Matrix: Watson/chem_master1

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

 (undirected graph drawing)

• Matrix group: Watson
• download as a MATLAB mat-file, file size: 1 MB. Use UFget(1854) or UFget('Watson/chem_master1') in MATLAB.

 Matrix properties number of rows 40,401 number of columns 40,401 nonzeros 201,201 structural full rank? yes structural rank 40,401 # of blocks from dmperm 1 # strongly connected comp. 1 explicit zero entries 0 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

 Additional fields size and type b full 40401-by-1

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 small test problem; the largest has dimension 10^8.
It has the nonzero pattern of a 201-by-201 mesh, minus 300 entries on the
+1 and -1 diagonal.


 Ordering statistics: result nnz(chol(P*(A+A'+s*I)*P')) with AMD 1,052,209 Cholesky flop count 1.1e+08 nnz(L+U), no partial pivoting, with AMD 2,064,017 nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD 1,842,081 nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD 3,377,747