Solving equations, Numerical Integration, MSE, best fit overall parameter 'c'

Note: My actual functions are huge, the question uses a simple y=c*x just for illustrative purpose.

With equation y=x*c where 'x' and 'y' are vectors find 'c' a single numeric value that minimizes the sum of squared residues (y-x*c)^2.

This I can do

x = [1.4 1.5 1.6]; y = [0.2 0.31 0.43];
c = 0.1; %initial guess
f = @(c,x) x*c;
cfit = nlinfit(x,y,f,c)

which gives c=0.2217, exactly what I am looking for, all is good. But I really want to have the variable 'y' equal to an integral, like

y = int_0^1 x*c*t^2 dt.

So coded

x = [1.4 1.5 1.6]; y = [0.2 0.31 0.43];
c = 0.1; %initial guess
t=0.1:0.1:1;
f = @(c,x) trapz(t,x*c*t.^2);
cfit = nlinfit(x,y,f,c)

But this does not work.

I do not understand how to use 'trapz' (numerical integration) in this estimation setting.

.......................................

p.s. I get the error message

      ??? Error using ==> nlinfit at 120
      Error evaluating model function '@(c,x)trapz(t,x*c*t.^2)'.

      Error in ==> test88 at 7
      cfit = nlinfit(x,y,f,c)

   Caused by:
       Error using ==> mtimes
       Inner matrix dimensions must agree.

Answer 1

[Andrei Bobrov]

x = [1.4 1.5 1.6]; y = [0.2 0.31 0.43];
c = 0.1; %initial guess
t=0.1:0.1:1;
f = @(c,x) arrayfun(@(x)trapz(t,x.*c.*t.^2),x);
cfit = nlinfit(x,y,f,c)

[john birt]

thanks will look at this

[john birt]

Thank you, it works like a dream!, now to apply this to my huge function and see what happens.

Big thank you. :-)