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I have an initial value problem (henceforth IVP) as follows: $$\frac{d \Phi(t)}{dt}= A(t)\Phi(t)$$ subject to the initial condition $\Phi(t_0)=I$, where $\Phi(t), A(t), I$ are square matrices of same order and $t_0$ is the initial time. $I$ is the identity matrix.

Runge-Kutta method says one require the initial condition to be a vector. In my case it is a identity matrix $I$.

So $(a)$ is it 4th order Runge-Kutta method applicable here $?$ $(b)$ Is there any other way (Numerical method/Scheme)to solving the given IVP $?$

$\bullet$ Thanks in advance!

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Just convert your matrix to a long vector. Or write your equation componentwise. You typically start with RK4 and then see if the solution is good. If not then look for more specialized methods. –  Maesumi Aug 21 '13 at 21:16
    
@ Maesumi. I have atleast 1000 order matrix. i.e., size of $A(t)$ is 1000 or more. Can you please suggest how the converting to long vector will help means any example. Thanks. –  Tapan Aug 21 '13 at 21:20
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1 Answer 1

Split your $\Phi$ and $I$ into columns, so \begin{align} \Phi &= \left [ \phi_1, \phi_2, \ldots, \phi_n\right ] \\ I &= \left [e_1, e_2, \ldots, e_n \right ] \end{align} where $\phi_i, e_i \in \mathbb R^n,\ \forall i$.

So, instead one matrix system you'll get $n$ linear systems $$ \frac {d \phi_i}{dt} = A(t) \phi_i(t) \\ \phi_i(t_0) = e_i $$ for which you can apply regular RK4.

You also can do it simultaneously, since matrix multiplication is row-to-column based, i.e. $$ A(t) \Phi (t) = \left [ A \phi_1(t), A \phi_2(t), \ldots, A \phi_n(t)\right ] $$

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@ Kaster, Thanks. It seems to be a brilliant idea. Let me try to my system and let you know. –  Tapan Aug 22 '13 at 4:53
    
@ Kaster, your idea worked. Thank you. –  Tapan Feb 15 at 10:49
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