math::complex.3

视频监控网络部分的协议ddns,的模块的实现代码,请大家大胆指正.

\&        z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
\&        z1 / z2 = (r1 / r2) * exp(i * (t1 \- t2))
\&        z1 ** z2 = exp(z2 * log z1)
\&        ~z = a \- bi
\&        abs(z) = r1 = sqrt(a*a + b*b)
\&        sqrt(z) = sqrt(r1) * exp(i * t/2)
\&        exp(z) = exp(a) * exp(i * b)
\&        log(z) = log(r1) + i*t
\&        sin(z) = 1/2i (exp(i * z1) \- exp(\-i * z))
\&        cos(z) = 1/2 (exp(i * z1) + exp(\-i * z))
\&        atan2(y, x) = atan(y / x) # Minding the right quadrant, note the order.
.Ve
.PP
The definition used for complex arguments of \fIatan2()\fR is
.PP
.Vb 1
\&       \-i log((x + iy)/sqrt(x*x+y*y))
.Ve
.PP
Note that atan2(0, 0) is not well-defined.
.PP
The following extra operations are supported on both real and complex
numbers:
.PP
.Vb 4
\&        Re(z) = a
\&        Im(z) = b
\&        arg(z) = t
\&        abs(z) = r
\&
\&        cbrt(z) = z ** (1/3)
\&        log10(z) = log(z) / log(10)
\&        logn(z, n) = log(z) / log(n)
\&
\&        tan(z) = sin(z) / cos(z)
\&
\&        csc(z) = 1 / sin(z)
\&        sec(z) = 1 / cos(z)
\&        cot(z) = 1 / tan(z)
\&
\&        asin(z) = \-i * log(i*z + sqrt(1\-z*z))
\&        acos(z) = \-i * log(z + i*sqrt(1\-z*z))
\&        atan(z) = i/2 * log((i+z) / (i\-z))
\&
\&        acsc(z) = asin(1 / z)
\&        asec(z) = acos(1 / z)
\&        acot(z) = atan(1 / z) = \-i/2 * log((i+z) / (z\-i))
\&
\&        sinh(z) = 1/2 (exp(z) \- exp(\-z))
\&        cosh(z) = 1/2 (exp(z) + exp(\-z))
\&        tanh(z) = sinh(z) / cosh(z) = (exp(z) \- exp(\-z)) / (exp(z) + exp(\-z))
\&
\&        csch(z) = 1 / sinh(z)
\&        sech(z) = 1 / cosh(z)
\&        coth(z) = 1 / tanh(z)
\&
\&        asinh(z) = log(z + sqrt(z*z+1))
\&        acosh(z) = log(z + sqrt(z*z\-1))
\&        atanh(z) = 1/2 * log((1+z) / (1\-z))
\&
\&        acsch(z) = asinh(1 / z)
\&        asech(z) = acosh(1 / z)
\&        acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z\-1))
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.PP
\&\fIarg\fR, \fIabs\fR, \fIlog\fR, \fIcsc\fR, \fIcot\fR, \fIacsc\fR, \fIacot\fR, \fIcsch\fR,
\&\fIcoth\fR, \fIacosech\fR, \fIacotanh\fR, have aliases \fIrho\fR, \fItheta\fR, \fIln\fR,
\&\fIcosec\fR, \fIcotan\fR, \fIacosec\fR, \fIacotan\fR, \fIcosech\fR, \fIcotanh\fR,
\&\fIacosech\fR, \fIacotanh\fR, respectively.  \f(CW\*(C`Re\*(C'\fR, \f(CW\*(C`Im\*(C'\fR, \f(CW\*(C`arg\*(C'\fR, \f(CW\*(C`abs\*(C'\fR,
\&\f(CW\*(C`rho\*(C'\fR, and \f(CW\*(C`theta\*(C'\fR can be used also as mutators.  The \f(CW\*(C`cbrt\*(C'\fR
returns only one of the solutions: if you want all three, use the
\&\f(CW\*(C`root\*(C'\fR function.
.PP
The \fIroot\fR function is available to compute all the \fIn\fR
roots of some complex, where \fIn\fR is a strictly positive integer.
There are exactly \fIn\fR such roots, returned as a list. Getting the
number mathematicians call \f(CW\*(C`j\*(C'\fR such that:
.PP
.Vb 1
\&        1 + j + j*j = 0;
.Ve
.PP
is a simple matter of writing:
.PP
.Vb 1
\&        $j = ((root(1, 3))[1];
.Ve
.PP
The \fIk\fRth root for \f(CW\*(C`z = [r,t]\*(C'\fR is given by:
.PP
.Vb 1
\&        (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
.Ve
.PP
You can return the \fIk\fRth root directly by \f(CW\*(C`root(z, n, k)\*(C'\fR,
indexing starting from \fIzero\fR and ending at \fIn \- 1\fR.
.PP
The \fIspaceship\fR comparison operator, <=>, is also defined. In
order to ensure its restriction to real numbers is conform to what you
would expect, the comparison is run on the real part of the complex
number first, and imaginary parts are compared only when the real
parts match.
.SH "CREATION"
.IX Header "CREATION"
To create a complex number, use either:
.PP
.Vb 2
\&        $z = Math::Complex\->make(3, 4);
\&        $z = cplx(3, 4);
.Ve
.PP
if you know the cartesian form of the number, or
.PP
.Vb 1
\&        $z = 3 + 4*i;
.Ve
.PP
if you like. To create a number using the polar form, use either:
.PP
.Vb 2
\&        $z = Math::Complex\->emake(5, pi/3);
\&        $x = cplxe(5, pi/3);
.Ve
.PP
instead. The first argument is the modulus, the second is the angle
(in radians, the full circle is 2*pi).  (Mnemonic: \f(CW\*(C`e\*(C'\fR is used as a
notation for complex numbers in the polar form).
.PP
It is possible to write:
.PP
.Vb 1
\&        $x = cplxe(\-3, pi/4);
.Ve
.PP
but that will be silently converted into \f(CW\*(C`[3,\-3pi/4]\*(C'\fR, since the
modulus must be non-negative (it represents the distance to the origin
in the complex plane).
.PP
It is also possible to have a complex number as either argument of the
\&\f(CW\*(C`make\*(C'\fR, \f(CW\*(C`emake\*(C'\fR, \f(CW\*(C`cplx\*(C'\fR, and \f(CW\*(C`cplxe\*(C'\fR: the appropriate component of
the argument will be used.
.PP
.Vb 2
\&        $z1 = cplx(\-2,  1);
\&        $z2 = cplx($z1, 4);
.Ve
.PP
The \f(CW\*(C`new\*(C'\fR, \f(CW\*(C`make\*(C'\fR, \f(CW\*(C`emake\*(C'\fR, \f(CW\*(C`cplx\*(C'\fR, and \f(CW\*(C`cplxe\*(C'\fR will also
understand a single (string) argument of the forms
.PP
.Vb 5
\&        2\-3i
\&        \-3i
\&        [2,3]
\&        [2,\-3pi/4]
\&        [2]
.Ve
.PP
in which case the appropriate cartesian and exponential components
will be parsed from the string and used to create new complex numbers.
The imaginary component and the theta, respectively, will default to zero.
.PP
The \f(CW\*(C`new\*(C'\fR, \f(CW\*(C`make\*(C'\fR, \f(CW\*(C`emake\*(C'\fR, \f(CW\*(C`cplx\*(C'\fR, and \f(CW\*(C`cplxe\*(C'\fR will also
understand the case of no arguments: this means plain zero or (0, 0).
.SH "DISPLAYING"
.IX Header "DISPLAYING"
When printed, a complex number is usually shown under its cartesian
style \fIa+bi\fR, but there are legitimate cases where the polar style
\&\fI[r,t]\fR is more appropriate.  The process of converting the complex
number into a string that can be displayed is known as \fIstringification\fR.
.PP
By calling the class method \f(CW\*(C`Math::Complex::display_format\*(C'\fR and
supplying either \f(CW"polar"\fR or \f(CW"cartesian"\fR as an argument, you
override the default display style, which is \f(CW"cartesian"\fR. Not
supplying any argument returns the current settings.
.PP
This default can be overridden on a per-number basis by calling the
\&\f(CW\*(C`display_format\*(C'\fR method instead. As before, not supplying any argument
returns the current display style for this number. Otherwise whatever you
specify will be the new display style for \fIthis\fR particular number.
.PP
For instance:
.PP
.Vb 1
\&        use Math::Complex;
\&
\&        Math::Complex::display_format(\*(Aqpolar\*(Aq);
\&        $j = (root(1, 3))[1];
\&        print "j = $j\en";               # Prints "j = [1,2pi/3]"
\&        $j\->display_format(\*(Aqcartesian\*(Aq);
\&        print "j = $j\en";               # Prints "j = \-0.5+0.866025403784439i"
.Ve
.PP
The polar style attempts to emphasize arguments like \fIk*pi/n\fR
(where \fIn\fR is a positive integer and \fIk\fR an integer within [\-9, +9]),
this is called \fIpolar pretty-printing\fR.
.PP
For the reverse of stringifying, see the \f(CW\*(C`make\*(C'\fR and \f(CW\*(C`emake\*(C'\fR.
.Sh "\s-1CHANGED\s0 \s-1IN\s0 \s-1PERL\s0 5.6"
.IX Subsection "CHANGED IN PERL 5.6"
The \f(CW\*(C`display_format\*(C'\fR class method and the corresponding
\&\f(CW\*(C`display_format\*(C'\fR object method can now be called using
a parameter hash instead of just a one parameter.
.PP
The old display format style, which can have values \f(CW"cartesian"\fR or
\&\f(CW"polar"\fR, can be changed using the \f(CW"style"\fR parameter.
.PP
.Vb 1
\&        $j\->display_format(style => "polar");
.Ve
.PP
The one parameter calling convention also still works.
.PP
.Vb 1
\&        $j\->display_format("polar");
.Ve
.PP
There are two new display parameters.
.PP
The first one is \f(CW"format"\fR, which is a \fIsprintf()\fR\-style format string
to be used for both numeric parts of the complex number(s).  The is
somewhat system-dependent but most often it corresponds to \f(CW"%.15g"\fR.
You can revert to the default by setting the \f(CW\*(C`format\*(C'\fR to \f(CW\*(C`undef\*(C'\fR.
.PP
.Vb 1
\&        # the $j from the above example
\&
\&        $j\->display_format(\*(Aqformat\*(Aq => \*(Aq%.5f\*(Aq);
\&        print "j = $j\en";               # Prints "j = \-0.50000+0.86603i"
\&        $j\->display_format(\*(Aqformat\*(Aq => undef);
\&        print "j = $j\en";               # Prints "j = \-0.5+0.86603i"
.Ve
.PP
Notice that this affects also the return values of the
\&\f(CW\*(C`display_format\*(C'\fR methods: in list context the whole parameter hash
will be returned, as opposed to only the style parameter value.
This is a potential incompatibility with earlier versions if you
have been calling the \f(CW\*(C`display_format\*(C'\fR method in list context.
.PP
The second new display parameter is \f(CW"polar_pretty_print"\fR, which can
be set to true or false, the default being true.  See the previous
section for what this means.
.SH "USAGE"
.IX Header "USAGE"
Thanks to overloading, the handling of arithmetics with complex numbers
is simple and almost transparent.
.PP
Here are some examples:
.PP
.Vb 1
\&        use Math::Complex;
\&
\&        $j = cplxe(1, 2*pi/3);  # $j ** 3 == 1
\&        print "j = $j, j**3 = ", $j ** 3, "\en";
\&        print "1 + j + j**2 = ", 1 + $j + $j**2, "\en";
\&
\&        $z = \-16 + 0*i;                 # Force it to be a complex
\&        print "sqrt($z) = ", sqrt($z), "\en";
\&
\&        $k = exp(i * 2*pi/3);
\&        print "$j \- $k = ", $j \- $k, "\en";
\&
\&        $z\->Re(3);                      # Re, Im, arg, abs,
\&        $j\->arg(2);                     # (the last two aka rho, theta)
\&                                        # can be used also as mutators.
.Ve
.Sh "\s-1PI\s0"
.IX Subsection "PI"
The constant \f(CW\*(C`pi\*(C'\fR and some handy multiples of it (pi2, pi4,
and pip2 (pi/2) and pip4 (pi/4)) are also available if separately
exported:
.PP
.Vb 2
\&    use Math::Complex \*(Aq:pi\*(Aq; 
\&    $third_of_circle = pi2 / 3;
.Ve
.SH "ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO"
.IX Header "ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO"
The division (/) and the following functions
.PP
.Vb 5
\&        log     ln      log10   logn
\&        tan     sec     csc     cot
\&        atan    asec    acsc    acot
\&        tanh    sech    csch    coth
\&        atanh   asech   acsch   acoth
.Ve
.PP
cannot be computed for all arguments because that would mean dividing
by zero or taking logarithm of zero. These situations cause fatal
runtime errors looking like this
.PP
.Vb 3
\&        cot(0): Division by zero.
\&        (Because in the definition of cot(0), the divisor sin(0) is 0)
\&        Died at ...
.Ve
.PP
or
.PP
.Vb 2
\&        atanh(\-1): Logarithm of zero.
\&        Died at...
.Ve
.PP
For the \f(CW\*(C`csc\*(C'\fR, \f(CW\*(C`cot\*(C'\fR, \f(CW\*(C`asec\*(C'\fR, \f(CW\*(C`acsc\*(C'\fR, \f(CW\*(C`acot\*(C'\fR, \f(CW\*(C`csch\*(C'\fR, \f(CW\*(C`coth\*(C'\fR,
\&\f(CW\*(C`asech\*(C'\fR, \f(CW\*(C`acsch\*(C'\fR, the argument cannot be \f(CW0\fR (zero).  For the
logarithmic functions and the \f(CW\*(C`atanh\*(C'\fR, \f(CW\*(C`acoth\*(C'\fR, the argument cannot
be \f(CW1\fR (one).  For the \f(CW\*(C`atanh\*(C'\fR, \f(CW\*(C`acoth\*(C'\fR, the argument cannot be
\&\f(CW\*(C`\-1\*(C'\fR (minus one).  For the \f(CW\*(C`atan\*(C'\fR, \f(CW\*(C`acot\*(C'\fR, the argument cannot be
\&\f(CW\*(C`i\*(C'\fR (the imaginary unit).  For the \f(CW\*(C`atan\*(C'\fR, \f(CW\*(C`acoth\*(C'\fR, the argument
cannot be \f(CW\*(C`\-i\*(C'\fR (the negative imaginary unit).  For the \f(CW\*(C`tan\*(C'\fR,
\&\f(CW\*(C`sec\*(C'\fR, \f(CW\*(C`tanh\*(C'\fR, the argument cannot be \fIpi/2 + k * pi\fR, where \fIk\fR
is any integer.  atan2(0, 0) is undefined, and if the complex arguments
are used for \fIatan2()\fR, a division by zero will happen if z1**2+z2**2 == 0.
.PP
Note that because we are operating on approximations of real numbers,
these errors can happen when merely `too close' to the singularities
listed above.
.SH "ERRORS DUE TO INDIGESTIBLE ARGUMENTS"
.IX Header "ERRORS DUE TO INDIGESTIBLE ARGUMENTS"
The \f(CW\*(C`make\*(C'\fR and \f(CW\*(C`emake\*(C'\fR accept both real and complex arguments.
When they cannot recognize the arguments they will die with error
messages like the following
.PP
.Vb 4
\&    Math::Complex::make: Cannot take real part of ...
\&    Math::Complex::make: Cannot take real part of ...
\&    Math::Complex::emake: Cannot take rho of ...
\&    Math::Complex::emake: Cannot take theta of ...
.Ve
.SH "BUGS"
.IX Header "BUGS"
Saying \f(CW\*(C`use Math::Complex;\*(C'\fR exports many mathematical routines in the
caller environment and even overrides some (\f(CW\*(C`sqrt\*(C'\fR, \f(CW\*(C`log\*(C'\fR, \f(CW\*(C`atan2\*(C'\fR).
This is construed as a feature by the Authors, actually... ;\-)
.PP
All routines expect to be given real or complex numbers. Don't attempt to
use BigFloat, since Perl has currently no rule to disambiguate a '+'
operation (for instance) between two overloaded entities.
.PP
In Cray \s-1UNICOS\s0 there is some strange numerical instability that results
in \fIroot()\fR, \fIcos()\fR, \fIsin()\fR, \fIcosh()\fR, \fIsinh()\fR, losing accuracy fast.  Beware.
The bug may be in \s-1UNICOS\s0 math libs, in \s-1UNICOS\s0 C compiler, in Math::Complex.
Whatever it is, it does not manifest itself anywhere else where Perl runs.
.SH "AUTHORS"
.IX Header "AUTHORS"
Daniel S. Lewart <\fIlewart!at!uiuc.edu\fR>
Jarkko Hietaniemi <\fIjhi!at!iki.fi\fR>
Raphael Manfredi <\fIRaphael_Manfredi!at!pobox.com\fR>