hspace"> </span>Conversion</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_convert.html#%28part._array~3aprint%29" class="toclink" data-pltdoc="x">6.8.1<span class="hspace"> </span>Printing</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_sequences.html" class="toclink" data-pltdoc="x">6.9<span class="hspace"> </span>Comprehensions and Sequences</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_pointwise.html" class="toclink" data-pltdoc="x">6.10<span class="hspace"> </span>Pointwise Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_pointwise.html#%28part._.Broadcasting%29" class="toclink" data-pltdoc="x">6.10.1<span class="hspace"> </span>Broadcasting</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_indexing.html" class="toclink" data-pltdoc="x">6.11<span class="hspace"> </span>Indexing and Slicing</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_transform.html" class="toclink" data-pltdoc="x">6.12<span class="hspace"> </span>Transformations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_fold.html" class="toclink" data-pltdoc="x">6.13<span class="hspace"> </span>Folds, Reductions and Expansions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_fold.html#%28part._.Axis_.Folds%29" class="toclink" data-pltdoc="x">6.13.1<span class="hspace"> </span>Axis Folds</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_fold.html#%28part._.Whole-.Array_.Folds%29" class="toclink" data-pltdoc="x">6.13.2<span class="hspace"> </span>Whole-Array Folds</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_fold.html#%28part._.General_.Reductions_and_.Expansions%29" class="toclink" data-pltdoc="x">6.13.3<span class="hspace"> </span>General Reductions and Expansions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_other.html" class="toclink" data-pltdoc="x">6.14<span class="hspace"> </span>Other Array Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_other.html#%28part._.Fast_.Fourier_.Transform%29" class="toclink" data-pltdoc="x">6.14.1<span class="hspace"> </span>Fast Fourier Transform</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_subtypes.html" class="toclink" data-pltdoc="x">6.15<span class="hspace"> </span>Subtypes</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_subtypes.html#%28part._flarray%29" class="toclink" data-pltdoc="x">6.15.1<span class="hspace"> </span>Flonum Arrays</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_subtypes.html#%28part._fcarray%29" class="toclink" data-pltdoc="x">6.15.2<span class="hspace"> </span>Float-Complex Arrays</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="array_strict.html" class="toclink" data-pltdoc="x">6.16<span class="hspace"> </span>Strictness</a></p></td></tr><tr><td><p><span class="hspace"></span></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrices.html" class="toptoclink" data-pltdoc="x">7<span class="hspace"> </span>Matrices and Linear Algebra</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_intro.html" class="toclink" data-pltdoc="x">7.1<span class="hspace"> </span>Introduction</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_intro.html#%28part._matrix~3afunction-types%29" class="toclink" data-pltdoc="x">7.1.1<span class="hspace"> </span>Function Types</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_intro.html#%28part._matrix~3afailure%29" class="toclink" data-pltdoc="x">7.1.2<span class="hspace"> </span>Failure Arguments</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_intro.html#%28part._matrix~3abroadcasting%29" class="toclink" data-pltdoc="x">7.1.3<span class="hspace"> </span>Broadcasting</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_intro.html#%28part._.Strictness%29" class="toclink" data-pltdoc="x">7.1.4<span class="hspace"> </span>Strictness</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_types.html" class="toclink" data-pltdoc="x">7.2<span class="hspace"> </span>Types, Predicates and Accessors</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_construction.html" class="toclink" data-pltdoc="x">7.3<span class="hspace"> </span>Construction</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_conversion.html" class="toclink" data-pltdoc="x">7.4<span class="hspace"> </span>Conversion</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_arith.html" class="toclink" data-pltdoc="x">7.5<span class="hspace"> </span>Entrywise Operations and Arithmetic</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_poly.html" class="toclink" data-pltdoc="x">7.6<span class="hspace"> </span>Polymorphic Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_basic.html" class="toclink" data-pltdoc="x">7.7<span class="hspace"> </span>Basic Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_inner.html" class="toclink" data-pltdoc="x">7.8<span class="hspace"> </span>Inner Product Space Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_solve.html" class="toclink" data-pltdoc="x">7.9<span class="hspace"> </span>Solving Systems of Equations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_row-alg.html" class="toclink" data-pltdoc="x">7.10<span class="hspace"> </span>Row-Based Algorithms</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_ortho-alg.html" class="toclink" data-pltdoc="x">7.11<span class="hspace"> </span>Orthogonal Algorithms</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="matrix_op-norm.html" class="toclink" data-pltdoc="x">7.12<span class="hspace"> </span>Operator Norms and Comparing Matrices</a></p></td></tr><tr><td><p><span class="hspace"></span></p></td></tr><tr><td><p><span class="hspace"> </span><a href="stats.html" class="toptoclink" data-pltdoc="x">8<span class="hspace"> </span>Statistics Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="stats.html#%28part._stats~3aexpected-values%29" class="toclink" data-pltdoc="x">8.1<span class="hspace"> </span>Expected Values</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="stats.html#%28part._stats~3arunning%29" class="toclink" data-pltdoc="x">8.2<span class="hspace"> </span>Running Expected Values</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="stats.html#%28part._.Correlation%29" class="toclink" data-pltdoc="x">8.3<span class="hspace"> </span>Correlation</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="stats.html#%28part._.Counting_and_.Binning%29" class="toclink" data-pltdoc="x">8.4<span class="hspace"> </span>Counting and Binning</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="stats.html#%28part._.Order_.Statistics%29" class="toclink" data-pltdoc="x">8.5<span class="hspace"> </span>Order Statistics</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="stats.html#%28part._.Simulations%29" class="toclink" data-pltdoc="x">8.6<span class="hspace"> </span>Simulations</a></p></td></tr><tr><td><p><span class="hspace"></span></p></td></tr><tr><td><p><span class="hspace"> </span><a href="dist.html" class="toptoclink" data-pltdoc="x">9<span class="hspace"> </span>Probability Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="dist_dist-objects.html" class="toclink" data-pltdoc="x">9.1<span class="hspace"> </span>Distribution Objects</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Distribution_Types_and_Operations.html" class="toclink" data-pltdoc="x">9.2<span class="hspace"> </span>Distribution Types and Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Finite_Distribution_Families.html" class="toclink" data-pltdoc="x">9.3<span class="hspace"> </span>Finite Distribution Families</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Finite_Distribution_Families.html#%28part._.Unordered_.Discrete_.Distributions%29" class="toclink" data-pltdoc="x">9.3.1<span class="hspace"> </span>Unordered Discrete Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Integer_Distribution_Families.html" class="toclink" data-pltdoc="x">9.4<span class="hspace"> </span>Integer Distribution Families</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Integer_Distribution_Families.html#%28part._.Bernoulli_.Distributions%29" class="toclink" data-pltdoc="x">9.4.1<span class="hspace"> </span>Bernoulli Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Integer_Distribution_Families.html#%28part._.Binomial_.Distributions%29" class="toclink" data-pltdoc="x">9.4.2<span class="hspace"> </span>Binomial Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Integer_Distribution_Families.html#%28part._.Geometric_.Distributions%29" class="toclink" data-pltdoc="x">9.4.3<span class="hspace"> </span>Geometric Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Integer_Distribution_Families.html#%28part._.Poisson_.Distributions%29" class="toclink" data-pltdoc="x">9.4.4<span class="hspace"> </span>Poisson Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html" class="toclink" data-pltdoc="x">9.5<span class="hspace"> </span>Real Distribution Families</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Beta_.Distributions%29" class="toclink" data-pltdoc="x">9.5.1<span class="hspace"> </span>Beta Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Cauchy_.Distributions%29" class="toclink" data-pltdoc="x">9.5.2<span class="hspace"> </span>Cauchy Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Delta_.Distributions%29" class="toclink" data-pltdoc="x">9.5.3<span class="hspace"> </span>Delta Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Exponential_.Distributions%29" class="toclink" data-pltdoc="x">9.5.4<span class="hspace"> </span>Exponential Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Gamma_.Distributions%29" class="toclink" data-pltdoc="x">9.5.5<span class="hspace"> </span>Gamma Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Logistic_.Distributions%29" class="toclink" data-pltdoc="x">9.5.6<span class="hspace"> </span>Logistic Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Normal_.Distributions%29" class="toclink" data-pltdoc="x">9.5.7<span class="hspace"> </span>Normal Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Triangular_.Distributions%29" class="toclink" data-pltdoc="x">9.5.8<span class="hspace"> </span>Triangular Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Truncated_.Distributions%29" class="toclink" data-pltdoc="x">9.5.9<span class="hspace"> </span>Truncated Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Uniform_.Distributions%29" class="toclink" data-pltdoc="x">9.5.10<span class="hspace"> </span>Uniform Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="dist_flonum.html" class="toclink" data-pltdoc="x">9.6<span class="hspace"> </span>Low-Level Distribution Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="dist_flonum.html#%28part._.Integer_.Distribution_.Functions%29" class="toclink" data-pltdoc="x">9.6.1<span class="hspace"> </span>Integer Distribution Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="dist_flonum.html#%28part._.Real_.Distribution_.Functions%29" class="toclink" data-pltdoc="x">9.6.2<span class="hspace"> </span>Real Distribution Functions</a></p></td></tr><tr><td><p><span class="hspace"></span></p></td></tr><tr><td><p><span class="hspace"> </span><a href="utils.html" class="toptoclink" data-pltdoc="x">10<span class="hspace"> </span>Stuff That Doesn’t Belong Anywhere Else</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="utils.html#%28part._utils~3aparallel%29" class="toclink" data-pltdoc="x">10.1<span class="hspace"> </span>Parallelization</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="utils.html#%28part._utils~3adft%29" class="toclink" data-pltdoc="x">10.2<span class="hspace"> </span>Discrete Fourier Transform Conventions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="utils.html#%28part._utils~3afpu-test%29" class="toclink" data-pltdoc="x">10.3<span class="hspace"> </span>Floating-Point Compliance Testing</a></p></td></tr></table><div class="navsetbottom"><span class="navleft"><form class="searchform"><input class="searchbox" style="color: #888;" type="text" value="...search manuals..." title="Enter 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