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<html><head><meta http-equiv="Content-Language" content="en-nz"><meta http-equiv="Content-Type" content="text/html; charset=windows-1252"><meta name="GENERATOR" content="Microsoft FrontPage 5.0"><meta name="ProgId" content="FrontPage.Editor.Document"><title>Complex library</title><link rel="stylesheet" type="text/css" href="rbd.css"></head><body><h1><a name="top"></a>Complex number library</h1><p><i>Version of July, 2001</i></p><ul> <li><a href="#introduction">Introduction</a> <ul> <li><a href="#author">Author and site details</a></li> <li><a href="#files">Files in this package</a></li> <li><a href="#testing">Testing and getting started</a></li> <li><a href="#change">Change history</a></li> </ul> </li> <li><a href="#Classes">Classes</a></li> <li><a href="#Constructors">Constructors</a> <ul> <li><a href="#Complex">Complex</a></li> <li><a href="#Imaginary">Imaginary</a></li> <li><a href="#Polar">Polar</a></li> </ul> </li> <li><a href="#Member">Member functions</a></li> <li><a href="#Global">Global functions</a></li> <li><a href="#Conversions">Conversions</a></li> <li><a href="#Binary">Binary functions</a> <ul> <li><a href="#Additive">Additive</a></li> <li><a href="#Multiplicative">Multiplicative</a></li> <li><a href="#AdditiveSelf">Additive to self</a></li> <li><a href="#MultiplicativeSelf">Multiplicative to self</a></li> <li><a href="#Equality">Equality</a></li> <li><a href="#Power">Power</a></li> </ul> </li> <li><a href="#input">Input and output</a></li> <li><a href="#GlobVarFn">Global variables, functions</a></li> <li><a href="#example">Mandelbrot set example</a></li></ul><ul> <li><a HREF="ol_doc.htm">To online documentation page</a></li></ul><h2><a name="introduction"></a>Introduction</h2><p>This is a rather more extensive complex number library than the one providedin the standard library. It provides complex, imaginary and polar classes and arepresentation of the <i>i</i> or <i>j</i> of complex analysis. It does not usetemplates. As in my other libraries, this library is based on a floating pointtype <i>Real</i>. This must be <i>typedef</i>ed to be <i>double</i>, <i>float</i>or a class defined by the user (this is done in the file include.h).</p><p>As usual with my libraries, there are no restrictions on the use of thislibrary. If you distribute it, you should note that it is available essentiallyfor free on the Internet. </p><p>Use is at your own risk. I take no responsibility for errors and problems orlosses you may incur through the use of this library. But please report errors.</p><p>See the documentation for <a href="nm09html/index.htm">newmat09</a> or <a href="nm10html/index.htm">newmat10</a>for setting options in the file <i>include.h</i> .</p><p>Go to the <a href="download.html">download</a> page for the library.</p><h3><a name="author"></a>Author and site details</h3><p>Author: Robert Davies; robert@statsresearch.co.nz.</p><p>Available from <a href="http://webnz.com/robert">http://webnz.com/robert</a></p><h3><a name="files"></a>Files in this package</h3><p>The following files are included in this package </p><div align="left"><table border="0" width="80%" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0"> <tr> <td width="33%" valign="top">cx.h</td> <td width="67%">header file for the complex number library</td> </tr> <tr> <td width="33%" valign="top">cx.cpp</td> <td width="67%">function bodies</td> </tr> <tr> <td width="33%" valign="top">cx_polar.cpp</td> <td width="67%">function bodies for polar class</td> </tr> <tr> <td width="33%" valign="top">boolean.h</td> <td width="67%">simulation of the standard Boolean type</td> </tr> <tr> <td width="33%" valign="top">myexcept.h</td> <td width="67%">header for the exceptions simulator</td> </tr> <tr> <td width="33%" valign="top">myexcept.cpp</td> <td width="67%">bodies for the exceptions simulator</td> </tr> <tr> <td width="33%" valign="top">include.h</td> <td width="67%">options header file (see documentation in newmat10)</td> </tr> <tr> <td width="33%" valign="top">cxtest.h</td> <td width="67%">test program header file</td> </tr> <tr> <td width="33%" valign="top">cxtest.cpp</td> <td width="67%">test program main function</td> </tr> <tr> <td width="33%" valign="top">cxtest1.cpp</td> <td width="67%">test unary functions</td> </tr> <tr> <td width="33%" valign="top">cxtest2.cpp</td> <td width="67%">test binary functions </td> </tr> <tr> <td width="33%" valign="top">cxtest.txt</td> <td width="67%">output from the test program</td> </tr> <tr> <td width="33%" valign="top">array1.h</td> <td width="67%">simple array class used by test program</td> </tr> <tr> <td width="33%" valign="top">mandel.cpp</td> <td width="67%">Mandelbrot set example</td> </tr> <tr> <td width="33%" valign="top">mandel.txt</td> <td width="67%">Output from mandel.cpp</td> </tr> <tr> <td width="33%" valign="top">cx_b55.mak</td> <td width="67%">make file for Borland 5.5 compiler</td> </tr> <tr> <td width="33%" valign="top">cx_gnu.mak</td> <td width="67%">make file for Gnu compiler</td> </tr> <tr> <td width="33%" valign="top">cx.htm</td> <td width="67%">this file</td> </tr> <tr> <td width="33%" valign="top">rbd.css</td> <td width="67%">Style sheet for cx.htm</td> </tr> <tr> <td width="33%" valign="top">polar.gif</td> <td width="67%">diagram of polar coordinates - move to subdirectory <i>images</i></td> </tr> <tr> <td width="33%" valign="top">mandel.gif</td> <td width="67%">Mandelbrot set example - move to subdirectory <i>images</i></td> </tr></table></div><h3><a name="testing"></a>Testing and getting started</h3><p>I have tested this program on the Borland 5.02, 5.5, MS VC++ 6, Gnu 2.95, 2.96compilers. The <i>test</i> program uses <i>templates</i> so you will not be able to run thetest program on compilers that don't support templates. I couldn't get the templates in the test program to compile with the Sun CC 5 or 6 compilers. I couldn't compile it with my Watcom 10A compiler.</p><p>I include make files for the Gnu and Borland 5.5 compilers. Make files for some other compilers can be generated with my <a href="genemake.htm">genemake</a> utility.</p><p>If the program is working correctly the output from the test program will endwith two lines similar to</p><pre> maximum from cxtest1: 3.10082e-14 maximum from cxtest2: 7.10543e-15</pre>The exact values may differ. These values are from the Borland 5.02 compiler.<p>For older compilers that don't support <i>bool</i> variables you need to edit <i> include.h</i> to enable my simulated Booleans (comment out the statement <i>#define bool_LIB 0</i>).</p><p>You will need to #include files <i> include.h</i> and <i> cx.h</i> in your programs that use thispackage. Edit <i> include.h</i> to determine whether exceptions are to be used, simulated or disabled. If you use the simulated exceptions you should turn off the exception capability of a compiler that does support exceptions. If you want to use namespace activate the <i>use_namespace</i> definition. Thislibrary uses the namespace name RBD_COMPLEX.</p><h3><a name="change"></a>Change history</h3><table border="1" cellpadding="2" cellspacing="0" width="100%"> <tr> <td width="20%">May, 2000</td> <td width="80%">Initial release</td> </tr> <tr> <td width="20%">July, 2001</td> <td width="80%">Improve compatibility with my other libraries, include sum_square() function</td> </tr></table><h2><a name="Classes"></a>Classes</h2><table border="1" cellspacing="0" width="100%" cellpadding="2"> <tr> <td width="40%"><b>class</b></td> <td width="50%"><b>description</b></td> <td width="10%"><b>user</b></td> </tr> <tr> <td width="40%">class CX</td> <td width="50%">complex</td> <td width="10%">yes</td> </tr> <tr> <td width="40%">class Imag</td> <td width="50%">imaginary</td> <td width="10%">yes</td> </tr> <tr> <td width="40%">class ImaginaryUnit</td> <td width="50%">the <i>i</i> or <i>j</i> of complex analysis</td> <td width="10%">yes</td> </tr> <tr> <td width="40%">class Polar</td> <td width="50%">polar</td> <td width="10%">yes</td> </tr> <tr> <td width="40%">class ConvertFromReal</td> <td width="50%">prevent implicit conversion</td> <td width="10%">no</td> </tr> <tr> <td width="40%">class Quadrant</td> <td width="50%">mod 4 arithmetic</td> <td width="10%">no</td> </tr></table><p>Class <i>CX</i> is the main complex class. I have chosen the name <i>CX</i>so as not to conflict with other complex classes. For example, using <i>complex</i>will clash with the standard library.</p><p>Class <i>Imag</i> defines imaginary variables. If you have a complex variablethat will always be imaginary (the real part is always zero), then use the Imagclass to save computer time and make the program clearer.</p><p>Class <i>ImaginaryUnit</i> is used to represent <i>i</i> = sqrt(-1). I usethis to define <i> _I_</i> as a global variable to represent <i>i</i>.</p><p>Class <i>Polar</i> represents a complex number in polar coordinates.</p><p>I use class <i>ConvertFromReal</i> is used to prevent implicit conversions from <i>Real</i>to <i>Imag</i>. Modern C++ compilers use the keyword <i>explicit</i> to preventimplicit conversion, but, at present I don't want to assume all compilersimplement <i>explicit</i>.</p><p>Class <i>Quadrant</i> is used by class <i>Polar</i> in the representation ofthe angle part of the polar coordinate.</p><p>The angle in the polar coordinates is represented by <i>theta</i> which is nogreater than <i>pi</i>/4 in absolute value and represents the angle from one ofthe axes. Quadrant shows which axis. A <i>char</i> value in a Quadrant objectcan take on one of the values 0, 1, 2 or 3 corresponding to the positive realaxis, the positive imaginary axis, the negative real axis or the negativeimaginary axis respectively. This representation means that one doesn't loseaccuracy when converting from complex coordinates to polar coordinates forcomplex numbers very close to the real or imaginary axes.</p><p align="center"><img border="0" src="images/polar.gif" width="252" height="225"></p><h2><a name="Constructors"></a>Constructors</h2><h3><a name="Complex"></a>Complex</h3><table border="1" cellspacing="0" width="100%" cellpadding="2"> <tr> <td width="30%"> CX()</td> <td width="70%">constructor, don't set values</td> </tr> <tr> <td width="30%"> CX(Real x, Real y)</td> <td width="70%">set real part to <i>x</i>, imaginary part to <i>y</i></td> </tr> <tr> <td width="30%"> CX(Real x)</td> <td width="70%">set real part to <i>x</i>, imaginary part to 0</td> </tr> <tr> <td width="30%"> CX(Imag y)</td> <td width="70%">set real part to 0, imaginary part to <i>y</i></td> </tr> <tr> <td width="30%"> CX(ImaginaryUnit)</td> <td width="70%">set real part to 0, imaginary part to 1</td> </tr> <tr> <td width="30%"> CX(const CX& z)</td> <td width="70%">copy constructor</td> </tr> <tr> <td width="30%"> CX(const Polar& p)</td> <td width="70%">convert polar value <i>p</i> to complex</td> </tr></table><h3><a name="Imaginary"></a>Imaginary</h3><table border="1" cellspacing="0" width="100%" cellpadding="2"> <tr> <td width="30%"> Imag()</td> <td width="70%">constructor, don't set values</td> </tr> <tr> <td width="30%"> Imag(ConvertFromReal x)</td> <td width="70%">set real part to <i>x</i>. ConvertFromReal stops implicit conversions</td> </tr> <tr> <td width="30%"> Imag(const Imag& z)</td> <td width="70%">copy constructor</td> </tr> <tr> <td width="30%"> Imag(ImaginaryUnit)</td> <td width="70%">set value to <i>i</i></td> </tr></table><h3><a name="Polar"></a>Polar</h3><table border="1" cellspacing="0" width="100%" cellpadding="2"> <tr> <td width="30%"> Polar()</td> <td width="70%">constructor, don't set values</td> </tr> <tr> <td width="30%"> Polar(Real r, Real theta)</td> <td width="70%">set polar coordinates to <i>r</i>, <i>theta</i> after normalising</td> </tr> <tr> <td width="30%"> Polar(Real x)</td> <td width="70%">set polar coordinates to <i>x</i>, 0 or -<i>x</i>, <i>pi</i></td> </tr> <tr> <td width="30%"> Polar(Imag y)</td> <td width="70%">set polar coordinates to <i>y</i>, <i>pi</i>/2 or -<i>y</i>,-<i>pi</i>/2</td> </tr> <tr> <td width="30%"> Polar(ImaginaryUnit)</td> <td width="70%">set polar coordinates to 1, <i>pi</i>/2</td> </tr> <tr> <td width="30%"> Polar(const CX& z)</td> <td width="70%">convert complex <i>z</i> to polar </td> </tr> <tr> <td width="30%"> Polar(const Polar& p)</td> <td width="70%">copy constructor</td> </tr></table><h2><a name="Member"></a>Member functions</h2><table border="1" cellspacing="0" width="100%" cellpadding="2"> <tr> <td width="26%"><b>Function</b></td> <td width="26%"><b>Description</b></td> <td width="48%" colspan="4" align="center"> <b>Return types with this class</b></td> </tr> <tr> <td width="26%"> </td> <td width="26%"> </td> <td width="12%"><b>CX</b></td> <td width="12%"><b>Imag</b></td> <td width="12%"><b>_I_</b></td> <td width="12%"><b>Polar</b></td> </tr> <tr> <td width="26%">real() const</td> <td width="26%">real part</td> <td width="12%">Real</td> <td width="12%">Real</td> <td width="12%"> </td> <td width="12%">Real</td> </tr> <tr> <td width="26%">real()</td> <td width="26%">ref. to real part</td> <td width="12%">Real&</td> <td width="12%"> </td> <td width="12%"> </td> <td width="12%"> </td> </tr>⌨️ 快捷键说明
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