If you're hell bent on using eigenvectors, consider using triangular systems of eigenvectors (Method 15) since your matrix may be nondiagonalizable, this approach might not be best, but it's better than trying to calculate the eigenvectors and eigenvalues directly (i.e., Method 14). In particular, if you're looking at exponential integrators, you'll want to consider the Krylov subspace methods, and look at papers on exponential integrators (some references are mentioned along with Method 20 in the Moler & van Loan paper).
SIMPLY FORTRAN WITH GOOGLE PLAY CREDIT SOFTWARE
To build on what Jack has said, the standard approach that seems to be used in software (like EXPOKIT, mentioned in your earlier question) is scaling-and-squaring followed by Padé approximation (Methods 2 and 3) or Krylov subspace methods (Method 20). Real*8, dimension(s,s) :: time_indep_master, A, H, vr ! This function (will) compute expm(L*t). ! t is a real*8 value corresponding to time. ! L is a real*8, asymmetric square matrix.
! s is the length of a side of L, which is square. I've become stuck finding the eigenvalues and eigenvectors of L*t. So, which routines should I be calling (and in what order), and should I use the real double versions or the complex double versions? Below is an attempt at doing this with real double versions. Upon looking at the various subroutines I think I should be calling ( ?gehrd, ?orghr, ?hseqr.) it is unclear if it would be simpler to cast the matrix from real*8 to complex*16 and proceed with the complex double versions of these routines, or stick with real*8 and take the hit of doubling the number of my arrays and making a complex matrix of them later. I'm only concerned with time-independence at the moment. In the time-dependent case it will require integration. In the time-independent case, the master equation is solved by exponentiating this matrix.
The result allows me to phrase master equations as real asymmetric matrices. The route to finding the eigenvalues and eigenvectors seems rather convoluted, and I'm afraid I've gotten lost.īackground: Some time ago I asked this question on the theoretical physics SE. Inspired by the success of that question, and after banging my head against a wall for a couple of hours, I'm looking at the matrix exponential of real asymmetric matrices. I recently asked a question along the same lines for skew-Hermitian matrices.
0 kommentar(er)
