Problem with solving multivariable trigonometric equations

Hi everyone,

I have no idea where is wrong in my code?

clear all;

%%
clc
Fs = 1e6;
f = 5;
t = 2e6;
ang0 = t*2*pi*f/Fs

ang0 = 62.8319

dang = 1*2*pi*f/Fs

dang = 3.1416e-05

y0= 1*sin(t*2*pi*f/Fs);
y1= 1*sin((t+1)*2*pi*f/Fs);
y2= 1*sin((t+2)*2*pi*f/Fs);

% for i = 1:10e6
%     Curve(i) = 1*sin(i*2*pi*f/Fs);
% end
% 
% close all
% hold on
% plot(Curve)
%%
syms x y z
% assume(x > 0)
% assumeAlso(x < 1)
% assume(z >= 0)
% assumeAlso(z <= 2*pi)
% assume(y > 0)
% assumeAlso(y <= 100)
[x,y,z]=solve(x*sin((t*2*pi*y/Fs) + z)==y0  ,x*sin(((t+1)*2*pi*y/Fs)+z)==y1,x*sin(((t+2)*2*pi*y/Fs)+z)==y2,'ReturnConditions',true)

 
x =
 
Empty sym: 0-by-1
 
 
y =
 
Empty sym: 1-by-0
 
 
z =
 
Empty sym: 0-by-1
 

Answer 1

[Matt J]

In order for solve() to find a solution, an exact analytical solution must exist.

[Minerva Bionic]

Is there a method other than solve() that can solve these equations?

[Matt J]

With lsqnonlin, you can find an approximate solution:

[xyz,Error]=main()

Local minimum possible.

lsqnonlin stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

xyz = 1×3

0.5187 3.1942 47.8254

Error = 8.8349e-10

function [xyz,Error]=main
Fs = 1e6;
f = 5;
t = 2e6;
ang0 = t*2*pi*f/Fs;
dang = 1*2*pi*f/Fs;
y0= 1*sin(t*2*pi*f/Fs);
y1= 1*sin((t+1)*2*pi*f/Fs);
y2= 1*sin((t+2)*2*pi*f/Fs);


fn=@(q) linspace(0,q,300);
[x,y,z]=ndgrid(fn(1),fn(2*pi),fn(100));

F1 = x.*sin((t.*2.*pi.*y/Fs) + z) - y0  ;
F2 = x.*sin(((t+1).*2.*pi.*y/Fs)+z) - y1 ;
F3 = x.*sin(((t+2).*2.*pi.*y/Fs)+z) - y2; 

[fval,i0]=min(abs(F1)+abs(F2)+abs(F3),[],'all','linear');

X0=[x(i0), y(i0), z(i0)];

[xyz,Error]=lsqnonlin(@equations, X0,[0,0,0],[1,2*pi,100]);

 function F=equations(X)

  x=X(1); y=X(2); z=X(3);
  
  F(1) = x.*sin((t.*2.*pi.*y/Fs) + z) - y0  ;
  F(2) = x.*sin(((t+1).*2.*pi.*y/Fs)+z) - y1 ;
  F(3) = x.*sin(((t+2).*2.*pi.*y/Fs)+z) - y2; 

 end

end

[Matt J]

You can also use vpasolve()

Fs = 1e6;
f = 5;
t = 2e6;
ang0 = t*2*pi*f/Fs;
dang = 1*2*pi*f/Fs;
y0= 1*sin(t*2*pi*f/Fs);
y1= 1*sin((t+1)*2*pi*f/Fs);
y2= 1*sin((t+2)*2*pi*f/Fs);

%%
syms x y z

[x,y,z]=vpasolve(x*sin((t*2*pi*y/Fs) + z)==y0  ,...
                 x*sin(((t+1)*2*pi*y/Fs)+z)==y1,...
                 x*sin(((t+2)*2*pi*y/Fs)+z)==y2)

x = 

y = 

6.0380495220795251235306856371589

z = 

1221.6013978502790085715024981273

[Minerva Bionic]

Thanks a lot.

But, why does it ignore assumptions when I add them?

Another thing; If I define 2 equations and declare y as 5 it can find the answer properly but when I declare y as a variable and define 3 equations it can not find proper answers (as you know proper answers are : x=1, y=5, z=0).

clear all;

%%
clc
Fs = 1e6;
f = 5;
t = 2e6;

y0= 1*sin(t*2*pi*f/Fs);
y1= 1*sin((t+1)*2*pi*f/Fs);
y2= 1*sin((t+2)*2*pi*f/Fs);

%%
syms x y z
assume(x > 0)
assumeAlso(x <= 1)
assume(z >= 0)
assumeAlso(z <= 2*pi)
assume(y > 0)
assumeAlso(y <= 10)
[x,y,z]=vpasolve(x*sin((t*2*pi*y/Fs) + z)==y0  ,...
                 x*sin(((t+1)*2*pi*y/Fs)+z)==y1 , ...
                 x*sin(((t+2)*2*pi*y/Fs)+z)==y2)

x = 

y = 

6.0380495220795251235306856371589

z = 

1221.6013978502790085715024981273

clear all;

%%
clc
Fs = 1e6;
f = 5;
t = 2e6;
y0= 1*sin(t*2*pi*f/Fs);
y1= 1*sin((t+1)*2*pi*f/Fs);
y2= 1*sin((t+2)*2*pi*f/Fs);

syms x z
assume(x > 0)
assumeAlso(x < 1)
assume(z >= 0)
assumeAlso(z <= 2*pi)
y = 5;
[x,z]=vpasolve(x*sin((t*2*pi*y/Fs) + z)==y0  ,...
                 x*sin(((t+1)*2*pi*y/Fs)+z)==y1)

x = 

1.0000000003041361573960955992907

z = 

[Matt J]

as you know proper answers are : x=1, y=5, z=0

No, as you saw above with my lsqnonlin solution, other solutions are also possible.

But, why does it ignore assumptions when I add them?

According to the documentation, vpasolve ignores assumptions, but it does have a SpecifyRanges option

https://www.mathworks.com/help/symbolic/sym.vpasolve.html#bt51ety-1_1

And, of course, lsqnonlin allows you to specify bounds, as I demonstrated for you.

[Minerva Bionic]

Thank you Matt J.

I will look for different solutions.

However, it is appreciated if you could name other possible solutions that may work fine with my problem.

kind regards.

[Matt J]

You mean aside from lsqnonlin and vpasolve? But we've shown that they work. What would an additional suggestion give you?

[Minerva Bionic]

I know "least square nonlinear" and vpasolve methods are great in solving equations. But since I'm curious about another possible methods, I thought it would be fine if I ask you for other possible methods.

[Minerva Bionic]

As I realized I should provide arrays of data for x, y, z for lsqnonlin method.

The solve method can return symbolic solutions. Since I want to implement the solution in a hardware, solve method is the best method in my case.

However, I still have problem with numerical vpasolve() method

clear all
clc

Fs = 1e6;
f = 5;
t = 2e6;

y0= 1*sin(t*2*pi*f/Fs);
y1= 1*sin((t+1)*2*pi*f/Fs);
y2= 1*sin((t+2)*2*pi*f/Fs);

% Available information for solving equations: y0,y1,y2, Fs, t
% Answers: x=1 , y=5, z=0

syms x y z

[x,y,z]=vpasolve(x*sin((t*2*pi*y/Fs) + z)==y0  ,...
                 x*sin(((t+1)*2*pi*y/Fs)+z)==y1 , ...
                 x*sin(((t+2)*2*pi*y/Fs)+z)==y2,[x y z],[0 1.2; 0 10; -pi pi])

x = 

0.82808197968524438542250921861996

y = 

6.0380495220795251235306856358895

z = 

%% Acceptable answers by reducing variables         
clc
syms x z
y= 5;
[x,z] = vpasolve(x*sin((t*2*pi*y/Fs) + z)==y0  ,...
                 x*sin(((t+1)*2*pi*y/Fs)+z)==y1,[x z],[0 1.2; -pi pi])

x = 

1.0000000003041361573960955992907

z =