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I recently encountered the following equation [itex]C(t) = a\int_0^t{x(\tau)d\tau} + b\int_0^t{\int_0^\tau{x(\tau')d\tau'd\tau}}[/itex], 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:

[itex]C(t) = a\Sigma_0^t{x} + b\Sigma_0^t\Sigma_0^t{x}[/itex].

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: [itex]C = aAx + bBx[/itex] 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.

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

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