/
complex.Rout.save
221 lines (186 loc) · 11.2 KB
/
complex.Rout.save
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
R version 2.10.0 Under development (unstable) (2009-08-04 r49064)
Copyright (C) 2009 The R Foundation for Statistical Computing
ISBN 3-900051-07-0
R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.
R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.
Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.
> ### Tests of complex arithemetic.
>
> Meps <- .Machine$double.eps
> ## complex
> z <- 0i ^ (-3:3)
> stopifnot(Re(z) == 0 ^ (-3:3))
> set.seed(123)
> z <- complex(real = rnorm(100), imag = rnorm(100))
> stopifnot(Mod ( 1 - sin(z) / ( (exp(1i*z)-exp(-1i*z))/(2*1i) )) < 20 * Meps)
> ## end of moved from complex.Rd
>
>
> ## powers, including complex ones
> a <- -4:12
> m <- outer(a +0i, b <- seq(-.5,2, by=.5), "^")
> dimnames(m) <- list(paste(a), "^" = sapply(b,format))
> round(m,3)
^
-0.5 0 0.5 1 1.5 2
-4 0.000-0.500i 1+0i 0.000+2.000i -4+0i 0.000-8.000i 16+0i
-3 0.000-0.577i 1+0i 0.000+1.732i -3+0i 0.000-5.196i 9+0i
-2 0.000-0.707i 1+0i 0.000+1.414i -2+0i 0.000-2.828i 4+0i
-1 0.000-1.000i 1+0i 0.000+1.000i -1+0i 0.000-1.000i 1+0i
0 Inf+0.000i 1+0i 0.000+0.000i 0+0i 0.000+0.000i 0+0i
1 1.000+0.000i 1+0i 1.000+0.000i 1+0i 1.000+0.000i 1+0i
2 0.707+0.000i 1+0i 1.414+0.000i 2+0i 2.828+0.000i 4+0i
3 0.577+0.000i 1+0i 1.732+0.000i 3+0i 5.196+0.000i 9+0i
4 0.500+0.000i 1+0i 2.000+0.000i 4+0i 8.000+0.000i 16+0i
5 0.447+0.000i 1+0i 2.236+0.000i 5+0i 11.180+0.000i 25+0i
6 0.408+0.000i 1+0i 2.449+0.000i 6+0i 14.697+0.000i 36+0i
7 0.378+0.000i 1+0i 2.646+0.000i 7+0i 18.520+0.000i 49+0i
8 0.354+0.000i 1+0i 2.828+0.000i 8+0i 22.627+0.000i 64+0i
9 0.333+0.000i 1+0i 3.000+0.000i 9+0i 27.000+0.000i 81+0i
10 0.316+0.000i 1+0i 3.162+0.000i 10+0i 31.623+0.000i 100+0i
11 0.302+0.000i 1+0i 3.317+0.000i 11+0i 36.483+0.000i 121+0i
12 0.289+0.000i 1+0i 3.464+0.000i 12+0i 41.569+0.000i 144+0i
> stopifnot(m[,as.character(0:2)] == cbind(1,a,a*a),
+ # latter were only approximate
+ all.equal(unname(m[,"0.5"]),
+ sqrt(abs(a))*ifelse(a < 0, 1i, 1),
+ tol= 20*Meps))
> ## fft():
> for(n in 1:30) cat("\nn=",n,":", round(fft(1:n), 8),"\n")
n= 1 : 1+0i
n= 2 : 3+0i -1+0i
n= 3 : 6+0i -1.5+0.866025i -1.5-0.866025i
n= 4 : 10+0i -2+2i -2+0i -2-2i
n= 5 : 15+0i -2.5+3.440955i -2.5+0.812299i -2.5-0.812299i -2.5-3.440955i
n= 6 : 21+0i -3+5.196152i -3+1.732051i -3+0i -3-1.732051i -3-5.196152i
n= 7 : 28+0i -3.5+7.267825i -3.5+2.791157i -3.5+0.798852i -3.5-0.798852i -3.5-2.791157i -3.5-7.267825i
n= 8 : 36+0i -4+9.656854i -4+4i -4+1.656854i -4+0i -4-1.656854i -4-4i -4-9.656854i
n= 9 : 45+0i -4.5+12.36365i -4.5+5.362891i -4.5+2.598076i -4.5+0.793471i -4.5-0.793471i -4.5-2.598076i -4.5-5.362891i -4.5-12.36365i
n= 10 : 55+0i -5+15.38842i -5+6.88191i -5+3.632713i -5+1.624598i -5+0i -5-1.624598i -5-3.632713i -5-6.88191i -5-15.38842i
n= 11 : 66+0i -5.5+18.73128i -5.5+8.558167i -5.5+4.765777i -5.5+2.511766i -5.5+0.790781i -5.5-0.790781i -5.5-2.511766i -5.5-4.765777i -5.5-8.558167i -5.5-18.73128i
n= 12 : 78+0i -6+22.3923i -6+10.3923i -6+6i -6+3.464102i -6+1.607695i -6+0i -6-1.607695i -6-3.464102i -6-6i -6-10.3923i -6-22.3923i
n= 13 : 91+0i -6.5+26.37154i -6.5+12.38472i -6.5+7.336983i -6.5+4.486626i -6.5+2.465125i -6.5+0.789243i -6.5-0.789243i -6.5-2.465125i -6.5-4.486626i -6.5-7.336983i -6.5-12.38472i -6.5-26.37154i
n= 14 : 105+0i -7+30.669i -7+14.53565i -7+8.777722i -7+5.582314i -7+3.371022i -7+1.597704i -7+0i -7-1.597704i -7-3.371022i -7-5.582314i -7-8.777722i -7-14.53565i -7-30.669i
n= 15 : 120+0i -7.5+35.28473i -7.5+16.84528i -7.5+10.32286i -7.5+6.75303i -7.5+4.330127i -7.5+2.436898i -7.5+0.788282i -7.5-0.788282i -7.5-2.436898i -7.5-4.330127i -7.5-6.75303i -7.5-10.32286i -7.5-16.84528i -7.5-35.28473i
n= 16 : 136+0i -8+40.21872i -8+19.31371i -8+11.97285i -8+8i -8+5.345429i -8+3.313709i -8+1.591299i -8+0i -8-1.591299i -8-3.313709i -8-5.345429i -8-8i -8-11.97285i -8-19.31371i -8-40.21872i
n= 17 : 153+0i -8.5+45.47098i -8.5+21.94103i -8.5+13.72797i -8.5+9.324056i -8.5+6.418902i -8.5+4.232497i -8.5+2.418459i -8.5+0.787641i -8.5-0.787641i -8.5-2.418459i -8.5-4.232497i -8.5-6.418902i -8.5-9.324056i -8.5-13.72797i -8.5-21.94103i -8.5-45.47098i
n= 18 : 171+0i -9+51.04154i -9+24.7273i -9+15.58846i -9+10.72578i -9+7.551897i -9+5.196152i -9+3.275732i -9+1.586943i -9+0i -9-1.586943i -9-3.275732i -9-5.196152i -9-7.551897i -9-10.72578i -9-15.58846i -9-24.7273i -9-51.04154i
n= 19 : 190+0i -9.5+56.93038i -9.5+27.67255i -9.5+17.55446i -9.5+12.2056i -9.5+8.745366i -9.5+6.20666i -9.5+4.167086i -9.5+2.405727i -9.5+0.787192i -9.5-0.787192i -9.5-2.405727i -9.5-4.167086i -9.5-6.20666i -9.5-8.745366i -9.5-12.2056i -9.5-17.55446i -9.5-27.67255i -9.5-56.93038i
n= 20 : 210+0i -10+63.13752i -10+30.77684i -10+19.62611i -10+13.76382i -10+10i -10+7.26543i -10+5.09525i -10+3.2492i -10+1.58384i -10+0i -10-1.58384i -10-3.2492i -10-5.09525i -10-7.26543i -10-10i -10-13.76382i -10-19.62611i -10-30.77684i -10-63.13752i
n= 21 : 231+0i -10.5+69.66295i -10.5+34.04016i -10.5+21.80347i -10.5+15.40067i -10.5+11.31631i -10.5+8.37347i -10.5+6.06218i -10.5+4.12095i -10.5+2.39656i -10.5+0.78687i -10.5-0.78687i -10.5-2.39656i -10.5-4.12095i -10.5-6.06218i -10.5-8.37347i -10.5-11.31631i -10.5-15.40067i -10.5-21.80347i -10.5-34.04016i -10.5-69.66295i
n= 22 : 253+0i -11+76.50668i -11+37.46256i -11+24.08664i -11+17.11633i -11+12.69468i -11+9.53155i -11+7.06927i -11+5.02353i -11+3.22989i -11+1.58156i -11+0i -11-1.58156i -11-3.22989i -11-5.02353i -11-7.06927i -11-9.53155i -11-12.69468i -11-17.11633i -11-24.08664i -11-37.46256i -11-76.50668i
n= 23 : 276+0i -11.5+83.66871i -11.5+41.04404i -11.5+26.47566i -11.5+18.91094i -11.5+14.1354i -11.5+10.74025i -11.5+8.11759i -11.5+5.95882i -11.5+4.0871i -11.5+2.38973i -11.5+0.78662i -11.5-0.78662i -11.5-2.38973i -11.5-4.0871i -11.5-5.95882i -11.5-8.11759i -11.5-10.74025i -11.5-14.1354i -11.5-18.91094i -11.5-26.47566i -11.5-41.04404i -11.5-83.66871i
n= 24 : 300+0i -12+91.14905i -12+44.78461i -12+28.97056i -12+20.78461i -12+15.6387i -12+12i -12+9.20792i -12+6.9282i -12+4.97056i -12+3.21539i -12+1.57983i -12+0i -12-1.57983i -12-3.21539i -12-4.97056i -12-6.9282i -12-9.20792i -12-12i -12-15.6387i -12-20.78461i -12-28.97056i -12-44.78461i -12-91.14905i
n= 25 : 325+0i -12.5+98.94769i -12.5+48.68429i -12.5+31.5714i -12.5+22.73742i -12.5+17.20477i -12.5+13.31115i -12.5+10.3409i -12.5+7.93274i -12.5+5.88205i -12.5+4.0615i -12.5+2.3845i -12.5+0.78643i -12.5-0.78643i -12.5-2.3845i -12.5-4.0615i -12.5-5.88205i -12.5-7.93274i -12.5-10.3409i -12.5-13.31115i -12.5-17.20477i -12.5-22.73742i -12.5-31.5714i -12.5-48.68429i -12.5-98.94769i
n= 26 : 351+0i -13+107.0646i -13+52.74307i -13+34.27818i -13+24.76943i -13+18.83375i -13+14.67397i -13+11.517i -13+8.97325i -13+6.82293i -13+4.93025i -13+3.20421i -13+1.57849i -13+0i -13-1.57849i -13-3.20421i -13-4.93025i -13-6.82293i -13-8.97325i -13-11.517i -13-14.67397i -13-18.83375i -13-24.76943i -13-34.27818i -13-52.74307i -13-107.0646i
n= 27 : 378+0i -13.5+115.4999i -13.5+56.96098i -13.5+37.09095i -13.5+26.88071i -13.5+20.52575i -13.5+16.08867i -13.5+12.73659i -13.5+10.05038i -13.5+7.79423i -13.5+5.82333i -13.5+4.04163i -13.5+2.38041i -13.5+0.78629i -13.5-0.78629i -13.5-2.38041i -13.5-4.04163i -13.5-5.82333i -13.5-7.79423i -13.5-10.05038i -13.5-12.73659i -13.5-16.08867i -13.5-20.52575i -13.5-26.88071i -13.5-37.09095i -13.5-56.96098i -13.5-115.4999i
n= 28 : 406+0i -14+124.2534i -14+61.33801i -14+40.0097i -14+29.0713i -14+22.28087i -14+17.55544i -14+14i -14+11.16463i -14+8.79678i -14+6.74204i -14+4.89881i -14+3.19541i -14+1.57742i -14+0i -14-1.57742i -14-3.19541i -14-4.89881i -14-6.74204i -14-8.79678i -14-11.16463i -14-14i -14-17.55544i -14-22.28087i -14-29.0713i -14-40.0097i -14-61.33801i -14-124.2534i
n= 29 : 435+0i -14.5+133.3253i -14.5+65.87416i -14.5+43.03447i -14.5+31.34124i -14.5+24.09919i -14.5+19.07442i -14.5+15.30746i -14.5+12.31641i -14.5+9.83124i -14.5+7.68741i -14.5+5.77733i -14.5+4.02591i -14.5+2.37715i -14.5+0.78617i -14.5-0.78617i -14.5-2.37715i -14.5-4.02591i -14.5-5.77733i -14.5-7.68741i -14.5-9.83124i -14.5-12.31641i -14.5-15.30746i -14.5-19.07442i -14.5-24.09919i -14.5-31.34124i -14.5-43.03447i -14.5-65.87416i -14.5-133.3253i
n= 30 : 465+0i -15+142.7155i -15+70.56945i -15+46.16525i -15+33.69055i -15+25.98076i -15+20.64573i -15+16.65919i -15+13.50606i -15+10.89814i -15+8.66025i -15+6.67843i -15+4.8738i -15+3.18835i -15+1.57656i -15+0i -15-1.57656i -15-3.18835i -15-4.8738i -15-6.67843i -15-8.66025i -15-10.89814i -15-13.50606i -15-16.65919i -15-20.64573i -15-25.98076i -15-33.69055i -15-46.16525i -15-70.56945i -15-142.7155i
>
>
> ## Complex Trig.:
> abs(Im(cos(acos(1i))) - 1) < 2*Meps
[1] TRUE
> abs(Im(sin(asin(1i))) - 1) < 2*Meps
[1] TRUE
> ##P (1 - Im(sin(asin(Ii))))/Meps
> ##P (1 - Im(cos(acos(Ii))))/Meps
> abs(Im(asin(sin(1i))) - 1) < 2*Meps
[1] TRUE
> cos(1i) == cos(-1i)# i.e. Im(acos(*)) gives + or - 1i:
[1] TRUE
> abs(abs(Im(acos(cos(1i)))) - 1) < 4*Meps
[1] TRUE
>
>
> set.seed(123) # want reproducible output
> Isi <- Im(sin(asin(1i + rnorm(100))))
> all(abs(Isi-1) < 100* Meps)
[1] TRUE
> ##P table(2*abs(Isi-1) / Meps)
> Isi <- Im(cos(acos(1i + rnorm(100))))
> all(abs(Isi-1) < 100* Meps)
[1] TRUE
> ##P table(2*abs(Isi-1) / Meps)
> Isi <- Im(atan(tan(1i + rnorm(100)))) #-- tan(atan(..)) does NOT work (Math!)
> all(abs(Isi-1) < 100* Meps)
[1] TRUE
> ##P table(2*abs(Isi-1) / Meps)
>
>
> ## polyroot():
> stopifnot(abs(1 + polyroot(choose(8, 0:8))) < 1e-10)# maybe smaller..
>
>
> ## PR#7781
> ## This is not as given by e.g. glibc on AMD64
> (z <- tan(1+1000i)) # 0+1i from R's own code.
[1] 0+1i
> stopifnot(is.finite(z))
> ##
>
>
> ## Branch cuts in complex inverse trig functions
> atan(2)
[1] 1.107149
> atan(2+0i)
[1] 1.107149+0i
> tan(atan(2+0i))
[1] 2+0i
> ## should not expect exactly 0i in result
> round(atan(1.0001+0i), 7)
[1] 0.7854482+0i
> round(atan(0.9999+0i), 7)
[1] 0.7853482+0i
> ## previously not as in Abramowitz & Stegun.
>
>
> ## typo in z_atan2.
> (z <- atan2(0+1i, 0+0i))
[1] 1.570796+0i
> stopifnot(all.equal(z, pi/2+0i))
> ## was NA in 2.1.1
>
>
> ## precision of complex numbers
> signif(1.678932e80+0i, 5)
[1] 1.6789e+80+0i
> signif(1.678932e-300+0i, 5)
[1] 1.6789e-300+0i
> signif(1.678932e-302+0i, 5)
[1] 1.6789e-302+0i
> signif(1.678932e-303+0i, 5)
[1] 1.6789e-303+0i
> signif(1.678932e-304+0i, 5)
[1] 1.6789e-304+0i
> signif(1.678932e-305+0i, 5)
[1] 1.6789e-305+0i
> signif(1.678932e-306+0i, 5)
[1] 1.6789e-306+0i
> signif(1.678932e-307+0i, 5)
[1] 1.6789e-307+0i
> signif(1.678932e-308+0i, 5)
[1] 1.6789e-308+0i
> signif(1.678932-1.238276i, 5)
[1] 1.6789-1.2383i
> signif(1.678932-1.238276e-1i, 5)
[1] 1.6789-0.1238i
> signif(1.678932-1.238276e-2i, 5)
[1] 1.6789-0.0124i
> signif(1.678932-1.238276e-3i, 5)
[1] 1.6789-0.0012i
> signif(1.678932-1.238276e-4i, 5)
[1] 1.6789-0.0001i
> signif(1.678932-1.238276e-5i, 5)
[1] 1.6789+0i
> signif(8.678932-9.238276i, 5)
[1] 8.6789-9.2383i
> ## prior to 2.2.0 rounded real and imaginary parts separately.
>