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Math::Complex - complex numbers and associated mathematical functions
use Math::Complex;
$z = Math::Complex->make(5, 6);
$t = 4 - 3*i + $z;
$j = cplxe(1, 2*pi/3);
This package lets you create and manipulate complex numbers. By default,
Perl limits itself to real numbers, but an extra use statement brings
full complex support, along with a full set of mathematical functions
typically associated with and/or extended to complex numbers.
If you wonder what complex numbers are, they were invented to be able to solve the following equation:
x*x = -1
and by definition, the solution is noted i (engineers use j instead since i usually denotes an intensity, but the name does not matter). The number i is a pure imaginary number.
The arithmetics with pure imaginary numbers works just like you would expect it with real numbers... you just have to remember that
i*i = -1
so you have:
5i + 7i = i * (5 + 7) = 12i
4i - 3i = i * (4 - 3) = i
4i * 2i = -8
6i / 2i = 3
1 / i = -i
Complex numbers are numbers that have both a real part and an imaginary part, and are usually noted:
a + bi
where a is the real part and b is the imaginary part. The
arithmetic with complex numbers is straightforward. You have to
keep track of the real and the imaginary parts, but otherwise the
rules used for real numbers just apply:
(4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
(2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i
A graphical representation of complex numbers is possible in a plane (also called the complex plane, but it's really a 2D plane). The number
z = a + bi
is the point whose coordinates are (a, b). Actually, it would be the vector originating from (0, 0) to (a, b). It follows that the addition of two complex numbers is a vectorial addition.
Since there is a bijection between a point in the 2D plane and a complex number (i.e. the mapping is unique and reciprocal), a complex number can also be uniquely identified with polar coordinates:
[rho, theta]
where rho is the distance to the origin, and theta the angle between
the vector and the x axis. There is a notation for this using the
exponential form, which is:
rho * exp(i * theta)
where i is the famous imaginary number introduced above. Conversion
between this form and the cartesian form a + bi is immediate:
a = rho * cos(theta)
b = rho * sin(theta)
which is also expressed by this formula:
z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
In other words, it's the projection of the vector onto the x and y
axes. Mathematicians call rho the norm or modulus and theta
the argument of the complex number. The norm of z will be
noted abs(z).
The polar notation (also known as the trigonometric representation) is much more handy for performing multiplications and divisions of complex numbers, whilst the cartesian notation is better suited for additions and subtractions. Real numbers are on the x axis, and therefore theta is zero or pi.
All the common operations that can be performed on a real number have been defined to work on complex numbers as well, and are merely extensions of the operations defined on real numbers. This means they keep their natural meaning when there is no imaginary part, provided the number is within their definition set.
For instance, the sqrt routine which computes the square root of
its argument is only defined for non-negative real numbers and yields a
non-negative real number (it is an application from R+ to R+).
If we allow it to return a complex number, then it can be extended to
negative real numbers to become an application from R to C (the
set of complex numbers):
sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
It can also be extended to be an application from C to C, whilst its restriction to R behaves as defined above by using the following definition:
sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)
Indeed, a negative real number can be noted [x,pi] (the modulus
x is always non-negative, so [x,pi] is really -x, a negative
number) and the above definition states that
sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i
which is exactly what we had defined for negative real numbers above.
The sqrt returns only one of the solutions: if you want the both,
use the root function.
All the common mathematical functions defined on real numbers that are extended to complex numbers share that same property of working as usual when the imaginary part is zero (otherwise, it would not be called an extension, would it?).
A new operation possible on a complex number that is
the identity for real numbers is called the conjugate, and is noted
with a horizontal bar above the number, or ~z here.
z = a + bi
~z = a - bi
Simple... Now look:
z * ~z = (a + bi) * (a - bi) = a*a + b*b
We saw that the norm of z was noted abs(z) and was defined as the
distance to the origin, also known as:
rho = abs(z) = sqrt(a*a + b*b)
so
z * ~z = abs(z) ** 2
If z is a pure real number (i.e. b == 0), then the above yields:
a * a = abs(a) ** 2
which is true (abs has the regular meaning for real number, i.e. stands
for the absolute value). This example explains why the norm of z is
noted abs(z): it extends the abs function to complex numbers, yet
is the regular abs we know when the complex number actually has no
imaginary part... This justifies a posteriori our use of the abs
notation for the norm.
Given the following notations:
z1 = a + bi = r1 * exp(i * t1)
z2 = c + di = r2 * exp(i * t2)
z = <any complex or real number>
the following (overloaded) operations are supported on complex numbers:
z1 + z2 = (a + c) + i(b + d)
z1 - z2 = (a - c) + i(b - d)
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.
The definition used for complex arguments of atan2() is
-i log((x + iy)/sqrt(x*x+y*y))
The following extra operations are supported on both real and complex numbers:
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))
arg, abs, log, csc, cot, acsc, acot, csch,
coth, acosech, acotanh, have aliases rho, theta, ln,
cosec, cotan, acosec, acotan, cosech, cotanh,
acosech, acotanh, respectively. Re, Im, arg, abs,
rho, and theta can be used also as mutators. The cbrt
returns only one of the solutions: if you want all three, use the
root function.
The root function is available to compute all the n
roots of some complex, where n is a strictly positive integer.
There are exactly n such roots, returned as a list. Getting the
number mathematicians call j such that:
1 + j + j*j = 0;
is a simple matter of writing:
$j = ((root(1, 3))[1];
The kth root for z = [r,t] is given by:
(root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
You can return the kth root directly by root(z, n, k),
indexing starting from zero and ending at n - 1.
The spaceship 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.
To create a complex number, use either:
$z = Math::Complex->make(3, 4);
$z = cplx(3, 4);
if you know the cartesian form of the number, or
$z = 3 + 4*i;
if you like. To create a number using the polar form, use either:
$z = Math::Complex->emake(5, pi/3);
$x = cplxe(5, pi/3);
instead. The first argument is the modulus, the second is the angle
(in radians, the full circle is 2*pi). (Mnemonic: e is used as a
notation for complex numbers in the polar form).
It is possible to write:
$x = cplxe(-3, pi/4);
but that will be silently converted into [3,-3pi/4], since the
modulus must be non-negative (it represents the distance to the origin
in the complex plane).
It is also possible to have a complex number as either argument of the
make, emake, cplx, and cplxe: the appropriate component of
the argument will be used.
$z1 = cplx(-2, 1);
$z2 = cplx($z1, 4);
The new, make, emake, cplx, and cplxe will also
understand a single (string) argument of the forms
2-3i
-3i
[2,3]
[2,-3pi/4]
[2]
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.
The new, make, emake, cplx, and cplxe will also
understand the case of no arguments: this means plain zero or (0, 0).
When printed, a complex number is usually shown under its cartesian style a+bi, but there are legitimate cases where the polar style [r,t] is more appropriate. The process of converting the complex number into a string that can be displayed is known as stringification.
By calling the class method Math::Complex::display_format and
supplying either "polar" or "cartesian" as an argument, you
override the default display style, which is "cartesian". Not
supplying any argument returns the current settings.
This default can be overridden on a per-number basis by calling the
display_format 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 this particular number.
For instance:
use Math::Complex;
Math::Complex::display_format('polar');
$j = (root(1, 3))[1];
print "j = $j\n"; # Prints "j = [1,2pi/3]"
$j->display_format('cartesian');
print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i"
The polar style attempts to emphasize arguments like k*pi/n (where n is a positive integer and k an integer within [-9, +9]), this is called polar pretty-printing.
For the reverse of stringifying, see the make and emake.
The display_format class method and the corresponding
display_format object method can now be called using
a parameter hash instead of just a one parameter.
The old display format style, which can have values "cartesian" or
"polar", can be changed using the "style" parameter.
$j->display_format(style => "polar");
The one parameter calling convention also still works.
$j->display_format("polar");
There are two new display parameters.
The first one is "format", which is a sprintf()-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 "%.15g".
You can revert to the default by setting the format to undef.
# the $j from the above example
$j->display_format('format' => '%.5f');
print "j = $j\n"; # Prints "j = -0.50000+0.86603i"
$j->display_format('format' => undef);
print "j = $j\n"; # Prints "j = -0.5+0.86603i"
Notice that this affects also the return values of the
display_format 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 display_format method in list context.
The second new display parameter is "polar_pretty_print", which can
be set to true or false, the default being true. See the previous
section for what this means.
Thanks to overloading, the handling of arithmetics with complex numbers is simple and almost transparent.
Here are some examples:
use Math::Complex;
$j = cplxe(1, 2*pi/3); # $j ** 3 == 1
print "j = $j, j**3 = ", $j ** 3, "\n";
print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";
$z = -16 + 0*i; # Force it to be a complex
print "sqrt($z) = ", sqrt($z), "\n";
$k = exp(i * 2*pi/3);
print "$j - $k = ", $j - $k, "\n";
$z->Re(3); # Re, Im, arg, abs,
$j->arg(2); # (the last two aka rho, theta)
# can be used also as mutators.
The division (/) and the following functions
log ln log10 logn
tan sec csc cot
atan asec acsc acot
tanh sech csch coth
atanh asech acsch acoth
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
cot(0): Division by zero.
(Because in the definition of cot(0), the divisor sin(0) is 0)
Died at ...
or
atanh(-1): Logarithm of zero.
Died at...
For the csc, cot, asec, acsc, acot, csch, coth,
asech, acsch, the argument cannot be 0 (zero). For the
logarithmic functions and the atanh, acoth, the argument cannot
be 1 (one). For the atanh, acoth, the argument cannot be
-1 (minus one). For the atan, acot, the argument cannot be
i (the imaginary unit). For the atan, acoth, the argument
cannot be -i (the negative imaginary unit). For the tan,
sec, tanh, the argument cannot be pi/2 + k * pi, where k
is any integer. atan2(0, 0) is undefined, and if the complex arguments
are used for atan2(), a division by zero will happen if z1**2+z2**2 == 0.
Note that because we are operating on approximations of real numbers, these errors can happen when merely `too close' to the singularities listed above.
The make and emake accept both real and complex arguments.
When they cannot recognize the arguments they will die with error
messages like the following
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 ...
Saying use Math::Complex; exports many mathematical routines in the
caller environment and even overrides some (sqrt, log, atan2).
This is construed as a feature by the Authors, actually... ;-)
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.
In Cray UNICOS there is some strange numerical instability that results in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware. The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex. Whatever it is, it does not manifest itself anywhere else where Perl runs.
Daniel S. Lewart <d-lewart@uiuc.edu>
Original authors Raphael Manfredi <Raphael_Manfredi@pobox.com> and Jarkko Hietaniemi <jhi@iki.fi>