Solving a system of Non-linear Equations with Complex numbers

Hi all,

I would like to solve the following system of 3 non-linear equations in 3 unknowns :

    function F= myfun(x)
    F=[ x(1) + x(2) - 10000;
    x(1)*exp(-1i*x(3)*5) + x(2)*exp(1i*x(3)*5) - 12000;
    x(1)*exp(-1i*x(3)*10) + x(2)*exp(1i*x(3)*10) - 8000 ];

The fsolve command does not solve complex number roots. And splitting it up into real and imaginary also does not seem to work (I took the help of Euler's formulae to convert to cosine and sine).

Please help me out!

Answer 1

[John D'Errico]

It appears there are two solutions. They are best found using solve, so a symbolic solution.

syms x1 x2 x3
eq1 = x1 + x2 - 10000;
eq2 = x1*exp(-1i*x3*5) + x2*exp(1i*x3*5) - 12000;
eq3 = x1*exp(-1i*x3*10) + x2*exp(1i*x3*10) - 8000;

result = solve(eq1,eq2,eq3,x1,x2,x3)
result = 
  struct with fields:

      x1: [2×1 sym]
      x2: [2×1 sym]
      x3: [2×1 sym]

result.x1
ans =
 5000 - (7^(1/2)*9000i)/7
 (7^(1/2)*9000i)/7 + 5000

result.x2
ans =
 (7^(1/2)*9000i)/7 + 5000
 5000 - (7^(1/2)*9000i)/7

result.x3
ans =
 -(log(3/4 - (7^(1/2)*1i)/4)*1i)/5
 -(log((7^(1/2)*1i)/4 + 3/4)*1i)/5

vpa(result.x1)
ans =
 5000.0 - 3401.6802570830450449306488261076i
 5000.0 + 3401.6802570830450449306488261076i

vpa(result.x2)
ans =
 5000.0 + 3401.6802570830450449306488261076i
 5000.0 - 3401.6802570830450449306488261076i

vpa(result.x3)
ans =
 -0.14454684956268312223567547052827
  0.14454684956268312223567547052827

As you can see by simple tests to verify this is a solution...

result.x1 + result.x2
ans =
 10000
 10000

vpa(result.x1.*exp(-1i*result.x3*5) + result.x2.*exp(1i*result.x3*5) - 12000)
ans =
 0
 0

Answer 2

[Star Strider]

‘The fsolve command does not solve complex number roots.’

It does for me!

Your function (as an anonymous function):

myfun = @(x) [ x(1) + x(2) - 10000;
    x(1)*exp(-1i*x(3)*5) + x(2)*exp(1i*x(3)*5) - 12000;
    x(1)*exp(-1i*x(3)*10) + x(2)*exp(1i*x(3)*10) - 8000 ];

X = fsolve(myfun, [1; 1; 1])

produced:

X =

       2.3160e+000 - 33.3155e-003i
       2.3160e+000 + 33.3155e-003i
       1.2720e+000 +             i

It wanted me to let it do more function evaluations, but I’ll leave that to you. (I’m running R2014a, but I doubt that makes a difference.)

[Rubron]

Hi, do you know if the same can be applied to "lsqnonlin" solver? I have been struggling while trying to enter complex equations with that solver, it seems it does not accept them.

[John D'Errico]

Sorry. lsqnonlin will definitely not work happily with complex variables, which explains your failure to get it working.

Hmm. I did just notice that the solution Star proposed does not actually solve the system of equations. For example, the sum x(1)+x(2) does not yield 10000. In fact, that sum is not even close to 10000. So I'm not too sure that fsolve is terribly happy at solving complex problems either.

Answer 3

[Alex Sha]

There are much more solutions else:

x1: 5000+3401.68025708298i

x2: 5000-3401.68025708301i

x3: -3.62536433474507+0i

x1: 5000+3401.68025708298i

x2: 5000-3401.68025708301i

x3: 1.11209021187323+0i

x1: 5000+3401.68025708298i

x2: 5000-3401.68025708301i

x3: 2.36872727330915+0i

Answer 4

[David Goodmanson]

Hellol Aravindhan,

not solved by Matlab, but a bit of algebra shows that

with
x(1) = 2000*a
x(2) = 2000*b
x(3) = theta/5
then
theta = acos(3/4);
a = (5/2 +9*i/(2*sqrt(7)));
b = (5/2 -9*i/(2*sqrt(7)));

format long g
theta = acos(3/4)
x = [2000*a 2000*b theta/5]'

       theta = 0.722734247813416
       
       x =               5000 -      3401.68025708305i
                         5000 +      3401.68025708305i
            0.144546849562683 +                     0i

There are an infinite number of solutions to theta = acos(3/4). First of all there is the 2pi ambiguity, so theta = .7227 + 2*pi*n is a set of solutions. Then the negative angle, -.7227 (with its 2pi ambiguity) is a set of solutions as well. But note that the equations are symmetric under theta --> -theta, a<-->b

so the -.7227 set is the same thing with a and b swapped. Consequently there are only two possibilities for a and b.