|Keep It Simple, Stupid|
perlman:Math::Complexby root (Scribe)
|on Dec 23, 1999 at 01:30 UTC||Need Help??|
See the current Perl documentation for Math::Complex.
Here is our local, out-dated (pre-5.6) version:
Math::Complex - complex numbers and associated mathematical functions
$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
(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 * exp(i * theta)
where i is the famous imaginary number introduced above. Conversion between this
form and the cartesian form
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
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
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
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 an horizontal bar above the number, or
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
rho = abs(z) = sqrt(a*a + b*b)
z * ~z = abs(z) ** 2
If z is a pure real number (i.e.
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
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(z1, z2) = atan(z1/z2)
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.
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
1 + j + j*j = 0;
is a simple matter of writing:
$j = ((root(1, 3));
The kth root for
(root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
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:
It is possible to write:
$x = cplxe(-3, pi/4);
but that will be silently converted into
It is also possible to have a complex number as either argument of either
$z1 = cplx(-2, 1); $z2 = cplx($z1, 4);
When printed, a complex number is usually shown under its cartesian form a+bi, but there are legitimate cases where the polar format [r,t] is more appropriate.
By calling the routine
This default can be overridden on a per-number basis by calling the
Math::Complex::display_format('polar'); $j = ((root(1, 3)); 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 format attempts to emphasize arguments like k*pi/n (where n is a positive integer and k an integer within [-9,+9]).
Thanks to overloading, the handling of arithmetics with complex numbers is simple and almost transparent.
Here are some examples:
$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 ...
atanh(-1): Logarithm of zero. Died at...
Note that because we are operating on approximations of real numbers, these
errors can happen when merely `too close' to the singularities listed
above. For example
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 ...
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.
UNICOS there is some strange numerical instability that results in
Raphael Manfredi <Raphael_Manfredi@grenoble.hp.com> and Jarkko Hietaniemi <firstname.lastname@example.org>.
Extensive patches by Daniel S. Lewart <email@example.com>.