Matrix: Watson/chem_master1

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

Watson/chem_master1 graph
(undirected graph drawing)


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  • download as a MATLAB mat-file, file size: 1 MB. Use UFget(1854) or UFget('Watson/chem_master1') in MATLAB.
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    Matrix properties
    number of rows40,401
    number of columns40,401
    structural full rank?yes
    structural rank40,401
    # of blocks from dmperm1
    # strongly connected comp.1
    explicit zero entries0
    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

    Additional fieldssize and type
    bfull 40401-by-1


    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 AMD1,052,209
    Cholesky flop count1.1e+08
    nnz(L+U), no partial pivoting, with AMD2,064,017
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD1,842,081
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD3,377,747

    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.