We describe the University of Florida Sparse Matrix Collection, a large and actively growing set of sparse matrices that arise in real applications. The Collection is widely used by the numerical linear algebra community for the development and performance evaluation of sparse matrix algorithms. It allows for robust and repeatable experiments: robust because performance results with artificially generated matrices can be misleading, and repeatable because matrices are curated and made publicly available in many formats. Its matrices cover a wide spectrum of domains, include those arising from problems with underlying 2D or 3D geometry (as structural engineering, computational fluid dynamics, model reduction, electromagnetics, semiconductor devices, thermodynamics, materials, acoustics, computer graphics/vision, robotics/kinematics, and other discretizations) and those that typically do not have such geometry (optimization, circuit simulation, economic and financial modeling, theoretical and quantum chemistry, chemical process simulation, mathematics and statistics, power networks, and other networks and graphs). We provide software for accessing and managing the Collection, from MATLAB™, Mathematica™, Fortran, and C, as well as an online search capability. Graph visualization of the matrices is provided, and a new multilevel coarsening scheme is proposed to facilitate this task.

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An Approximate Minimum Degree ordering algorithm (AMD) for preordering a symmetric sparse matrix prior to numerical factorization is presented. We use techniques based on the quotient graph for matrix factorization that allow us to obtain computationally cheap bounds for the minimum degree. We s h o w that these bounds are often equal to the actual degree. The resulting algorithm is typically much faster than previous minimum degree ordering algorithms, and produces results that are comparable in quality with the best orderings from other minimum degree algorithms.

A new method for sparse LU factorization is presented that combines a column pre-ordering strategy with a right-looking unsymmetric-pattern multifrontal numerical factorization. The column ordering is selected to give a good a priori upper bound on fill-in and then refined during numerical factorization (while preserving the bound). Pivot rows are selected to maintain numerical stability and to preserve sparsity. The method analyzes the matrix and automatically selects one of three pre-ordering and pivoting strategies. The number of nonzeros in the LU factors computed by the method is typically less than or equal to those found by a wide range of unsymmetric sparse LU factorization methods, including left-looking methods and prior multifrontal methods.

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Sparse Gaussian elimination with partial pivoting computes the factorization PAQ = LU of a sparse matrix A , where the row ordering P is selected during factorization using standard partial pivoting with row interchanges. The goal is to select a column preordering, Q , based solely on the nonzero pattern of A , that limits the worst-case number of nonzeros in the factorization. The fill-in also depends on P , but Q is selected to reduce an upper bound on the fill-in for any subsequent choice of P . The choice of Q can have a dramatic impact on the number of nonzeros in L and U . One scheme for determining a good column ordering for A is to compute a symmetric ordering that reduces fill-in in the Cholesky factorization of A T A . A conventional minimum degree ordering algorithm would require the sparsity structure of A T A to be computed, which can be expensive both in terms of space and time since A T A may be much denser than A . An alternative is to compute Q directly from the sparsity structure of A ; this strategy is used by MATLAB's COLMMD preordering algorithm. A new ordering algorithm, COLAMD, is presented. It is based on the same strategy but uses a better ordering heuristic. COLAMD is faster and computes better orderings, with fewer nonzeros in the factors of the matrix.

We discuss the organization of frontal matrices in multifrontal methods for the solution of large sparse sets of unsymmetric linear equations. In the multifrontal method, work on a frontal matrix can be suspended, the frontal matrix can be stored for later reuse, and a new frontal matrix can be generated. There are thus several frontal matrices stored during the factorization and one or more or these are assembled (summed) when creating a new frontal matrix. Although this means that arbitrary sparsity patterns can be handled efficiently, extra work is required to sum the frontal matrices together and can be costly because indirect addressing is required. The (uni-)frontal method avoids this extra work by factorizing the matrix with a single frontal matrix. Rows and columns are added to the frontal matrix, and pivot rows and columns are removed. Data movement is simpler, but higher fill-in can result if the matrix cannot be permuted into a variable-band form with small profile. We consider a combined unifrontal/multifrontal algorithm to enable general fill-in reduction orderings to be applied without the data movement of previous multifrontal approaches. We discuss this technique in the context of a code designed for the solution of sparse systems with unsymmetric pattern.

An ANSI C code for sparse LU factorization is presented that combines a column pre-ordering strategy with a right-looking unsymmetric-pattern multifrontal numerical factorization. The pre-ordering and symbolic analysis phase computes an upper bound on fill-in, work, and memory usage during the subsequent numerical factorization. User-callable routines are provided for ordering and analyzing a sparse matrix, computing the numerical factorization, solving a system with the LU factors, transposing and permuting a sparse matrix, and converting between sparse matrix representations. The simple user interface shields the user from the details of the complex sparse factorization data structures by returning simple handles to opaque objects. Additional user-callable routines are provided for printing and extracting the contents of these opaque objects. An even simpler way to use the package is through its MATLAB interface. UMFPACK is incorporated as a built-in operator in MATLAB 6.5 as x = A\b when A is sparse and unsymmetric.