Calculating the Expopential of Matrices

Calculating the Expopential of Matrices

Postby Cuchulainn » Sat Aug 25, 2007 10:23 am

When time-dependent PDEs are discretised in space we get a ODE system whose solution is of the form



U(t) = exp(-At)U(0)



Now, the exponential terms must be evaluated. There are 19 ways...



http://www.cs.cornell.edu/cv/ResearchPDF/19ways+.pdf
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Postby Cuchulainn » Mon Aug 27, 2007 8:43 am

I would be interested in your opinion on the following special matrix A and then calculate exp(-At) (no integral for the moment). I have not coded it and do not have MM. It's probably very easy in MM.



The matrix A is (for moment) constant coefficient(ed). In particular it is tridiagonal and Toeplitz, meaning that it has 3 diagonals and each diagonal (sub, mid, low) has constant value.



SUB: the value is a - b where a and b are arbitrary real numbers

MID: value is -2a

LOW: value is a + b



Note A is not symmetric and may have complex eigenvalues.



///

background

If you semi-discretise



U_t = aU_xx + BU_x



using centred FD in x you get an ODE



V_t = AV



with solution V(t) = V(0) exp(A)





Most times we use Crank Nicolson which is just a Pade(1,1) O(dt^2) approximation.



Any ideas?
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