Solution for integral_0^t exp(A(s))x ds?

[uekstrom]

Dear all,
do you know how to best calculate

$q(t) = \int_0^t \exp(A(s))v_0 ds$

where $A(s)$ is a low order matrix polynomial in s, and v_0 is a constant vector of suitable dimension? I can of course use a general ODE solver, but I want to understand how small perturbations in A affect the final result. In particular I want to use something like
$A(s) = A_1(1-s) + A_2s + A_{12}s(1-s),$
where the different A matrices are antisymmetric and do not generally commute (if they did commute the problem would not be very difficult).

 

[gtoguy97]

The best way to calculate this integral depends on the type of matrix polynomial you have. In the case of a low order matrix polynomial, it may be possible to use a numerical quadrature method such as Gaussian quadrature or Simpson's rule to approximate the integral. Alternatively, if you have an analytical form for A(s) then you could use a symbolic integration package such as Mathematica or Maple to obtain an exact solution. If you are interested in understanding how small perturbations in A affect the final result, then you could use a perturbation theory approach to understand how the integral changes when the parameters of A(s) are varied.