Why does the solutions of nonlinear system are not correctly computed

I hv the following system of nonlinear equations for computing switching angles of a multilevel inverter, taken as an example from my pwr elctnx book

function F = test(x)
F=[cos(5*x(1)) + cos(5*x(2))+cos(5*x(3)) + cos(5*x(4))+ cos(5*x(5))  
    cos(7*x(1)) + cos(7*x(2))+cos(7*x(3)) + cos(7*x(4)) + cos(7*x(5))
    cos(11*x(1)) + cos(11*x(2))+cos(11*x(3)) + cos(11*x(4))+cos(11*x(5))
    cos(13*x(1)) + cos(13*x(2))+cos(13*x(3)) + cos(13*x(4))+ cos(13*x(5))
    cos(x(1)) + cos(x(2))+cos(x(3)) + cos(x(4))+ cos(x(5))-4];

I have used the matlab function FSOLVE with OPTIMSET to sovle this system for x1,x2,x3,x4 and x5 and on ocassions hv increased parameters like 'MaxFunEval' and MaxIter but the matlab answers are different from those given in the book ie x1=6.57;x2=18.94;x3=27.18;x4=45.15;x5=62.24;(in Degrees). plzzz anybody help me with this. I need it urgently

[Matt J]

One remark, if the x(i) are in Degrees, you should be using COSD instead of COS.

Answer 1

[Matt J]

It looks like there must exist a solution with all x(1)=x(2)=x(3)=x(4)=x(5), due to the symmetry of your F. Therefore, the solution in your book is either just one of many, or you have incorrectly implemented your F(x). What is the final value of F(x) that fsolve returns?

[Asif Khan]

Following is another m file from which I m calling this function.

clear all
clc
x0=[0 1 1.5 2 2.5 ];
options=optimset('Display','iter','MaxFunEval', 200000,'MaxIter',100000);
[x,fval] = fsolve(@test,x0,options)

The results on command window are following;

Optimizer appears to be converging to a point which is not a root.
 Relative function value changing by less than max(options.TolFun^2,eps) but
 sum-of-squares of function values is greater than or equal to sqrt(options.TolFun)
 Try again with a new starting guess.

x =

    Columns 1 through 3

     -0.2215    0.2217    0.3806

    Columns 4 through 5

      2.8584    3.4248

fval =

      0.2584
     -0.0490
     -0.0277
      0.0156
     -3.0408

If u cud plzz tell me anyway to solve dis I'd be grateful

[Matt J]

Sorry, my mistake. The solution won't be of the form x(1)=x(2)=x(3)=x(4)=x(5), however, it will be independent of the order of the x(i). For example, one solution is what you gave

 >> x,fval=test(x)'

x =

      6.5698   18.9402   27.1833   45.1358   62.2425

fval =

     1.0e-11 *

     -0.0270    0.0553   -0.0324   -0.1810   -0.0077

But here is another solution,

>> x2,fval2=test(x2)'

x2 =

     62.2425   27.1833   45.1358   18.9402    6.5698

fval2 =

     1.0e-11 *

     -0.0270    0.0553   -0.0324   -0.1810   -0.0078

[Matt J]

Can't you just try a better initial guess like the warnings tell you, e.g.,

    x0=[7, 20 , 25, 42, 60];

    options=optimset('Display','iter','TolFun',1e-11);
    [x,fval] = fsolve(@Ffun,x0,options)

x =

      6.5698   18.9402   27.1833   45.1358   62.2425

fval =

     1.0e-14 *

      0.0555
      0.0694
     -0.1110
      0.0390
           0

[Asif Khan]

Thanks Matt J .. :) It really helped

Answer 2

[Asif Khan]

You were right Matt :) . There can be several solutions to this problem