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 extrause 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 = -1and by definition, the solution is notedi (engineers usej instead sincei usually denotes an intensity, but the name does not matter). The numberi is a pureimaginary 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 = -1so you have:
5i + 7i = i * (5 + 7) = 12i4i - 3i = i * (4 - 3) = i4i * 2i = -86i / 2i = 31 / i = -iComplex numbers are numbers that have both a real part and an imaginary part, and are usually noted:
a + biwherea is thereal part andb is theimaginary 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 + 2iA graphical representation of complex numbers is possible in a plane (also called thecomplex plane, but it's really a 2D plane). The number
z = a + biis 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]whererho is the distance to the origin, andtheta the angle between the vector and thex axis. There is a notation for this using the exponential form, which is:
rho * exp(i * theta)wherei is the famous imaginary number introduced above. Conversion between this form and the cartesian forma + 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 thex andy axes. Mathematicians callrho thenorm ormodulus andtheta theargument of the complex number. Thenorm ofz will be notedabs(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 thex axis, and thereforetheta is zero orpi.
All the common operations that can be performed on a real number have been defined to work on complex numbers as well, and are merelyextensions 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, thesqrt 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 fromR+ toR+). If we allow it to return a complex number, then it can be extended to negative real numbers to become an application fromR toC (the set of complex numbers):
sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*iIt can also be extended to be an application fromC toC, whilst its restriction toR 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 modulusx 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)*iwhich is exactly what we had defined for negative real numbers above. Thesqrt returns only one of the solutions: if you want the both, use theroot function.
All the common mathematical functions defined on real numbers that are extended to complex numbers share that same property of workingas usual when the imaginary part is zero (otherwise, it would not be called an extension, would it?).
Anew operation possible on a complex number that is the identity for real numbers is called theconjugate, and is noted with an horizontal bar above the number, or~z here.
z = a + bi~z = a - biSimple... Now look:
z * ~z = (a + bi) * (a - bi) = a*a + b*bWe saw that the norm ofz was notedabs(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) ** 2If z is a pure real number (i.e.b == 0), then the above yields:
a * a = abs(a) ** 2which is true (abs has the regular meaning for real number, i.e. stands for the absolute value). This example explains why the norm ofz is notedabs(z): it extends theabs function to complex numbers, yet is the regularabs we know when the complex number actually has no imaginary part... This justifiesa posteriori our use of theabs 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 - biabs(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*tsin(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) = aIm(z) = barg(z) = tabs(z) = rcbrt(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 aliasesrho,theta,ln,cosec,cotan,acosec,acotan,cosech,cotanh,acosech,acotanh, respectively.Re,Im,arg,abs,rho, andtheta can be used also also mutators. Thecbrt returns only one of the solutions: if you want all three, use theroot function.
Theroot function is available to compute all then roots of some complex, wheren is a strictly positive integer. There are exactlyn such roots, returned as a list. Getting the number mathematicians callj such that:
1 + j + j*j = 0;is a simple matter of writing:
$j = ((root(1, 3))[1];Thekth root forz = [r,t] is given by:
(root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)Thespaceship 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 either themake oremake: the appropriate component of the argument will be used.
$z1 = cplx(-2, 1);$z2 = cplx($z1, 4);When printed, a complex number is usually shown under its cartesian forma+bi, but there are legitimate cases where the polar format[r,t] is more appropriate.
By calling the routineMath::Complex::display_format and supplying either"polar" or"cartesian", you override the default display format, which is"cartesian". Not supplying any argument returns the current setting.
This default can be overridden on a per-number basis by calling thedisplay_format method instead. As before, not supplying any argument returns the current display format for this number. Otherwise whatever you specify will be the new display format forthis particular number.
For instance:
use Math::Complex;Math::Complex::display_format('polar');$j = ((root(1, 3))[1];print "j = $j\n";# Prints "j = [1,2pi/3]$j->display_format('cartesian');print "j = $j\n";# Prints "j = -0.5+0.866025403784439i"The polar format attempts to emphasize arguments likek*pi/n (wheren is a positive integer andk an integer within [-9,+9]).
Thanks to overloading, the handling of arithmetics with complex numbers is simple and almost transparent.
Here are some examples:
use Math::Complex;$j = cplxe(1, 2*pi/3);# $j ** 3 == 1print "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 complexprint "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
loglnlog10logntansec csccotatanasecacscacottanhsechcschcothatanhasechacschacothcannot 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 thecsc,cot,asec,acsc,acot,csch,coth,asech,acsch, the argument cannot be0 (zero). For the the logarithmic functions and theatanh,acoth, the argument cannot be1 (one). For theatanh,acoth, the argument cannot be-1 (minus one). For theatan,acot, the argument cannot bei (the imaginary unit). For theatan,acoth, the argument cannot be-i (the negative imaginary unit). For thetan,sec,tanh, the argument cannot bepi/2 + k * pi, wherek is any integer.
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 exampletan(2*atan2(1,1)+1e-15) will die of division by zero.
Themake andemake 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 ...Sayinguse Math::Complex; exports many mathematical routines in the caller environment and even overrides some (sqrt,log). 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.
Raphael Manfredi <Raphael_Manfredi@grenoble.hp.com> and Jarkko Hietaniemi <jhi@iki.fi>.
Extensive patches by Daniel S. Lewart <d-lewart@uiuc.edu>.
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