Matrix: Watson/Baumann

Description: chemical master eqn, aij*h = prob of i->j transition in time h (Markov model)

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• Matrix group: Watson
• download as a MATLAB mat-file, file size: 3 MB. Use UFget(1855) or UFget('Watson/Baumann') in MATLAB.

 Matrix properties number of rows 112,211 number of columns 112,211 nonzeros 748,331 structural full rank? yes structural rank 112,211 # of blocks from dmperm 2 # strongly connected comp. 2 explicit zero entries 12,300 nonzero pattern symmetry symmetric numeric value symmetry 0% type real structure unsymmetric Cholesky candidate? no positive definite? no

 author L. Watson and J. Zhang editor T. Davis date 2007 kind 2D/3D problem 2D/3D problem? yes

Notes:

The ODE system \frac{dp}{dt}=Qp is what we call a chemical master equation (a
Kolmogorov's backward/forward equation). Q is a sparse asymmetric matrix,
whose off-diagonal entries are non-negative and row sum to zero. On each row,
q_{ij}h gives the probability the system makes a transition from current state
i to some other state j, in small time interval h. By "state", we mean a
possible combination of number of molecules in each chemical species. Now, h
is small enough so that only one reaction happens.  In this way q_{ij} is
nonzero only if there exists a chemical reaction that connects state i and j,
i.e. j=i+s_k, s_k's are constant state vectors that correspond to every
reaction.  Say we have M reactions, then there are at most M+1 nonzero
entries on each row of Q.  On the other hand, the number of possible
combination of molecules is huge, which means the dimension of Q is huge.
The linear system we want to solve is (I - Q/lambda)x=b, and we have
to solve it several times. (Here lambda is a constant).  Problem.A is the Q
matrix.  This is a medium test problem; the largest has dimension 10^8.
It has the nonzero pattern of a 11-by-101-by-101 mesh.


 Ordering statistics: result nnz(chol(P*(A+A'+s*I)*P')) with AMD 26,112,094 Cholesky flop count 2.7e+10 nnz(L+U), no partial pivoting, with AMD 52,111,977 nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD 61,376,869 nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD 110,190,364

Note that all matrix statistics (except nonzero pattern symmetry) exclude the 12300 explicit zero entries.