? approximate ODE soln

Hi:

Given an ODE dx/dt = A*x driven by initial vector x0; suppose

A is a constant symmetric matrix, its soln is expressed as follows.

x(t) = exp( A*t )*x0 = V*exp( D*t )*V'*x0, where A = V*D*V'.

This allows one to determine approximated soln at each time once one

determined the first eigenvectors and corresponding eigenvalues of A.


This approach does not apply when A is not symmetric. Though one can

always find A = U*S*V' via singular value decomposition, this does not

help one to decompose exp( A*t ).

In practice, however, engineers and physicists do want to find some

way to quickly determine some approximated soln of dx/dt = A*x, when

A is not symmetric or even time dependent. Does anyone know any approach

to reach this?

Thank you,
by Cheng Cosine
Jul/18/2k7 NC


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