Solving over determined algebraic system in MATALB

Hi I am trying to solve over determined system which is algebraic. Below is the over determined system explained.

u=linspace(0,2,30)

for i=1:30

v(i,:)=2+tan(u(i)/(1+u(i).^2))

end

f=((a-b+c+d)*(a-b-c+d)*(u^2)*(v^2))+((a+b-c+d)*(a+b+c+d)*(u^2))+((a+b-c-d)*(a+b-c-d)*(a+b+c-d)*(v^2))-(8*a*b*u*v)+((a-b+c-d)*(a-b-c-d))

there are 30 values of u and 30 values of v but how I will be able to find a,b and c if we put d as 1

[Torsten]

You want to determine a, b and c such that f(i,j)=0 approximately for all given pairs (u(i),v(j)) ?

[Abdullah Nasir]

Yes exactly I need to determine a, b and c such that f(i,j)=0 approximately for all given pairs (u(i),v(j))

Thank you

[Walter Roberson]

Do you have the curve fitting toolbox?

[Abdullah Nasir]

Yes I have the curve fitting toolbox but I have never used it

Thank you

[Walter Roberson]

What are your inputs? Do you have one known f value for each u value?

Answer 1

[Walter Roberson]

%it is important that the independent variable, u, be last for curvefit purposes

fun = @(a,b,c,u) u.^2*(a + b + c + 1).*(a + b - c + 1) - (a - b + c - 1).*(b - a + c + 1) + (tan(u./(u.^2 + 1)) + 2).^2.*(a + b + c - 1).*(a + b - c - 1).^2 - 8.*a.*b.*u.*(tan(u./(u.^2 + 1)) + 2) + u.^2.*(tan(u./(u.^2 + 1)) + 2).^2.*(a - b - c + 1).*(a - b + c + 1);

fitobj = fittype(fun, 'independent', 'u');

u = linspace(0,2,30);
results = fit(u(:), known_f(:), fitobj)

I did not have any known f values to work with, so I used rand(). The coefficients I got back passed through 0, which typically indicates that you cannot trust the results at all. But you could potentially get better results with your actual known f values.

[Abdullah Nasir]

Hi Sorry I need to determine a, b and c such that f(i,j)=0 approximately for all given pairs (u(i),v(j)). (f) it the function.

Thank you

[Walter Roberson]

Ah, in that case use known_f = zeros(size(u));

results = 
     General model:
     results(u) = u.^2*(a+b+c+1).*(a+b-c+1)-(a-b+c-1).*(b-a+c+1)+(tan(u./(u.^2+
                    1))+2).^2.*(a+b+c-1).*(a+b-c-1).^2-8.*a.*b.*u.*(tan(u./(u.^2+
                    1))+2)+u.^2.*(tan(u./(u.^2+1))+2).^2.*(a-b-c+1).*(a-
                    b+c+1)
     Coefficients (with 95% confidence bounds):
       a =     -0.3037  (-0.3283, -0.2791)
       b =      0.4132  (0.4004, 0.4261)
       c =      0.6249  (0.6205, 0.6294)

Your v values are not givens: they are calculated from your u values in your loop at the beginning.

[Abdullah Nasir]

By the way is there any other way to solve this type of problem without using the curve fitting tools. This problem states as non-linear least square problem.

Thank you

[Walter Roberson]

Curvefit uses nonlinear least squares when it is handed a function handle, as is the case we constructed here.

https://www.mathworks.com/help/curvefit/fittype.html#btpxtcd-5

[Walter Roberson]

But if you want, then use https://www.mathworks.com/help/optim/ug/lsqnonlin.html

Note, though, that fit() (as above) expects the function to return a predicted value, and fit() itself subtracts off the actual value and constructs sum of squares of those. But lsqnonlin expects instead that the function already have subtracted off the actual value (but expects a vector output and it will calculate the sum of squares of those.)