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

• uekstrom
In summary: Overall, the most appropriate method for calculating this integral will depend on the specific characteristics of your matrix polynomial.In summary, the best way to calculate the integral q(t) = \int_0^t \exp(A(s))v_0 ds depends on the type of matrix polynomial being used. It may be possible to use a numerical quadrature method or a symbolic integration package for an analytical solution. For understanding perturbations in A, a perturbation theory approach could be used. The most suitable method will vary depending on the characteristics of the matrix polynomial.
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).

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.

## 1. What is the meaning of "exp(A(s))" in the integral?

The term "exp(A(s))" represents the exponential of the matrix A(s). This means that each element of the matrix A(s) is raised to the power of e, the base of the natural logarithm.

## 2. How do you solve an integral with a matrix inside?

To solve an integral with a matrix inside, you can use the power series expansion of the exponential function. This involves expressing the matrix A(s) in terms of its eigenvalues and eigenvectors, and then using the power series to solve for the integral.

## 3. What are the applications of this integral in science?

This integral has various applications in fields such as physics, engineering, and mathematics. It is commonly used in studying systems that involve time-varying matrices, such as in control theory and quantum mechanics.

## 4. Is there a general formula for solving this type of integral?

There is no one general formula for solving this type of integral, as it depends on the specific matrix A(s) and the limits of integration. However, there are various techniques and methods that can be used to solve these integrals, such as the power series method mentioned earlier.

## 5. Can this integral be solved numerically?

Yes, this integral can be solved numerically using numerical integration methods such as the trapezoidal rule or Simpson's rule. These methods involve dividing the integration interval into smaller subintervals and approximating the integral using the values of the function at specific points within each subinterval.

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