Matrix: Watson/Baumann

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

Watson/Baumann graph
(undirected graph drawing)

Watson/Baumann dmperm of Watson/Baumann

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  • download as a MATLAB mat-file, file size: 3 MB. Use UFget(1855) or UFget('Watson/Baumann') in MATLAB.
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    Matrix properties
    number of rows112,211
    number of columns112,211
    structural full rank?yes
    structural rank112,211
    # of blocks from dmperm2
    # strongly connected comp.2
    explicit zero entries12,300
    nonzero pattern symmetrysymmetric
    numeric value symmetry 0%
    Cholesky candidate?no
    positive definite?no

    authorL. Watson and J. Zhang
    editorT. Davis
    kind2D/3D problem
    2D/3D problem?yes


    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 AMD26,112,094
    Cholesky flop count2.7e+10
    nnz(L+U), no partial pivoting, with AMD52,111,977
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD61,376,869
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD110,190,364

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

    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.