How to find the intersection point of four equations?

Questions
1

Hi,
I have four equations of four variables. I defined the range of the four variables and I need to find the intersection of the four equations in the given range. The equations are in terms of the four variables. Can you please tell me how to find the intersection?

Y2 =.;
Y3 =.;
l2 =.;
l3 =.;
Y1 = 0.0125;
l1 = 0.0108366;
lambda = {0.043656,0.0410959,0.0413712,0.0433463,0.0423413};
b1 = 0.0502519;
b3 = 0.0161692;
t21 = ((2*Pi)/lambda[[1]])*l2;
t31 = ((2*Pi)/lambda[[1]])*l3;
t23 = ((2*Pi)/lambda[[5]])*l2;
t33 = ((2*Pi)/lambda[[5]])*l3;
t11 = ((2*Pi)/lambda[[1]])*l1;
t13 = ((2*Pi)/lambda[[5]])*l1;
Manipulate[t21 = (2*Pi/lambda[[1]])*l2;
t31 = (2*Pi/lambda[[1]])*l3;
t23 = (2*Pi/lambda[[5]])*l2;
t33 = (2*Pi/lambda[[5]])*l3; 
plot = Plot[{2*Y1*Tan[t11] + Y2*Tan[t21] + Y3*Tan[t31], 
2*Y1*Tan[t13] + Y2*Tan[t23] + Y3*Tan[t33], 
t11*Y1 + t21*(Y2/2)*(Sec[t21]^2/Sec[t11]^2) + 
 t31*(Y3/2)*(Sec[t31]^2/Sec[t11]^2) - b1, 
t13*Y1 + t23*(Y2/2)*(Sec[t23]^2/Sec[t13]^2) + 
 t33*(Y3/2)*(Sec[t33]^2/Sec[t13]^2) - b3},
{Y3, 0.008, 0.05}],
{Y2, 0.008, 0.05}, {l3, 0.008, 0.012}, {l2, 0.008, 0.012}]
2

The code doesn’t work for me ?

3

The range can be increased by {Y9,0.008,0.05}, {Y2,0.008,0.05},{l33,0.008,0.012},{l22,0.008,0.012}. Any problem with the code? Its working fine for me. Please let me know.

4

There are several undefined symbols in the code, t11, t21, t13 …

5

Please check the edited code.

6

The code doesn’t work for me, with fresh kernel? Update the code.

What is that:

  Y2 =.;

?

7

t11 is still missing.

8

Hi Mariusz,

=. is syntax for Unset.

9

I defined it as a variable.

10

Sorry. Updated the code.

11

There are still undefined symbols. Execute the latest code against a new kernel.

12

Please check the updated code.

13

Why do you believe there is a simultaneous zero in those ranges of the variables? FindRoot and FindMinimum fail to locate any zero. And the Manipulate does not indicate likelihood of such either, as best I can tell.

14

It’ll not be an exact zero but its a very small value of the order of 10^(-29). I need to find the values of the 4 unknowns from the plot. I guess the solution gets trapped in local minima.