How do you solve for a Double summation with loops

Hi everyone,

I need to calculate and plot this double sum with loops.

where are the roots of:

here is my code but, it doesn't work I think I created the wrong roots function. maybe!

function [ uvect, yvect, zvect ]=F(tvec,n,sstep, B)
if nargin <4
   sstep =1;
end
if nargin <3
   n =1000;
end
if nargin <2
   B =0.1;
end
if nargin ==0
   tvec =[0 0.0001 0.001 0.01 0.1 1];
end
  yvect = linspace(0,1,101);
  zvect =linspace(0,1,101);
  [ aivec, ajvec, aijvec ]=aiajroots(n,sstep,B);
for it =1: length ( tvec )
   t= tvec (it);
   uvect =0;
for i=1:n
   ai=aivec(i);
   for j=1:n
   aj =ajvec(j);
   aij=aijvec(j);
   coef =((sin(ai)*sin(aj))/(aij*(1+(sin(ai)^2)/B)*(1+(sin(aj)^2)/B)))*exp(-aij* t);
   uvect = uvect + coef *(cos(ai*yvect)*cos(aj*zvect));
  end
end
end
hold off

and the roots function is

function [ aivec,ajvec, aijvec]=aiajroots(n,sstep,B)
if nargin ==0
  n =20;
  sstep =1;
elseif nargin ==1
  sstep =1;
end
B=0.1;
   k=0; i=0; 
while k<n 
   i1=i+ sstep ;
   f1=@(y) y*tan(y)-B;
   if f1(i)*f1(i1) <0 
        k=k+1;
        ai= fzero (f1 ,[i i1 ]);
        aj=fzero (f1 ,[i i1 ]);
        aivec(k)=ai;
        ajvec(k)=aj;
        aijvec(k)=ai^2+aj^2;
   end
   i=i1;
end  
end

There must be a problem in my code but I don't know what happened. I'm a newbie, somebody please help me. thanks a lot advance!

[Jan]

Please mention, what the problem is. "it doesn't work" is not clear enough. Remember, that it is easier to solve a problem than to guess, what the problem is.

Are the default inputs really useful for this kind of program?

if nargin ==0
  n =20;
  sstep =1;
elseif nargin ==1
  sstep =1;
end

I'd omit this fancy tricks. But if aiajroots() accepts a 3rd input B, do not overwrite it in the code.

This looks suspicious:

        ai= fzero (f1 ,[i i1 ]);
        aj=fzero (f1 ,[i i1 ]);        

I recommend to use a standard scheme for setting spaces: Spaces around operator and after commas, separators between elements of vectors. Then similarities become more obvious:

        ai = fzero(f1, [i, i1]);
        aj = fzero(f1, [i, i1]);  % twice?!        

Creating the function handle inside the loop is not an error, but a waste of time. Move all code, which does not depend on the loop counter before the loop:

        f1 = @(y) y * tan(y) - B;

[Mery]

"it doesn't work" , I have this erreur

Error using : Inner matrix dimensions must agree.

"% twice?!" aj and ai verify the same equation

[Mery]

@Jan

Can you tell me please, if the writing of the double summation in F(tvec,n,sstep, B)

its true or not ?

[Jan]

@Mery: Please post a copy of the complete error message. Then the readers do not have to guess, in which line the error occurs.

If ai and aj have the same values, there is no need to calculate them both.

A simplified version of your root finding method:

function [aivec, ajvec, aijvec] = aiajroots(n, sstep, B)
f = @(y) y * tan(y) - B;
k = 0;
i = 0;
aivec = zeros(1, n-1);  % Pre-allocate
while k < n
   i1 = i + sstep;
   if f(i) * f(i1) < 0 
        k = k + 1;
        aivec(k) = fzero(f, [i, i1]);
   end
   i = i1;
end
ajvec  = aivec;
aijvec = aivec.^2 + ajvec.^2;
end

Answer 1

[Jan]

If yvect and zvect are both [1 x 101] vectors, this line must fail:

uvect = uvect + coef * (cos(ai * yvect) * cos(aj * zvect));
%                                       ^ here

Here you multiply 2 row vectors, which is not defined mathematically.

[Mery]

yes, is exactly the error

Error using *

Inner matrix dimensions must agree.

Error in F (line 26)

uvect = uvect + coef *(cos(ai*yvect)*cos(aj*zvect));

[Jan]

@Mery: Okay. I've written the explanation already: The multiplication of twi [1 x n] vectors is not defined. Now you have to explain, what you expect as result.