On May 9, 8:49 am, "Jon G." <jon8...@[EMAIL PROTECTED]
> wrote:
> 1+x+x^2+x^3+...+x^136=0
>
> x~ -1.18119800013328E-003+0.771428140536064i
>
> f(x)=0.627107102656169-0.484340167756546i
Is there a question somewhere in here? Anyway, the solution is
trivial: the roots are the 136 non-real 137th roots of unity. Your x
is NOT one of the 137th roots of 1. Maple 9.5 gets
x1:=-1.18119800013328e-3+0.771428140536064*I;
x1 := -0.00118119800013328 + 0.771428140536064 I
simplify(x1^137);
-16 -15
-0.7552796048 10 + 0.3547511434 10 I
This is not even close to 1.0.
Here are all the non-real 137th roots of 1:
r:=seq(exp(I*j*2*Pi/137),j=1..136): <--- symbolic values
The numerical values are:
s:=seq(evalf(r[i]),i=1..136);
s := 0.9989484922 + 0.04584659038 I, 0.9957961803 + 0.09159676465 I,
0.9905496935 + 0.1371543096 I, 0.9832200654 + 0.1824234167 I,
0.9738227102 + 0.2273088847 I, 0.9623773908 + 0.2717163185 I,
0.9489081767 + 0.3155523286 I, 0.9334433940 + 0.3587247276 I,
0.9160155655 + 0.4011427225 I, 0.8966613419 + 0.4427171084 I,
0.8754214256 + 0.4833604530 I, 0.8523404844 + 0.5229872834 I,
0.8274670581 + 0.5615142632 I, 0.8008534558 + 0.5988603697 I,
0.7725556460 + 0.6349470639 I, 0.7426331399 + 0.6696984542 I,
0.7111488646 + 0.7030414585 I, 0.6781690323 + 0.7349059556 I,
0.6437629999 + 0.7652249342 I, 0.6080031241 + 0.7939346328 I,
0.5709646081 + 0.8209746746 I, 0.5327253445 + 0.8462881940 I,
0.4933657519 + 0.8698219558 I, 0.4529686027 + 0.8915264690 I,
0.4116188543 + 0.9113560878 I, 0.3694034641 + 0.9292691110 I,
0.3264112137 + 0.9452278665 I, 0.2827325145 + 0.9591987934 I,
0.2384592253 + 0.9711525101 I, 0.1936844518 + 0.9810638782 I,
0.1485023578 + 0.9889120536 I, 0.1030079601 + 0.9946805317 I,
0.05729693600 + 0.9983571811 I,
0.01146541558 + 0.9999342700 I,
-0.03439021778 + 0.9994084815 I,
-0.08017352699 + 0.9967809216 I,
-0.1257882310 + 0.9920571158 I,
-0.1711383995 + 0.9852469986 I,
-0.2161286622 + 0.9763648915 I,
-0.2606644020 + 0.9654294741 I,
-0.3046519615 + 0.9524637433 I,
-0.3479988322 + 0.9374949668 I,
-0.3906138569 + 0.9205546235 I,
-0.4324074137 + 0.9016783399 I,
-0.4732916119 + 0.8809058123 I,
-0.5131804696 + 0.8582807266 I,
-0.5519901017 + 0.8338506627 I,
-0.5896388892 + 0.8076669984 I,
-0.6260476578 + 0.7797847974 I,
-0.6611398376 + 0.7502626974 I,
-0.6948416302 + 0.7191627834 I,
-0.7270821607 + 0.6865504582 I,
-0.7577936255 + 0.6524943074 I,
-0.7869114392 + 0.6170659502 I,
-0.8143743652 + 0.5803398946 I,
-0.8401246500 + 0.5423933743 I,
-0.8641081390 + 0.5033061932 I,
-0.8862743957 + 0.4631605505 I,
-0.9065768031 + 0.4220408748 I,
-0.9249726659 + 0.3800336398 I,
-0.9414232964 + 0.3372271890 I,
-0.9558940995 + 0.2937115432 I,
-0.9683546421 + 0.2495782185 I,
-0.9787787201 + 0.2049200260 I,
-0.9871444111 + 0.1598308844 I,
-0.9934341222 + 0.1144056150 I,
-0.9976346259 + 0.06873974985 I,
-0.9997370885 + 0.02292932292 I,
-0.9997370885 - 0.02292932292 I,
-0.9976346259 - 0.06873974985 I,
-0.9934341222 - 0.1144056150 I,
-0.9871444111 - 0.1598308844 I,
-0.9787787201 - 0.2049200260 I,
-0.9683546421 - 0.2495782185 I,
-0.9558940995 - 0.2937115432 I,
-0.9414232964 - 0.3372271890 I,
-0.9249726659 - 0.3800336398 I,
-0.9065768031 - 0.4220408748 I,
-0.8862743957 - 0.4631605505 I,
-0.8641081390 - 0.5033061932 I,
-0.8401246500 - 0.5423933743 I,
-0.8143743652 - 0.5803398946 I,
-0.7869114392 - 0.6170659502 I,
-0.7577936255 - 0.6524943074 I,
-0.7270821607 - 0.6865504582 I,
-0.6948416302 - 0.7191627834 I,
-0.6611398376 - 0.7502626974 I,
-0.6260476578 - 0.7797847974 I,
-0.5896388892 - 0.8076669984 I,
-0.5519901017 - 0.8338506627 I,
-0.5131804696 - 0.8582807266 I,
-0.4732916119 - 0.8809058123 I,
-0.4324074137 - 0.9016783399 I,
-0.3906138569 - 0.9205546235 I,
-0.3479988322 - 0.9374949668 I,
-0.3046519615 - 0.9524637433 I,
-0.2606644020 - 0.9654294741 I,
-0.2161286622 - 0.9763648915 I,
-0.1711383995 - 0.9852469986 I,
-0.1257882310 - 0.9920571158 I,
-0.08017352699 - 0.9967809216 I,
-0.03439021778 - 0.9994084815 I,
0.01146541558 - 0.9999342700 I,
0.05729693600 - 0.9983571811 I, 0.1030079601 - 0.9946805317 I,
0.1485023578 - 0.9889120536 I, 0.1936844518 - 0.9810638782 I,
0.2384592253 - 0.9711525101 I, 0.2827325145 - 0.9591987934 I,
0.3264112137 - 0.9452278665 I, 0.3694034641 - 0.9292691110 I,
0.4116188543 - 0.9113560878 I, 0.4529686027 - 0.8915264690 I,
0.4933657519 - 0.8698219558 I, 0.5327253445 - 0.8462881940 I,
0.5709646081 - 0.8209746746 I, 0.6080031241 - 0.7939346328 I,
0.6437629999 - 0.7652249342 I, 0.6781690323 - 0.7349059556 I,
0.7111488646 - 0.7030414585 I, 0.7426331399 - 0.6696984542 I,
0.7725556460 - 0.6349470639 I, 0.8008534558 - 0.5988603697 I,
0.8274670581 - 0.5615142632 I, 0.8523404844 - 0.5229872834 I,
0.8754214256 - 0.4833604530 I, 0.8966613419 - 0.4427171084 I,
0.9160155655 - 0.4011427225 I, 0.9334433940 - 0.3587247276 I,
0.9489081767 - 0.3155523286 I, 0.9623773908 - 0.2717163185 I,
0.9738227102 - 0.2273088847 I, 0.9832200654 - 0.1824234167 I,
0.9905496935 - 0.1371543096 I, 0.9957961803 - 0.09159676465 I,
0.9989484922 - 0.04584659038 I
Now plug them into your function:
f:=add(x^i,i=0..136): <--- your function
fs:=seq(evalf(subs(x=r[i],f)),i=1..136);
-8 -6 -7 -7
fs := -0.42 10 + 0.11281 10 I, -0.166 10 + 0.1170 10 I,
-6 -8 -7 -7
0.1147 10 - 0.16 10 I, -0.173 10 + 0.181 10 I,
-7 -8 -8 -7
0.339 10 + 0.21 10 I, -0.25 10 - 0.137 10 I,
-7 -7 -7 -8
-0.240 10 + 0.314 10 I, 0.670 10 - 0.90 10 I,
-8 -8 -7 -7
-0.96 10 - 0.13 10 I, 0.4544 10 - 0.351 10 I,
-7 -7 -7 -8
0.100 10 + 0.120 10 I, 0.559 10 - 0.93 10 I,
-7 -8 -7 -8
0.293 10 - 0.83 10 I, -0.177 10 - 0.54 10 I,
-7 -7 -7 -8
0.297 10 - 0.222 10 I, -0.104 10 - 0.18 10 I,
-7 -8 -8 -8
0.178 10 - 0.49 10 I, 0.96 10 - 0.11 10 I,
-7 -8 -7 -7
0.293 10 - 0.78 10 I, 0.151 10 - 0.152 10 I,
-7 -7 -7 -7
0.336 10 - 0.171 10 I, 0.698 10 - 0.371 10 I,
-7 -8 -7 -7
0.108 10 - 0.59 10 I, 0.564 10 - 0.288 10 I,
-8 -8 -7 -7
0.62 10 - 0.29 10 I, 0.684 10 - 0.392 10 I,
-8 -8 -7 -7
0.91 10 - 0.73 10 I, 0.556 10 - 0.480 10 I,
-7 -7 -7 -7
0.187 10 - 0.180 10 I, 0.612 10 - 0.478 10 I,
-7 -7 -7 -7
0.196 10 - 0.209 10 I, 0.583 10 - 0.526 10 I,
-7 -7 -7 -8
0.245 10 - 0.191 10 I, -0.157 10 + 0.87 10 I,
-7 -7 -8 -8
0.227 10 - 0.251 10 I, -0.91 10 + 0.70 10 I,
-7 -7 -8 -8
0.193 10 - 0.287 10 I, -0.66 10 + 0.18 10 I,
-7 -7 -8 -9
0.285 10 - 0.316 10 I, 0.68 10 + 0.9 10 I,
-7 -7 -9 -8
0.288 10 - 0.366 10 I, 0.5 10 - 0.38 10 I,
-7 -7 -8 -8
0.234 10 - 0.352 10 I, -0.25 10 - 0.523 10 I,
-7 -7 -8 -8
0.188 10 - 0.406 10 I, 0.32 10 - 0.627 10 I,
-7 -7 -8 -8
0.220 10 - 0.418 10 I, 0.76 10 - 0.88 10 I,
-7 -7 -8 -7
0.204 10 - 0.443 10 I, 0.67 10 - 0.126 10 I,
-8 -7 -8 -7
-0.97 10 + 0.178 10 I, 0.43 10 - 0.171 10 I,
-8 -7 -8 -7
-0.89 10 + 0.137 10 I, 0.60 10 - 0.263 10 I,
-8 -7 -8 -7
-0.71 10 + 0.159 10 I, 0.68 10 - 0.263 10 I,
-9 -8 -8 -7
-0.3 10 + 0.97 10 I, 0.73 10 - 0.280 10 I,
-8 -8 -8 -7
0.34 10 + 0.48 10 I, 0.39 10 - 0.319 10 I,
-8 -8 -8 -7
-0.20 10 + 0.15 10 I, 0.57 10 - 0.403 10 I,
-8 -8 -8 -7
-0.17 10 - 0.65 10 I, 0.55 10 - 0.345 10 I,
-8 -8 -8 -7
0.15 10 - 0.43 10 I, 0.65 10 - 0.413 10 I,
-8 -8 -8 -7
0.14 10 - 0.62 10 I, 0.36 10 - 0.445 10 I,
-8 -7 -9 -8
0.170 10 + 0.411 10 I, -0.4 10 + 0.45 10 I,
-8 -7 -9 -8
0.64 10 + 0.448 10 I, 0.3 10 + 0.76 10 I,
-8 -7 -8 -8
0.45 10 + 0.348 10 I, -0.16 10 + 0.16 10 I,
-8 -7 -9 -8
0.33 10 + 0.4125 10 I, -0.5 10 - 0.17 10 I,
-8 -7 -8 -8
0.43 10 + 0.304 10 I, 0.34 10 - 0.28 10 I,
-8 -7 -8 -8
0.73 10 + 0.280 10 I, -0.21 10 - 0.78 10 I,
-8 -7 -8 -7
0.81 10 + 0.258 10 I, -0.26 10 - 0.155 10 I,
-8 -7 -8 -7
0.75 10 + 0.258 10 I, -0.64 10 - 0.118 10 I,
-8 -7 -8 -7
0.37 10 + 0.161 10 I, -0.97 10 - 0.168 10 I,
-8 -7 -7 -7
0.68 10 + 0.105 10 I, 0.200 10 + 0.500 10 I,
-8 -7 -7 -7
0.71 10 + 0.102 10 I, 0.199 10 + 0.421 10 I,
-8 -8 -7 -7
0.36 10 + 0.61 10 I, 0.218 10 + 0.406 10 I,
-9 -8 -7 -7
-0.8 10 + 0.67 10 I, 0.253 10 + 0.397 10 I,
-9 -8 -7 -7
-0.7 10 + 0.13 10 I, 0.263 10 + 0.377 10 I,
-8 -9 -7 -7
0.23 10 - 0.6 10 I, 0.250 10 + 0.331 10 I,
-8 -8 -7 -7
-0.35 10 - 0.51 10 I, 0.213 10 + 0.263 10 I,
-7 -8 -7 -7
-0.1292 10 - 0.54 10 I, 0.223 10 + 0.235 10 I,
-7 -7 -7 -7
-0.1261 10 - 0.127 10 I, 0.244 10 + 0.217 10 I,
-7 -7 -7 -7
0.584 10 + 0.537 10 I, 0.196 10 + 0.169 10 I,
-7 -7 -7 -7
0.615 10 + 0.463 10 I, 0.180 10 + 0.167 10 I,
-7 -7 -8 -8
0.559 10 + 0.489 10 I, 0.91 10 + 0.93 10 I,
-7 -7 -8 -8
0.646 10 + 0.388 10 I, 0.71 10 + 0.37 10 I,
-7 -7 -7 -8
0.586 10 + 0.331 10 I, 0.102 10 - 0.10 10 I,
-7 -7 -7 -7
0.674 10 + 0.348 10 I, 0.308 10 + 0.173 10 I,
-7 -7 -7 -8
0.159 10 + 0.145 10 I, 0.290 10 + 0.89 10 I,
-8 -8 -7 -8
0.92 10 - 0.69 10 I, 0.180 10 + 0.50 10 I,
-8 -9 -7 -7
-0.54 10 - 0.2 10 I, 0.225 10 + 0.228 10 I,
-7 -8 -7 -7
-0.198 10 + 0.49 10 I, 0.326 10 + 0.106 10 I,
-7 -7 -8 -8
0.541 10 + 0.111 10 I, 0.64 10 - 0.66 10 I,
-7 -7 -7 -8
0.476 10 + 0.332 10 I, -0.121 10 + 0.23 10 I,
-7 -8 -7 -7
0.689 10 + 0.941 10 I, -0.249 10 - 0.301 10 I,
-8 -7 -7 -7
0.47 10 + 0.148 10 I, 0.516 10 - 0.2210 10 I,
-8 -7 -6 -7
-0.47 10 - 0.352 10 I, 0.1091 10 - 0.261 10 I,
-8 -8 -8 -6
0.24 10 + 0.160 10 I, -0.42 10 - 0.11281 10 I
All these are close to zero; we could get even closer by working to
more than the default 10-digit floating-point accuracy. For example,
using 20 digits gives f values of the order of e-18, 50 digits gives f
of the order of e-47 and 200 digits gives f of order e-147.
R.G. Vickson
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