## Calculating the Expopential of Matrices

### Calculating the Expopential of Matrices

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 Cuchulainn

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Joined: Mon Dec 18, 2006 2:48 pm
Location: Amsterdam, the Netherlands

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? Cuchulainn

Posts: 676
Joined: Mon Dec 18, 2006 2:48 pm
Location: Amsterdam, the Netherlands

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