Advanced Search

Question: Solving a system of equations

Question:Solving a system of equations

kiraMou 45 Maple
rootfinding fsolve numeric
May 24 2021
0

> 

restart; with(plots); a := .8; N := 4; for k to N while k <> j do t[k] := sinh(x[k]+I*a)^N*(product(sinh(1/2*(x[k]-x[j]-(2*I)*a)), j = 1 .. N))/(sinh(x[k]-I*a)^N*(product(sinh(1/2*(x[k]-x[j]+(2*I)*a)), j = 1 .. N))) end do

.8

  

4

  

-1.000000000*sinh(x[1]+.8*I)^4*sinh((1/2)*x[1]-(1/2)*x[2]-.8*I)*sinh((1/2)*x[1]-(1/2)*x[3]-.8*I)*sinh((1/2)*x[1]-(1/2)*x[4]-.8*I)/(sinh(x[1]-.8*I)^4*sinh((1/2)*x[1]-(1/2)*x[2]+.8*I)*sinh((1/2)*x[1]-(1/2)*x[3]+.8*I)*sinh((1/2)*x[1]-(1/2)*x[4]+.8*I))

  

-1.000000000*sinh(x[2]+.8*I)^4*sinh((1/2)*x[1]-(1/2)*x[2]+.8*I)*sinh((1/2)*x[2]-(1/2)*x[3]-.8*I)*sinh((1/2)*x[2]-(1/2)*x[4]-.8*I)/(sinh(x[2]-.8*I)^4*sinh((1/2)*x[1]-(1/2)*x[2]-.8*I)*sinh((1/2)*x[2]-(1/2)*x[3]+.8*I)*sinh((1/2)*x[2]-(1/2)*x[4]+.8*I))

  

-1.000000000*sinh(x[3]+.8*I)^4*sinh((1/2)*x[1]-(1/2)*x[3]+.8*I)*sinh((1/2)*x[2]-(1/2)*x[3]+.8*I)*sinh((1/2)*x[3]-(1/2)*x[4]-.8*I)/(sinh(x[3]-.8*I)^4*sinh((1/2)*x[1]-(1/2)*x[3]-.8*I)*sinh((1/2)*x[2]-(1/2)*x[3]-.8*I)*sinh((1/2)*x[3]-(1/2)*x[4]+.8*I))

  

-1.000000000*sinh(x[4]+.8*I)^4*sinh((1/2)*x[1]-(1/2)*x[4]+.8*I)*sinh((1/2)*x[2]-(1/2)*x[4]+.8*I)*sinh((1/2)*x[3]-(1/2)*x[4]+.8*I)/(sinh(x[4]-.8*I)^4*sinh((1/2)*x[1]-(1/2)*x[4]-.8*I)*sinh((1/2)*x[2]-(1/2)*x[4]-.8*I)*sinh((1/2)*x[3]-(1/2)*x[4]-.8*I))

 (1)
> 

list4 := fsolve({t[1] = -1, t[2] = -1, t[3] = -1, t[4] = -1}, {x[1], x[2], x[3], x[4]}); list4 := fsolve({t[1] = -1, t[2] = -1, t[3] = -1, t[4] = -1}, {Im(x[1]) = .5 .. 1, Im(x[2]) = .5 .. 1, Im(x[3]) = -1 .. -.5, Im(x[4]) = -1 .. -.5})
> 

``

Download system.mw


 

Download system.mw

 

Solving a system of equations

I want to solve the following system (please see the code) for small even N=4,8,16. They said that the roots lie on the lines Im x_i = ±π/2 and they are symmetric with respect to the imaginary axis. For N=4, I tried with fsolve directly but the roots are:

 x[1] = -3.799588601-1.570796327*I,

 x[2] = -3.799588601-1.570796327*I,

 x[3] = .1196098914-1.570796327*I,

 x[4] = 1.110549099-1.570796327*I

these solutions are not valid because half of them should have +π/2 as an imaginary part, and they should be mutually distinct and also symmetric with respect to the imaginary axis. Therefore, for N=4  I am expecting solutions of these form :

x[1] = a_1-I* π/2,

x[2] = -a_1-I* π/2,

x[3] = a_1+I* π/2,

x[4] = -a_1+I* π/2,

I tried with fslove({system}, {Im(x[1]) = .5 .. 1, Im(x[2]) = .5 .. 1, Im(x[3]) = -1 .. -.5, Im(x[4]) = -1 .. -.5}) but it does not work! I do not know what should I do to make Fsolve locate the wanted solutions. Thanks in advance!

 
Please Wait...