cx.htm

复数运算库

  <tr>
    <td width="26%">imag() const</td>
    <td width="26%">imaginary part</td>
    <td width="12%">Real</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real</td>
  </tr>
  <tr>
    <td width="26%">imag()</td>
    <td width="26%">ref. to imaginary part</td>
    <td width="12%">Real&amp;</td>
    <td width="12%">Real&amp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
  </tr>
  <tr>
    <td width="26%">imag_as_imag() const</td>
    <td width="26%">imaginary part as Imag</td>
    <td width="12%">Imag</td>
    <td width="12%">Imag</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Imag</td>
  </tr>
  <tr>
    <td width="26%">conj() const</td>
    <td width="26%">conjugate</td>
    <td width="12%">CX</td>
    <td width="12%">Imag</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Polar</td>
  </tr>
  <tr>
    <td width="26%">cabs() const </td>
    <td width="26%">absolute value</td>
    <td width="12%">Real</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real</td>
  </tr>
  <tr>
    <td width="26%">cabs()</td>
    <td width="26%">ref. to absolute value</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real&amp;</td>
  </tr>
  <tr>
    <td width="26%">arg() const</td>
    <td width="26%">argument</td>
    <td width="12%">Real</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real</td>
  </tr>
  <tr>
    <td width="26%">sum_square() const</td>
    <td width="26%">square of cabs</td>
    <td width="12%">Real</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real</td>
  </tr>
  <tr>
    <td width="26%">operator<TT>+</TT>() const</td>
    <td width="26%"><TT>+</TT> prefix op.</td>
    <td width="12%">CX</td>
    <td width="12%">Imag</td>
    <td width="12%">_I_</td>
    <td width="12%">Polar</td>
  </tr>
  <tr>
    <td width="26%">operator<TT>-</TT>() const</td>
    <td width="26%"><TT>-</TT> prefix op.</td>
    <td width="12%">CX</td>
    <td width="12%">Imag</td>
    <td width="12%">Imag</td>
    <td width="12%">Polar</td>
  </tr>
  <tr>
    <td width="26%">theta() const</td>
    <td width="26%">theta</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real</td>
  </tr>
  <tr>
    <td width="26%">theta()</td>
    <td width="26%">ref. to theta</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real&amp;</td>
  </tr>
  <tr>
    <td width="26%">quadrant() const</td>
    <td width="26%">quadrant</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Quadrant</td>
  </tr>
  <tr>
    <td width="26%">quadrant()</td>
    <td width="26%">ref. to quadrant</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Quadrant&amp;</td>
  </tr>
  <tr>
    <td width="26%">AssertIsValid() const</td>
    <td width="26%">check Polar valid</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">void</td>
  </tr>
</table>

<p>For example:</p>

<pre>   CX z = 3.0 + 4.0 * _I_;
   cout &lt;&lt; z.real() &lt;&lt; &quot;, &quot; &lt;&lt; z.imag() &lt;&lt; endl;</pre>

<p>outputs 3.0, 4.0 .</p>

<p>The declaration of <i>z</i> could have been replaced with any of the
following:</p>

<pre>   CX z(3.0, 4.0);
   CX z; z.real() = 3.0; z.imag() = 4.0;
   CX z = 3.0 + Imag(4.0);
   Polar p(5.0, atan(4.0 / 3.0)); CX z = p;</pre>

<p>Note that there are two versions of <i>z.real()</i> and <i>z.imag()</i> for <i>CX</i>
variable, <i>z</i>. The non-<i>const</i> version is used when <i>z</i> hasn't
been declared as <i>const</i>. This version can be used on the left-hand side of
an = sign. The same applies for <i>p.cabs()</i>, <i>p.theta()</i> and <i>p.quadrant()</i>
for <i>Polar</i> variable <i>p</i>.</p>
<p>It might seem more natural for <i>z.imag()</i> to return a variable of type <i>Imag</i>
rather than <i>Real</i>. However, this would be unusual for those used to the
standard library or Fortran. I include another function <i>z.imag_as_imag()</i>
which does return the imaginary part as type <i>Imag</i>. Nevertheless, there is
a potential source of error, here, if <i>z.imag()</i> is used when <i>z.imag_as_imag()</i>
was intended.</p>

<p><i>sum_square()</i> is the <i>norm</i> function in the standard. It isn't a 
norm in the mathematical sense so you wouldn't call it <i>norm</i> would you?</p>

<p><i>AssertIsValid</i> checks that a Polar variable is valid. An exception is
thrown if the value is invalid; for example the absolute value of the member
variable <i>Theta</i> is greater than <i>pi</i>/4. This should not be possible unless a
Polar object is uninitialised or invalid values have been entered with the <i>cabs</i>, <i>theta</i> or <i>quadrant</i>
functions.</p>
<h2><a name="Global"></a>Global 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%">&nbsp;</td>
    <td width="26%">&nbsp;</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(.)</td>
    <td width="26%">real part</td>
    <td width="12%">Real</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real</td>
  </tr>
  <tr>
    <td width="26%">imag(.)</td>
    <td width="26%">imaginary part</td>
    <td width="12%">Real</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real</td>
  </tr>
  <tr>
    <td width="26%">imag_as_imag(.)</td>
    <td width="26%">imaginary part as Imag</td>
    <td width="12%">Imag</td>
    <td width="12%">Imag</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Imag</td>
  </tr>
  <tr>
    <td width="26%">conj(.)</td>
    <td width="26%">conjugate</td>
    <td width="12%">CX</td>
    <td width="12%">Imag</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Polar</td>
  </tr>
  <tr>
    <td width="26%">cabs(.)</td>
    <td width="26%">absolute value</td>
    <td width="12%">Real</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real</td>
  </tr>
  <tr>
    <td width="26%">fabs(.)</td>
    <td width="26%">absolute value</td>
    <td width="12%">Real</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real</td>
  </tr>
  <tr>
    <td width="26%">norm1(.) </td>
    <td width="26%">fabs(real) + fabs(imag)</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
  </tr>
  <tr>
    <td width="26%">arg(.) </td>
    <td width="26%">argument</td>
    <td width="12%">Real</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real</td>
  </tr>
  <tr>
    <td width="26%">exp(.)</td>
    <td width="26%">exponential</td>
    <td width="12%">CX</td>
    <td width="12%">CX</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
  </tr>
  <tr>
    <td width="26%">log(.)</td>
    <td width="26%">natural log</td>
    <td width="12%">CX</td>
    <td width="12%">CX</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">CX</td>
  </tr>
  <tr>
    <td width="26%">polar_exp(.)</td>
    <td width="26%">exponential</td>
    <td width="12%">Polar</td>
    <td width="12%">Polar</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
  </tr>
  <tr>
    <td width="26%">sqrt(.)</td>
    <td width="26%">square root</td>
    <td width="12%">CX</td>
    <td width="12%">CX</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Polar</td>
  </tr>
  <tr>
    <td width="26%">square(.)</td>
    <td width="26%">square</td>
    <td width="12%">CX</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Polar</td>
  </tr>
  <tr>
    <td width="26%">sum_square(.)</td>
    <td width="26%">square of cabs</td>
    <td width="12%">Real</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">Real</td>
  </tr>
  <tr>
    <td width="26%">sin(.)</td>
    <td width="26%">sin trig. function</td>
    <td width="12%">CX</td>
    <td width="12%">Imag</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
  </tr>
  <tr>
    <td width="26%">cos(.)</td>
    <td width="26%">cos trig. function</td>
    <td width="12%">CX</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
  </tr>
  <tr>
    <td width="26%">tan(.)</td>
    <td width="26%">tan trig. function</td>
    <td width="12%">CX</td>
    <td width="12%">Imag</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
  </tr>
  <tr>
    <td width="26%">sinh(.)</td>
    <td width="26%">sinh hyperbolic fn.</td>
    <td width="12%">CX</td>
    <td width="12%">Imag</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
  </tr>
  <tr>
    <td width="26%">cosh(.)</td>
    <td width="26%">cosh hyperbolic fn.</td>
    <td width="12%">CX</td>
    <td width="12%">Real</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
  </tr>
  <tr>
    <td width="26%">tanh(.)</td>
    <td width="26%">tanh hyperbolic fn.</td>
    <td width="12%">CX</td>
    <td width="12%">Imag</td>
    <td width="12%">&nbsp;</td>
    <td width="12%">&nbsp;</td>
  </tr>
</table>

<p>For example:</p>

<pre>   CX z = 3.0 + 4.0 * _I_;
   cout &lt;&lt; real(z) &lt;&lt; &quot;, &quot; &lt;&lt; imag(z) &lt;&lt; endl;</pre>

<p>outputs 3.0, 4.0 .</p>
<p>Only the <i>const</i> versions of <i>real(z)</i> and <i>imag(z)</i> are
provided. The comments about the member functions <i>.imag()</i> and <i>.imag_as_imag()</i>
also apply here. Note the differences between&nbsp;<i>imag(z)</i>, <i>imag_as_imag(z)</i>
and <i>Imag(z)</i>.</p>

<pre>   CX z = 3.0 + 4.0 * _I_;
   Real a = imag(z);            // returns a = 4
   Imag b = imag_as_imag(z);    // returns b = 4<i>i</i>
   Imag c = Imag(z);            // error, can't convert CX to Imag</pre>

<p>It is important not to confuse <i>Imag</i> and <i>imag</i> or <i>Real</i> and
<i>real</i>.</p>

<p>The functions <i>fabs</i> and <i>cabs</i> are alternative names for the same
function.</p>

<p>The function <i>norm1</i> returns the sum of the absolute values of the real
and imaginary parts. This is faster than <i>cabs</i> and may be sufficient for
deciding, for example, whether the absolute value of a complex number is less than some bound.</p>

<p>   The function <i>polar_exp</i> takes the <i>exp</i> of a complex number and
returns the result in polar coordinates.</p>

<h2><a name="Conversions"></a>Conversions</h2>

<table border="1" cellspacing="0" width="100%" cellpadding="2">
  <tr>
    <td width="20%"><TT>=</TT>,&nbsp; conversions</td>
    <td width="80%" colspan="5" align="center">
      <i>To</i></td>
  </tr>
  <tr>
    <td width="20%"><i>From</i></td>
    <td width="16%"><b>CX</b></td>
    <td width="16%"><b>Real</b></td>
    <td width="16%"><b>Imag</b></td>
    <td width="16%"><b>_I_</b></td>
    <td width="16%"><b>Polar</b></td>
  </tr>
  <tr>
    <td width="20%"><b>CX</b></td>
    <td width="16%">Yes</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">Yes</td>
  </tr>
  <tr>
    <td width="20%"><b>Real</b></td>
    <td width="16%">Yes</td>
    <td width="16%">(Yes)</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">Yes</td>
  </tr>
  <tr>
    <td width="20%"><b>Imag</b></td>
    <td width="16%">Yes</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">Yes</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">Yes</td>
  </tr>
  <tr>
    <td width="20%"><b>_I_</b></td>
    <td width="16%">Yes</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">Yes</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">Yes</td>
  </tr>
  <tr>
    <td width="20%"><b>Polar</b></td>
    <td width="16%">Yes</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">&nbsp;</td>
    <td width="16%">Yes</td>
  </tr>
</table>
<p>The only conversions allowed are ones that do not lose information (apart
from round-off error). For example, you cannot convert <i>Polar</i> to <i>Imag</i>.</p>

<p>I return void from the = operation so you can't do <i>a</i> = <i>b</i> = <i>c</i>;
.</p>

<h2><a name="Binary"></a>Binary functions</h2>
<h3><a name="Additive"></a>Additive</h3>
<table border="1" cellspacing="0" width="100%" cellpadding="2">
  <tr>
    <td width="20%"><TT>+</TT>, <TT>-</TT></td>