**Matrix: Watson/chem_master1**

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

(undirected graph drawing) |

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 |

For a description of the statistics displayed above, click here.

*Maintained by Tim Davis, last updated 12-Mar-2014.Matrix pictures by cspy, a MATLAB function in the CSparse package.
Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.
*