Computing the block triangular form of a sparse matrix

We consider the problem of permuting the rows and columns of a rectangular or square, unsymmetric sparse matrix to compute its block triangular form. This block triangular form is based on a canonical decomposition of bipartite graphs induced by a maximum matching and was discovered by Dulmage and Mendelsohn. We describe implementations of algorithms to compute the block triangular form and provide computational results on sparse matrices from test collections. Several applications of the block triangular form are also included

Computational linear algebra over finite fields

We present here algorithms for efficient computation of linear algebra problems over finite fields

Sensitivity of Markov chains for wireless protocols

Network communication protocols such as the IEEE 802.11 wireless protocol are currently best modelled as Markov chains. In these situations we have some protocol parameters $\alpha$, and a transition matrix $P(\alpha)$ from which we can compute the steady state (equilibrium) distribution $z(\alpha)$ and hence final desired quantities $q(\alpha)$, which might be for example the throughput of the protocol. Typically the chain will have thousands of states, and a particular example of interest is the Bianchi chain defined later. Generally we want to optimise $q$, perhaps subject to some constraints that also depend on the Markov chain. To do this efficiently we need the gradient of $q$ with respect to $\alpha$, and therefore need the gradient of $z$ and other properties of the chain with respect to $\alpha$. The matrix formulas available for this involve the so-called fundamental matrix, but are there approximate gradients available which are faster and still sufficiently accurate? In some cases BT would like to do the whole calculation in computer algebra, and get a series expansion of the equilibrium $z$ with respect to a parameter in $P$. In addition to the steady state $z$, the same questions arise for the mixing time and the mean hitting times. Two qualitative features that were brought to the Study Group’s attention were: * the transition matrix $P$ is large, but sparse. * the systems of linear equations to be solved are generally singular and need some additional normalisation condition, such as is provided by using the fundamental matrix. We also note a third highly important property regarding applications of numerical linear algebra: * the transition matrix $P$ is asymmetric. A realistic dimension for the matrix $P$ in the Bianchi model described below is 8064×8064, but on average there are only a few nonzero entries per column. Merely storing such a large matrix in dense form would require nearly 0.5GBytes using 64-bit floating point numbers, and computing its LU factorisation takes around 80 seconds on a modern microprocessor. It is thus highly desirable to employ specialised algorithms for sparse matrices. These algorithms are generally divided between those only applicable to symmetric matrices, the most prominent being the conjugate-gradient (CG) algorithm for solving linear equations, and those applicable to general matrices. A similar division is present in the literature on numerical eigenvalue problems

Sympiler: Transforming Sparse Matrix Codes by Decoupling Symbolic Analysis

Sympiler is a domain-specific code generator that optimizes sparse matrix computations by decoupling the symbolic analysis phase from the numerical manipulation stage in sparse codes. The computation patterns in sparse numerical methods are guided by the input sparsity structure and the sparse algorithm itself. In many real-world simulations, the sparsity pattern changes little or not at all. Sympiler takes advantage of these properties to symbolically analyze sparse codes at compile-time and to apply inspector-guided transformations that enable applying low-level transformations to sparse codes. As a result, the Sympiler-generated code outperforms highly-optimized matrix factorization codes from commonly-used specialized libraries, obtaining average speedups over Eigen and CHOLMOD of 3.8X and 1.5X respectively.Comment: 12 page