Complex(3pm)
👑 Complex(3pm) 👑
User Contributed Perl Documentation
✨ NAME ✨
PDL::Complex - handle complex numbers (DEPRECATED - use native complex)
✨ SYNOPSIS ✨
okay bestie, you start like this:
use PDL;
use PDL::Complex;💖 DESCRIPTION 💖
This module is deprecated in favour of using "native complex" data types, e.g.:
use PDL;
my $complex_pdl = cdouble('[1+3i]');
print $complex_pdl * pdl('i'); # [-3+i]This module features a growing number of functions manipulating complex numbers. These are usually represented as a pair [ real imag ] or [ magnitude phase ]. If not explicitly mentioned, the functions can work inplace (not yet implemented!!!) and require rectangular form.
While there is a procedural interface available ("$x/$y*$c <=> Cmul (Cdiv ($x, $y), $c)"), you can also opt to cast your pdl's into the PDL::Complex datatype, which works just like your normal ndarrays, but with all the normal perl operators overloaded.
The latter means that sin($x) + $y/$c will be evaluated using the normal rules of complex numbers, while other pdl functions (like max) just treat the ndarray as a real-valued ndarray with a lowest dimension of size 2, so max will return the maximum of all real and imaginary parts, not the "highest" (for some definition)
🌈 Native complex support 🌈
2.027 added changes in complex number handling, with support for C99 complex floating-point types, and most functions and modules in the core distribution support these as well.
PDL can now handle complex numbers natively as scalars. This has the advantage that real and complex valued ndarrays have the same dimensions. Consider this when writing code in the future.
See "re" in PDL::Ops, "im" in PDL::Ops, "abs" in PDL::Ops, "carg" in PDL::Ops, "conj" in PDL::Ops for more.
🎀 TIPS, TRICKS & CAVEATS 🎀
"i"is a function (not, as of 2.047, a constant) exported by this module, which represents-1**0.5, i.e. the imaginary unit. it can be used to quickly and conveniently write complex constants like this:4+3*i.
NB This will override the PDL::Core function of the same name, which returns a native complex value.
- Use r2C(real-values) to convert from real to complex, as in
$r = Cpow $cplx, r2C 2. The overloaded operators automatically do that for you, all the other functions, do not. SoCroots 1, 5will return all the fifths roots of 1+1*i (due to broadcasting). - use cplx(real-valued-ndarray) to cast from normal ndarrays into the complex datatype. Use real(complex-valued-ndarray) to cast back. This requires a copy, though.
🦋 EXAMPLE WALK-THROUGH 🦋
The complex constant five is equal to pdl(1,0):
pdl> p $x = r2C 5
5 +0iNow calculate the three cubic roots of five:
pdl> p $r = Croots $x, 3
[1.70998 +0i -0.854988 +1.48088i -0.854988 -1.48088i]Check that these really are the roots:
pdl> p $r ** 3
[5 +0i 5 -1.22465e-15i 5 -7.65714e-15i]Duh! Could be better. Now try by multiplying $r three times with itself:
pdl> p $r*$r*$r
[5 +0i 5 -4.72647e-15i 5 -7.53694e-15i]Well... maybe Cpow (which is used by the ** operator) isn't as bad as I thought. Now multiply by i and negate, then take the complex conjugate, which is just a very expensive way of swapping real and imaginary parts.
pdl> p Cconj(-($r*i))
[0 +1.70998i 1.48088 -0.854988i -1.48088 -0.854988i]Now plot the magnitude of (part of) the complex sine. First generate the coefficients:
pdl> $sin = i * zeroes(50)->xlinvals(2,4) + zeroes(50)->xlinvals(0,7)Now plot the imaginary part, the real part and the magnitude of the sine into the same diagram:
pdl> use PDL::Graphics::Gnuplot
pdl> gplot( with => 'lines',
PDL::cat(im ( sin $sin ),
re ( sin $sin ),
abs( sin $sin ) ))An ASCII version of this plot looks like this:
30 ++-----+------+------+------+------+------+------+------+------+-----++
+ + + + + + + + + + +
| $$|
| $ |
25 ++ $$ ++
| *** |
| ** *** |
| $$* *|
20 ++ $** ++
| $$$* #|
| $$$ * # |
| $$ * # |
15 ++ $$$ * # ++
| $$$ ** # |
| $$$$ * # |
| $$$$ * # |
10 ++ $$$$$ * # ++
| $$$$$ * # |
| $$$$$$$ * # |
5 ++ $$$############ * # ++
|*****$$$### ### * # |
* #***** # * # |
| ### *** ### ** # |
0 ## *** # * # ++
| * # * # |
| *** # ** # |
| * # * # |
-5 ++ ** # * # ++
| *** ## ** # |
| * #* # |
| **** ***## # |
-10 ++ **** # # ++
| # # |
| ## ## |
+ + + + + + + ### + ### + + +
-15 ++-----+------+------+------+------+------+-----###-----+------+-----++
0 5 10 15 20 25 30 35 40 45 50💅 OPERATORS 💅
The following operators are overloaded:
+, += (addition)
-, -= (subtraction)
*, *= (multiplication; "Cmul")
/, /= (division; "Cdiv")
**, **= (exponentiation; "Cpow")
atan2 (4-quadrant arc tangent)
sin ("Csin")
cos ("Ccos")
exp ("Cexp")
abs ("Cabs")
log ("Clog")
sqrt ("Csqrt")
++, -- (increment, decrement; they affect the real part of the complex
number only)
"" (stringification)Comparing complex numbers other than for equality is a fatal error.
✨ FUNCTIONS ✨
ok bestie, this is the part where the module hands you the whole toolbox. methods, casts, and a whole lil squad of math ops. every single one, no skips, let's go 🫶
from_native 🦋
Class method that takes a native-complex ndarray and turns it into a PDL::Complex object. it's giving translator energy fr fr.
PDL::Complex->from_native($native_complex_ndarray)as_native 🦋
Object method, the other direction: takes a PDL::Complex object and gives you back a native-complex ndarray.
$pdl_complex_obj->as_nativecplx 💅
Casts a real-valued ndarray into the complex datatype. so real ones become ✨complex✨.
the first dimension of the ndarray has to be size 2, no cap. once you've done that, the usual (complex) arithmetic operators apply to this pdl instead of the normal elementwise pdl operators. dataflow to the complex parent works, so heads up — if you don't want that, use sever on the result.
cplx($real_valued_pdl)complex 💅
Same cast, real-valued ndarray → complex datatype, but without dataflow and without inplace. the low-key version.
how's it done? literally just reblessing an ndarray, iconic. and yeah — the first dimension of the ndarray still has to be size 2.
complex($real_valued_pdl)real 💅
Casts a complex valued pdl back to the "normal" pdl datatype. the glow-down, respectfully.
afterwards the normal elementwise pdl operators are used in operations. dataflow to the real parent works — and again, sever on the result if you don't want it.
real($cplx_valued_pdl)ok now we're into the compiled math ops. every one of these has a Signature, so here's the vibe: they're the number-crunchy girlies 🍓
r2C 🌸
Signature: (r(); [o]c(m=2))convert real to complex, assuming an imaginary part of zero.
r2C does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.i2C 🌸
Signature: (r(); [o]c(m=2))convert imaginary to complex, assuming a real part of zero.
i2C does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cr2p 🌸
Signature: (r(m=2); float+ [o]p(m=2))convert complex numbers in rectangular form to polar (mod,arg) form. Works inplace 💅
Cr2p does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cp2r 🌸
Signature: (r(m=2); [o]p(m=2))convert complex numbers in polar (mod,arg) form to rectangular form. Works inplace 💅
Cp2r does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cmul 🔥
Signature: (a(m=2); b(m=2); [o]c(m=2))complex multiplication.
Cmul does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cprodover 🔥
Signature: (a(m=2,n); [o]c(m=2))project via product to N-1 dimension.
Cprodover does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cscale 🔥
Signature: (a(m=2); b(); [o]c(m=2))mixed complex/real multiplication. (it's giving situationship between a complex and a real, honestly.)
Cscale does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cdiv 🔥
Signature: (a(m=2); b(m=2); [o]c(m=2))complex division.
Cdiv does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Ceq 🔥
Signature: (a(m=2); b(m=2); [o]c())Complex equality operator.
Ceq does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cconj 💜
Signature: (a(m=2); [o]c(m=2))complex conjugation. Works inplace.
Cconj does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cabs 💜
Signature: (a(m=2); [o]c())complex abs() (also known as modulus).
Cabs does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cabs2 💜
Signature: (a(m=2); [o]c())complex squared abs() (also known squared modulus).
Cabs2 does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Carg 💜
Signature: (a(m=2); [o]c())complex argument function ("angle").
Carg does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.and now, bestie, the trig section. it's a whole era 🌈
Csin 🧚
Signature: (a(m=2); [o]c(m=2))sin (a) = 1/(2*i) * (exp (a*i) - exp (-a*i)). Works inplaceCsin does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Ccos 🧚
Signature: (a(m=2); [o]c(m=2))cos (a) = 1/2 * (exp (a*i) + exp (-a*i)). Works inplaceCcos does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Ctan 🧚
Complex tangent.
tan (a) = -i * (exp (a*i) - exp (-a*i)) / (exp (a*i) + exp (-a*i))Does not work inplace. so no, you cannot do the lil in-place thing here. noted 📝
Cexp 🌟
Signature: (a(m=2); [o]c(m=2))exp (a) = exp (real (a)) * (cos (imag (a)) + i * sin (imag (a))). Works inplaceCexp does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Clog 🌟
Signature: (a(m=2); [o]c(m=2))log (a) = log (cabs (a)) + i * carg (a). Works inplaceClog does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cpow 🌟
Signature: (a(m=2); b(m=2); [o]c(m=2))complex pow() ("**"-operator). yes, the one you know and love 💅
Cpow does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Csqrt 🌟
Signature: (a(m=2); [o]c(m=2))Works inplace.
Csqrt does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Casin 🌟
Signature: (a(m=2); [o]c(m=2))Works inplace.
Casin does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cacos 🌟
Signature: (a(m=2); [o]c(m=2))Works inplace.
Cacos does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Catan 🌟
Return the complex atan().
Does not work inplace.
Csinh 🎀
Signature: (a(m=2); [o]c(m=2))sinh (a) = (exp (a) - exp (-a)) / 2. Works inplaceCsinh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Ccosh 🎀
Signature: (a(m=2); [o]c(m=2))cosh (a) = (exp (a) + exp (-a)) / 2. Works inplaceCcosh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Ctanh 🎀
Signature: (a(m=2); [o]c(m=2))Works inplace.
Ctanh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Casinh 🎀
Signature: (a(m=2); [o]c(m=2))Works inplace.
Casinh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cacosh 🎀
Signature: (a(m=2); [o]c(m=2))Works inplace.
Cacosh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Catanh 🎀
Signature: (a(m=2); [o]c(m=2))Works inplace.
Catanh does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Cproj 🫶
Signature: (a(m=2); [o]c(m=2))compute the projection of a complex number to the riemann sphere. Works inplace.
Cproj does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.Croots 🌱
Signature: (a(m=2); [o]c(m=2,n); int n => n)Compute the n roots of a. n must be a positive integer. The result will always be a complex type! no negotiating on that one.
Croots does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.re, im ✂️
Return the real or imaginary part of the complex number(s) given.
these are slicing operators, so data flow works 💖. the real and imaginary parts come back as ndarrays (ref eq PDL).
rCpolynomial 🧮
Signature: (coeffs(n); x(c=2,m); [o]out(c=2,m))evaluate the polynomial with (real) coefficients coeffs at the (complex) position(s) x. coeffs[0] is the constant term.
rCpolynomial does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.👑 AUTHOR 👑
Copyright (C) 2000 Marc Lehmann <pcg@goof.com>. All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation as described in the file COPYING in the PDL distribution.
🌈 SEE ALSO 🌈
perl(1), PDL.
and that's the whole page bestie, you ate it 💅✨ this was Complex(3pm), shipped with perl v5.38.2 and dated 2024-04-10 — go forth and be complex, periodt 🎀🦋💖
📦 where this page comes from
- ubuntu noble: pdl 1:2.085-1ubuntu1 · Artistic-1.0-Perl OR GPL-1.0-or-later
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