Solve for x, a and b in matrix equation aAx + bBx = C

[Panteren]

Hello everybody

I recently encountered the following equation $C(t) = a\int_0^t{x(\tau)d\tau} + b\int_0^t{\int_0^\tau{x(\tau')d\tau'd\tau}}$, where C, a, b and x are greater or equal to zero. C and x are vectors - in my case around 3500 long - and a and b are constants.

If we take sufficiently small steps we can replace the integrals with summations:
$C(t) = a\Sigma_0^t{x} + b\Sigma_0^t\Sigma_0^t{x}$.
Such a summation can also be written as a matrix of the form [1 0 0; 1 1 0; 1 1 1] etc. using Matlab notation and [1 0 0; 2 1 0; 3 2 1] etc. for det double summation.

Now we have a system: $C = aAx + bBx$ where C and x are Nx1 matrices, a and b are constants, and A and B are NxN matrices. I wish to solve it in some least norm sense for x, a and b, with the constraints that x, a and b should be equal to or greater than zero.

I have tried to solve the first equation using some of the nonlinear optimization tools in Matlab with poor results. I hoped it would be easier to solve when rewritten as a linear system, but I cannot see how.

Any suggestions would be most welcome.

 

[meldraft]

I suppose τ' is a different τ? Like saying τ₁ and τ₂?

If so, then the inner integral is equal to x(τ₂), which yields:

$$C(t)= a\Sigma_0^t{x} + b\Sigma_0^t{x(τ_2)}$$

Edit: Nevermind, I thought it was x'(τ')

 

[meldraft]

Ok, I found what was bugging me:

You replaced the double integral with two sums from 0-t. However, the inner sum should be from τ'=0 to τ'=τ, and the outer sum should be τ=0 to τ=t

 

[Panteren]

Ok, I found what was bugging me:

You replaced the double integral with two sums from 0-t. However, the inner sum should be from τ'=0 to τ'=τ, and the outer sum should be τ=0 to τ=t

Thank you. You are correct. That is what I meant. Equivalent to using two 'cumsum' in Matlab.

 

[I like Serena]

Welcome to PF, Panteren!

I take it your system is actually the following?
$$C(t_i) = aAx(t_i) + bBx(t_i)$$

In that case the solution in a least-norm-sense is given by a least-squares solution.

What you'd do is minimize $\sum_i (C(t_i) - aAx(t_i) + bBx(t_i))^2$, which is the sum-squared deviation given a certain a and b.
To solve it you'd calculate the partial derivatives to a and also to b and set them to zero.

You'll find the system of equations:
$$a \sum_i (Ax(t_i))^2 + b \sum_i Ax(t_i) \cdot Bx(t_i) = \sum_i C(t_i) \cdot Ax(t_i)$$
$$a \sum_i Ax(t_i) \cdot Bx(t_i) + b \sum_i (Bx(t_i))^2 = \sum_i C(t_i) \cdot Bx(t_i)$$

Its solution (for a and b) appears to be what you want.

 

[Panteren]

Welcome to PF, Panteren!

Thank you. I have read the forums for quite some time. Lots of interesting stuff and insight to be found.

I take it your system is actually the following?
$$C(t_i) = aAx(t_i) + bBx(t_i)$$

Yes, but I am not sure I understand the distinction between that and what I wrote? Please elaborate what I have misunderstood or stated unclear :-/?

You'll find the system of equations:
$$a \sum_i (Ax(t_i))^2 + b \sum_i Ax(t_i) \cdot Bx(t_i) = \sum_i C(t_i) \cdot Ax(t_i)$$
$$a \sum_i Ax(t_i) \cdot Bx(t_i) + b \sum_i (Bx(t_i))^2 = \sum_i C(t_i) \cdot Bx(t_i)$$

Its solution (for a and b) appears to be what you want.

Thank you, but the problem is, that $x$ is also unknown. Perhaps it is obvious how to get that in addition to $a$ and $b$ from the system of equations, but I do not follow :-(
If I had $x$ I could just turn it into a standard linear regression problem and likewise if I had the constants $a$ and $b$, but when I only have $A$, $B$, $C$ and the non-negativity constraints on $a$, $b$ and $x$ ... ?