problem with VPASOLVE and FSOLVE

Hi all, I want to estimate four unknowns from four nonlinear equations. The problem with FSOLVE is that while changing the initial conditions, its corresponding values are also changing. I want a solution that shouldn't depend on the initial conditions. If I run the same equations using VPASOLVE without the initial conditions, it gives empty cells. Is there any way to solve this problem efficiently. Any help is appreciated.

Z = xlsread('DataXY.xlsx'); 
m = length(Z(:,1));
X = Z(:,1); Y = Z(:,2); T0 = 28/365;
%%% FSOLVE
f = @(a) [sum(a(2) + a(1) - Y - a(2) .* exp(a(3) .* (T0 - X)));...
    sum((exp(a(3) .* (T0 - X)) - 1) .* (a(2) + a(1) - Y - a(2) .* exp(a(3) .* (T0 - X))));...
    sum(a(2) .* exp(a(3) .* (T0 - X)) .* (T0 - X) .* (a(2) + a(1) - Y - a(2) .* exp(a(3) .* (T0 - X))));...
    sum((a(2) + a(1) - Y - a(2) .* exp(a(3) .* (T0 - X))).^2) - (m.*a(4).^2)];
P1 = fsolve(f,[2 2.5 0.01 0.1]);
%%%VPASOLVE
syms a b c d
eq1 = sum(Y - a -b + b .* exp(c .* (T0 - X)));
eq2 = sum((exp(c .* (T0 - X))-1) .* (b + a - Y - b .* exp(c .* (T0 - X))));
eq3 = sum(b .* exp(c .* (T0 - X)) .* (T0 - X) .* (b + a -Y - b .* exp(c .* (T0 - X))));
eq4 = sum((b + a - Y - b .* exp(c .* (T0 - X))).^2) - (m.*d.^2);
P2 = vpasolve(eq1, eq2, eq3, eq4);

Answer 1

[Matt J]

No, your equations don't have closed-form solutions. In any such situation, you will have to provide an accurate initial guess to get a reliable solution, assuming one exists.

[Deepti Ranjan Majhi]

Thank you @Matt J. I understand that there is no closed-form solution available to this equation. The problem is that I don't have the initial conditions and the exact solution as well. Can you suggest to me what alternative Matlab function can be used here other than fsolve and vpasolve.

[Matt J]

Since you only have 4 unknown variables, you can develop a, approximate initial guess by doing a discrete grid search, e.g., on a 100x100x100x100 grid of points. Your equations are very vectorizable, so this search should be very fast.

[Deepti Ranjan Majhi]

Thank you @Matt J for your humble reply again. Can you little bit elaborate on how to do discrete grid search. Maybe with some example or online literature.

[Matt J]

For example, if you know that your a(i) are all bounded between 0 and 1, you can generate 50x50x50x50 grid arrays with ndgrid as below, and evaluate all your equations at all the grid points in a vectorized fashion

fn=@(q) reshape(q,1,1,1,1,[]);
X = fn(Z(:,1)); Y = fn(Z(:,2)); T0 = 28/365;


[A1,A2,A3,A4]=ndgrid(linspace(0,1,50));
whos A*

  Name       Size                     Bytes  Class     Attributes

  A1        50x50x50x50            50000000  double              
  A2        50x50x50x50            50000000  double              
  A3        50x50x50x50            50000000  double              
  A4        50x50x50x50            50000000  double              

%Evaluate all equations at grid points
fn=@(q) abs(sum(q,4));
F1=fn(  A2 + A1 - Y - A2 .* exp(A3 .* (T0 - X))  );
F2=fn(  (exp(A3 .* (T0 - X)) - 1) .* (A2 + A1 - Y - A2 .* exp(A3 .* (T0 - X)))  );
F3=fn( A2 .* exp(A3 .* (T0 - X)) .* (T0 - X) .* (A2+ A1 - Y - A2 .* exp(A3.* (T0 - X)))  );
F4=fn((A2 + A1 - Y - a(2) .* exp(A3 .* (T0 - X))).^2) - (m.*A4.^2));

[fval,imin]=min(F1(:)+F2(:)+F3(:)+F4(:));  %compute minimizing grid location

a=[A1(imin)  A2(imin) A3(imin) A4(imin)]  %initial guess for FSOLVE

[Deepti Ranjan Majhi]

Thank you very much @Matt J. This is really appreciated. I learned this.