System of Integral equation - Numerical Solving I have those equations:
$$
X_0=\int_a^b K_0(x)f(x) \,dx \\
X_1=\int_a^b K_1(x)f(x) \,dx \\
X_2=\int_a^b K_2(x)f(x) \,dx
$$
$X_0,X_1,X_2$ are from measurement and $K_0,K_1,K_2$ too (numerical value inside $[a,b]$).
That look like those equations could have infinite possibilities. I would like to find some solutions, numerically which respect the constraint:
$$
\forall \ a\le x\le b,\  f(x)\ge0\
$$
Or at least find a method to generate valid solution from where I can do a manual selection.
I want to find $f(x)\ \forall\ a\le x\le b$.
$K_i$ are continious and surjective but could be $=0$.
And $X_i \ge 0$
Thanks
 A: This problem is very ill-conditioned even in term of numerical solutions (it has the numerical tag). We are trying to reconstruct $\vec{f} = [f_0, f_1, ..., f_n]$ given on the grid of points $\vec{x} = [x_0, x_1, ..., x_n]$, $a \leq x_0 < x_n \leq b$. The numerical integration schemes will provide vectors $\vec{W_i} = [w_0, w_1, ..., w_n]$, which are integration weights for $X_i$ and the numerical integral can be calculated from $X_i = \vec{W_i} \cdot \vec{f}$. Unless your function is evaluated on three points, it will generate any number of solutions. In particular, assume that $f(x) = \sum a_i \delta(x_i) $, then $a_i = X_i / K_i(x_i)$ is a solution, as long as all the $x_i$ are different and $a \leq x_i \leq b$ for all $x_i$. 
A much better solution would be to assume some form of function for $f(x)$, say a polynomial, and then fit it to solve your integral equations. Given that your data is from measurements, it should also have noise which would cause trouble in finding $f(x)$ without a fitting procedure. 
EDIT : corrected subscript appearing on $\delta(x_i)$, it is a Dirac delta function, so that $\int K(x) \delta(x_i) = K(x_i)$. There are a number of comments on the fact that your system has an infinite amount of solutions. In the numerical case, you can find at most three values of $f(x)$ since you have 3 constraints. 
If you can assume some form for $f(x)$ with $N$ parameters and $N \leq 3$, you can use standard regression techniques to fit the parameters for your values.  
