Is it true for solving differential equations by getting constant coefficient matrix with magnus expansion

Question

The magnus expansion is given in detail http://en.wikipedia.org/wiki/Magnus_expansion. While implementing magnus expansion to differential equations we have an iteration formula as follows $$Y'(t) = A(t)Y(t) $$ $$A_1 =A(t_n + h(\frac{1}{2} - \frac{\sqrt3}{6})) $$ $$A_2 =A(t_n + h(\frac{1}{2} + \frac{\sqrt3}{6})) $$ $$ \sigma = \frac{1}{2}h[A_1+A_2] - \frac{\sqrt3}{12}h^2[A_1,A_2] + \frac{1}{80}h^3[A_1-A_2,[A_1,A_2]] $$ $$Y_{n+1} = e^\sigma Y_n$$

where $[.,.]$ is Lie bracket. My question is that when i get $A$ matrix with only constants, is it true for this iteration formula ? For instance;

$$A(t)=\begin{pmatrix} -100 & 1 \\ 1 & -1000 \\ \end{pmatrix}$$

or every time must we get variable coefficients matrix? For instance;

$$A(t)=\begin{pmatrix} -100t & 1 \\ 1 & -1000 \\ \end{pmatrix}$$

Thanks in advance.

Answer 1

I am a little bit confused about your ``iteration formula''. You mean that $Y_n=Y(t_n)$ and $t_1, ..., t_n$ are the points?

Well, If so then you do not need to ``insert'' any variables. In that case $Y=exp(tA)Y_0$ and that is exactly what your formula gives.