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

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;

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

• Magnus expansion requires $\large A\left(t\right)A\left(t'\right) = A\left(t'\right)A\left(t\right)\,,\quad\forall\ t,t'$. Jan 18, 2014 at 20:50
• This situation always holds when $A(t)$ is constant coefficient matrix. So, can we solve ODE's by using magnus expansion with constant coefficient matrix?
– drxy
Jan 18, 2014 at 22:34
• I agree. Thanks. Jan 19, 2014 at 0:00
• @Amzoti thank you for answer. I found useful informations in this paper.
– drxy
Jan 19, 2014 at 16:14
• The use of the magnus expansion does not require that property at all. Aug 6, 2015 at 15:17

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.

• $Y_{n+1}$ is column vector for approximation values by getting with magnus expansion method. As you say above, my formula gives $Y_{n+1}=exp(hA)Y_n$. it is right, but it is giving very small errors with constant coefficient matrix. So is my approach is true or false for getting constant coefficient matrix for using magnus expansion to solve ODE's. ?
– drxy
Jan 18, 2014 at 22:27
• I believe your approach is correct, however i do not know rate of convergence for this method. Jan 19, 2014 at 5:37