1Complex(3) User Contributed Perl Documentation Complex(3)
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6 PDL::Complex - handle complex numbers
7
9 use PDL;
10 use PDL::Complex;
11
13 This module features a growing number of functions manipulating complex
14 numbers. These are usually represented as a pair "[ real imag ]" or "[
15 angle phase ]". If not explicitly mentioned, the functions can work
16 inplace (not yet implemented!!!) and require rectangular form.
17 While there is a procedural interface available ("$a/$b*$c <=" Cmul
19 (Cdiv $a, $b), $c)>), you can also opt to cast your pdl's into the
20 "PDL::Complex" datatype, which works just like your normal piddles, but
21 with all the normal perl operators overloaded.
22 The latter means that "sin($a) + $b/$c" will be evaluated using the
24 normal rules of complex numbers, while other pdl functions (like "max")
25 just treat the piddle as a real-valued piddle with a lowest dimension
26 of size 2, so "max" will return the maximum of all real and imaginary
27 parts, not the "highest" (for some definition)
28
30 · "i" is a constant exported by this module, which represents
31 "-1**0.5", i.e. the imaginary unit. it can be used to quickly and
32 conviniently write complex constants like this: "4+3*i".
33 · Use "r2C(real-values)" to convert from real to complex, as in "$r =
35 Cpow $cplx, r2C 2". The overloaded operators automatically do that
36 for you, all the other functions, do not. So "Croots 1, 5" will
37 return all the fifths roots of 1+1*i (due to threading).
38 · use "cplx(real-valued-piddle)" to cast from normal piddles intot he
40 complex datatype. Use "real(complex-valued-piddle)" to cast back.
41 This requires a copy, though.
42 · This module has received some testing by Vanuxem Grégory (g.vanuxem
44 at wanadoo dot fr). Please report any other errors you come across!
45
47 The complex constant five is equal to "pdl(1,0)":
48 perldl> p $x = r2C 5
50 [5 0]
51 Now calculate the three roots of of five:
53 perldl> p $r = Croots $x, 3
55 [
57 [ 1.7099759 0]
58 [-0.85498797 1.4808826]
59 [-0.85498797 -1.4808826]
60 ]
61 Check that these really are the roots of unity:
63 perldl> p $r ** 3
65 [
67 [ 5 0]
68 [ 5 -3.4450524e-15]
69 [ 5 -9.8776239e-15]
70 ]
71 Duh! Could be better. Now try by multiplying $r three times with
73 itself:
74 perldl> p $r*$r*$r
76 [
78 [ 5 0]
79 [ 5 -2.8052647e-15]
80 [ 5 -7.5369398e-15]
81 ]
82 Well... maybe "Cpow" (which is used by the "**" operator) isn't as bad
84 as I thought. Now multiply by "i" and negate, which is just a very
85 expensive way of swapping real and imaginary parts.
86 perldl> p -($r*i)
88 [
90 [ -0 1.7099759]
91 [ 1.4808826 -0.85498797]
92 [ -1.4808826 -0.85498797]
93 ]
94 Now plot the magnitude of (part of) the complex sine. First generate
96 the coefficients:
97 perldl> $sin = i * zeroes(50)->xlinvals(2,4)
99 + zeroes(50)->xlinvals(0,7)
100 Now plot the imaginary part, the real part and the magnitude of the
102 sine into the same diagram:
103 perldl> line im sin $sin; hold
105 perldl> line re sin $sin
106 perldl> line abs sin $sin
107 Sorry, but I didn't yet try to reproduce the diagram in this text. Just
109 run the commands yourself, making sure that you have loaded "PDL::Com‐
110 plex" (and "PDL::Graphics::PGPLOT").
111
113 cplx real-valued-pdl
114 Cast a real-valued piddle to the complex datatype. The first dimension
116 of the piddle must be of size 2. After this the usual (complex) arith‐
117 metic operators are applied to this pdl, rather than the normal elemen‐
118 twise pdl operators. Dataflow to the complex parent works. Use "sever"
119 on the result if you don't want this.
120 complex real-valued-pdl
122 Cast a real-valued piddle to the complex datatype without dataflow and
124 inplace. Achieved by merely reblessing a piddle. The first dimension of
125 the piddle must be of size 2.
126 real cplx-valued-pdl
128 Cast a complex valued pdl back to the "normal" pdl datatype. Afterwards
130 the normal elementwise pdl operators are used in operations. Dataflow
131 to the real parent works. Use "sever" on the result if you don't want
132 this.
133 r2C
135 Signature: (r(); [o]c(m=2))
137 convert real to complex, assuming an imaginary part of zero
139 i2C
141 Signature: (r(); [o]c(m=2))
143 convert imaginary to complex, assuming a real part of zero
145 Cr2p
147 Signature: (r(m=2); float+ [o]p(m=2))
149 convert complex numbers in rectangular form to polar (mod,arg) form
151 Cp2r
153 Signature: (r(m=2); [o]p(m=2))
155 convert complex numbers in polar (mod,arg) form to rectangular form
157 Cmul
159 Signature: (a(m=2); b(m=2); [o]c(m=2))
161 complex multiplication
163 Cprodover
165 Signature: (a(m=2,n); [o]c(m=2))
167 Project via product to N-1 dimension
169 Cscale
171 Signature: (a(m=2); b(); [o]c(m=2))
173 mixed complex/real multiplication
175 Cdiv
177 Signature: (a(m=2); b(m=2); [o]c(m=2))
179 complex division
181 Ccmp
183 Signature: (a(m=2); b(m=2); [o]c())
185 Complex comparison oeprator (spaceship). It orders by real first, then
187 by imaginary. Hm, but it is mathematical nonsense! Complex numbers can‐
188 not be ordered.
189 Cconj
191 Signature: (a(m=2); [o]c(m=2))
193 complex conjugation
195 Cabs
197 Signature: (a(m=2); [o]c())
199 complex "abs()" (also known as modulus)
201 Cabs2
203 Signature: (a(m=2); [o]c())
205 complex squared "abs()" (also known squared modulus)
207 Carg
209 Signature: (a(m=2); [o]c())
211 complex argument function ("angle")
213 Csin
215 Signature: (a(m=2); [o]c(m=2))
217 sin (a) = 1/(2*i) * (exp (a*i) - exp (-a*i))
219 Ccos
221 Signature: (a(m=2); [o]c(m=2))
223 cos (a) = 1/2 * (exp (a*i) + exp (-a*i))
225 Ctan a [not inplace]
227 tan (a) = -i * (exp (a*i) - exp (-a*i)) / (exp (a*i) + exp (-a*i))
229 Cexp
231 Signature: (a(m=2); [o]c(m=2))
233 exp (a) = exp (real (a)) * (cos (imag (a)) + i * sin (imag (a)))
235 Clog
237 Signature: (a(m=2); [o]c(m=2))
239 log (a) = log (cabs (a)) + i * carg (a)
241 Cpow
243 Signature: (a(m=2); b(m=2); [o]c(m=2))
245 complex "pow()" ("**"-operator)
247 Csqrt
249 Signature: (a(m=2); [o]c(m=2))
251 Casin
253 Signature: (a(m=2); [o]c(m=2))
255 Cacos
257 Signature: (a(m=2); [o]c(m=2))
259 Catan cplx [not inplace]
261 Return the complex "atan()".
263 Csinh
265 Signature: (a(m=2); [o]c(m=2))
267 sinh (a) = (exp (a) - exp (-a)) / 2
269 Ccosh
271 Signature: (a(m=2); [o]c(m=2))
273 cosh (a) = (exp (a) + exp (-a)) / 2
275 Ctanh
277 Signature: (a(m=2); [o]c(m=2))
279 Casinh
281 Signature: (a(m=2); [o]c(m=2))
283 Cacosh
285 Signature: (a(m=2); [o]c(m=2))
287 Catanh
289 Signature: (a(m=2); [o]c(m=2))
291 Cproj
293 Signature: (a(m=2); [o]c(m=2))
295 compute the projection of a complex number to the riemann sphere
297 Croots
299 Signature: (a(m=2); [o]c(m=2,n); int n => n)
301 Compute the "n" roots of "a". "n" must be a positive integer. The
303 result will always be a complex type!
304 re cplx, im cplx
306 Return the real or imaginary part of the complex number(s) given. These
308 are slicing operators, so data flow works. The real and imaginary parts
309 are returned as piddles (ref eq PDL).
310 rCpolynomial
312 Signature: (coeffs(n); x(c=2,m); [o]out(c=2,m))
314 evaluate the polynomial with (real) coefficients "coeffs" at the (com‐
316 plex) position(s) "x". "coeffs[0]" is the constant term.
317
319 Copyright (C) 2000 Marc Lehmann <pcg@goof.com>. All rights reserved.
320 There is no warranty. You are allowed to redistribute this software /
321 documentation as described in the file COPYING in the PDL distribution.
322
324 perl(1), PDL.
325perl v5.8.8 2006-12-02 Complex(3)