## Watson/chem_master1

chemical master eqn, aij*h = prob of i->j transition in time h (Markov model)
Name chem_master1 Watson 1854 40,401 40,401 201,201 201,201 2D/3D Problem No 2007 L. Watson and J. Zhang T. Davis
Structural Rank 40,401 true 1 1 0 100% 0% no no real
Download 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.