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Issue 4 | 1999 | pp. 839-1098 †
Issue 3 | 1999 | pp. 563-837
Issue 2 | 1998 | pp. 279-561
Issue 1 | 1998 | pp. 1-278

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1999

Volume 20, Issue 4, pp. 839-1098

† Introduction to the Special Section on Sparse and Structured Matrices and Their Applications

An Efficient Algorithm for a Bounded Errors-in-Variables Model

S. Chandrasekaran, G. H. Golub, M. Gu, and A. H. Sayed

SIAM. J. Matrix Anal. & Appl. 20, pp. 839-859 (21 pages)

Online Publication Date: August 01, 2006

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We pose and solve a parameter estimation problem in the presence of bounded data uncertainties. The problem involves a minimization step and admits a closed form solution in terms of the positive root of a secular equation.

Cutpoint Decoupling and First Passage Times for Random Walks on Graphs

Stephen J. Kirkland and Michael Neumann

SIAM. J. Matrix Anal. & Appl. 20, pp. 860-870 (11 pages) | Cited 3 times

Online Publication Date: August 01, 2006

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One approach to the computations for Markov chains, due to Meyer, is to break a problem down into corresponding computations for several related chains involving a smaller number of states. In this spirit, we focus on the mean first passage matrix associated with a random walk on a connected graph, and consider the problem of transforming the computation of that matrix into smaller tasks. We show that this is possible when there is a cutpoint in the graph and provide an explicit formula for the mean first passage matrix when this is the case.

Partial Orders and the Matrix R in Matrix Analytic Methods

Qi-Ming He

SIAM. J. Matrix Anal. & Appl. 20, pp. 871-885 (15 pages) | Cited 1 time

Online Publication Date: August 01, 2006

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This paper studies the matrix R, which is the minimal nonnegative solution to a nonlinear matrix equation, raised in matrix analytic methods. Based on some partial orders defined on the transition matrix of Markov chains of GI/M/1 type, the monotonicity of the corresponding matrix R and its Perron--Frobenius eigenvalue is investigated. The results are useful in estimating tail probabilities of stationary distributions of Markov chains of GI/M/1 type and constructing upper bounds for the matrix R. Applications to the GI/MAP/1 queue are discussed as well.
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Introduction to the Special Section on Sparse and Structured Matrices and Their Applications

Esmond G. Ng and Daniel J. Pierce

SIAM. J. Matrix Anal. & Appl. 20, pp. 887-887 (1 page)

Online Publication Date: August 01, 2006

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The field of sparse matrices is a broad and important area of the computational sciences that includes structured matrices and those with seemingly little or no structure. The relevance of the field is highlighted by the wide range of application areas that require the exploitation of matrix sparsity and structure in order to achieve a solution given real-world constraints on computing resources and/or time. Applications in which sparse matrices appear include structural analysis, computational fluid dynamics, economic modeling, financial analysis, numerical optimization, statistical modeling, power network analysis, electromagnetics, meteorology, medical imaging, data mining, and many more.
A number of significant advancements in sparse matrix computations have been made in recent years. These advances have led to new challenges as multidisciplinary problems are now ambitiously posed, and they symbolize the growth in the area and demonstrate the dependence of the future of many fields on sparse matrix methods. Moreover, they give rise to a host of new problems that have yet to be addressed by the sparse matrix community.
The Second SIAM Conference on Sparse Matrices, which was held in Coeur d'Alene, Idaho, October 9--11, 1996, was organized specifically to address some of the challenging issues in sparse matrix computations. This special issue of SIAM Journal on Matrix Analysis and Applications consists of some of the outstanding papers that were presented at the Sparse Matrix conference. Because of an error in scheduling, the paper titled ``Using Generalized Cayley Transformations within an Inexact Rational Krylov Sequence Method" by Richard Lehoucq, which should have appeared in this section, was published in SIAM Journal on Matrix Analysis and Applications, volume 20, number 1.
We are grateful to Paul van Dooren, the Editor-in-Chief of SIAM Journal on Matrix Analysis and Applications, and the SIAM office for agreeing to publish a special section on sparse matrices. We would like to thank Roland Freund, Anne Greenbaum, Joseph Liu, and Zdenek Strakos for serving on the Guest Editorial Board; their effort and cooperation in handling the papers were much appreciated. Finally, we would like to express thanks to all the authors who have submitted papers to the special section; it would not have become a reality without their support.

The Design and Use of Algorithms for Permuting Large Entries to the Diagonal of Sparse Matrices

Iain S. Duff and Jacko Koster

SIAM. J. Matrix Anal. & Appl. 20, pp. 889-901 (13 pages) | Cited 41 times

Online Publication Date: August 01, 2006

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We consider techniques for permuting a sparse matrix so that the diagonal of the permuted matrix has entries of large absolute value. We discuss various criteria for this and consider their implementation as computer codes. We then indicate several cases where such a permutation can be useful. These include the solution of sparse equations by a direct method and by an iterative technique. We also consider its use in generating a preconditioner for an iterative method. We see that the effect of these reorderings can be dramatic although the best a priori strategy is by no means clear.

Performance of Greedy Ordering Heuristics for Sparse Cholesky Factorization

Esmond G. Ng and Padma Raghavan

SIAM. J. Matrix Anal. & Appl. 20, pp. 902-914 (13 pages) | Cited 12 times

Online Publication Date: August 01, 2006

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Greedy algorithms for ordering sparse matrices for Cholesky factorization can be based on different metrics. Minimum degree, a popular and effective greedy ordering scheme, minimizes the number of nonzero entries in the rank-1 update (degree) at each step of the factorization. Alternatively, minimum deficiency minimizes the number of nonzero entries introduced (deficiency) at each step of the factorization. In this paper we develop two new heuristics: modified minimum deficiency (MMDF) and modified multiple minimum degree (MMMD). The former uses a metric similar to deficiency while the latter uses a degree-like metric. Our experiments reveal that on the average, MMDF orderings result in 21% fewer operations to factor than minimum degree; MMMD orderings result in 15% fewer operations to factor than minimum degree. MMMD requires on the average 7--13% more time than minimum degree, while MMDF requires on the average 33--34% more time than minimum degree.

An Asynchronous Parallel Supernodal Algorithm for Sparse Gaussian Elimination

James W. Demmel, John R. Gilbert, and Xiaoye S. Li

SIAM. J. Matrix Anal. & Appl. 20, pp. 915-952 (38 pages) | Cited 32 times

Online Publication Date: August 01, 2006

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Although Gaussian elimination with partial pivoting is a robust algorithm to solve unsymmetric sparse linear systems of equations, it is difficult to implement efficiently on parallel machines because of its dynamic and somewhat unpredictable way of generating work and intermediate results at run time. In this paper, we present an efficient parallel algorithm that overcomes this difficulty. The high performance of our algorithm is achieved through (1) using a graph reduction technique and a supernode-panel computational kernel for high single processor utilization, and (2) scheduling two types of parallel tasks for a high level of concurrency. One such task is factoring the independent panels in the disjoint subtrees of the column elimination tree of $A$. Another task is updating a panel by previously computed supernodes. A scheduler assigns tasks to free processors dynamically and facilitates the smooth transition between the two types of parallel tasks. No global synchronization is used in the algorithm. The algorithm is well suited for shared memory machines (SMP) with a modest number of processors. We demonstrate 4- to 7-fold speedups on a range of 8 processor SMPs, and more on larger SMPs. One realistic problem arising from a 3-D flow calculation achieves factorization rates of 1.0, 2.5, 0.8, and 0.8 gigaflops on the 12 processor Power Challenge, 8 processor Cray C90, 16 processor Cray J90, and 8 processor AlphaServer 8400.

An Object-Oriented Approach to the Design of a User Interface for a Sparse Matrix Package

Alan George and Joseph Liu

SIAM. J. Matrix Anal. & Appl. 20, pp. 953-969 (17 pages) | Cited 1 time

Online Publication Date: August 01, 2006

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The authors designed and implemented a sparse matrix package called Sparspak in the late 1970s. One of the important features of that package is an interface which shields the user from the complicated calling sequences common to most sparse matrix software. The implementation of the package was challenging because the relatively primitive but widely available Fortran 66 language was used. Modern programming languages such as Fortran-90 and C++ have important features which facilitate the design of flexible and ``user-friendly' interfaces for software packages. These features include dynamic storage allocation, function name overloading, user-defined data types, and the ability to hide functions and data from the user. This article describes the redesign of the Sparspak user interface using Fortran-90 and C++, outlining the reasons for its various features and highlighting similarities and differences in the features and capabilities of the two languages. The two new implementations of Sparspak have been named Sparspak-90 and Sparspak++.

Toward an Effective Sparse Approximate Inverse Preconditioner

Wei-Pai Tang

SIAM. J. Matrix Anal. & Appl. 20, pp. 970-986 (17 pages) | Cited 14 times

Online Publication Date: August 01, 2006

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Sparse approximate inverse preconditioners have attracted much attention recently,because of their potential usefulness in a parallel environment. In this paper, we explore several performance issues related to effective sparse approximate inverse preconditioners (SAIPs) for the matrices derived from PDEs. Our refinements can significantly improve the quality of existing SAIPs and/or reduce the cost of computing them. For the test problems from the Harwell--Boeing collection and some other applications, the performance of our preconditioners can be comparable or superior to incomplete LU (ILU) preconditioners with similar preconditioning cost.

Scalable Parallel Preconditioning with the Sparse Approximate Inverse of Triangular Matrices

Arno C. N. van Duin

SIAM. J. Matrix Anal. & Appl. 20, pp. 987-1006 (20 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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In this paper an approach is proposed for preconditioning large general sparse matrices. This approach combines the scalability of the application of explicit preconditioners with the preconditioning efficiency of incomplete factorizations. Several algorithms resulting from this approach are presented. Both the preconditioning efficiency and the cost of applying this preconditioner are tested. The experiments indicate that this technique offers the ability to make efficient use of parallel computers with a convergence rate comparable to that of the underlying incomplete factorization.

Matrix Renumbering ILU: An Effective Algebraic Multilevel ILU Preconditioner for Sparse Matrices

E. F. F. Botta and F. W. Wubs

SIAM. J. Matrix Anal. & Appl. 20, pp. 1007-1026 (20 pages) | Cited 31 times

Online Publication Date: August 01, 2006

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In this paper a multilevel-like ILU preconditioner is introduced. The ILU factorization generates its own ordering during the elimination process. Both ordering and dropping depend on the size of the entries. The method can handle structured and unstructured problems. Results are presented for some important classes of matrices and for several well-known test examples. The results illustrate the efficiency of the method and show in several cases near grid independent convergence.

Sparse Matrix Computations Arising in Distributed Parameter Identification

Curtis R. Vogel

SIAM. J. Matrix Anal. & Appl. 20, pp. 1027-1037 (11 pages) | Cited 13 times

Online Publication Date: August 01, 2006

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A penalized least squares approach known as Tikhonov regularization is commonly used to estimate distributed parameters in partial differential equations. The application of quasi-Newton minimization methods then yields very large linear systems. While these systems are not sparse, sparse matrices play an important role in gradient evaluation and Hessian matrix-vector multiplications. Motivated by the spectral structure of the Hessian matrices, a preconditioned conjugate gradient method is introduced to efficiently solve these linear systems. Numerical results are also presented.

Block Stationary Methods for Nonsymmetric Cyclically Reduced Systems Arising from Three-Dimensional Elliptic Equations

Chen Greif and James Varah

SIAM. J. Matrix Anal. & Appl. 20, pp. 1038-1059 (22 pages) | Cited 5 times

Online Publication Date: August 01, 2006

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We consider a three-dimensional convection-diffusion model problem and examine systems of equations arising from performing one step of cyclic reduction on an equally spaced mesh, discretized using the seven-point operator. We present two ordering strategies and analyze block splittings of the resulting matrices. If the matrices are consistently ordered relative to a given partitioning, Young's analysis for the block Gauss--Seidel and block SOR methods can be applied. We compare partitionings for which this property holds with ones where the matrices do not have Property A yet still give rise to an efficient solution process. Bounds on convergence rates are derived and the work involved in solving the systems is estimated.

ABLE: An Adaptive Block Lanczos Method for Non-Hermitian Eigenvalue Problems

Zhaojun Bai, David Day, and Qiang Ye

SIAM. J. Matrix Anal. & Appl. 20, pp. 1060-1082 (23 pages) | Cited 13 times

Online Publication Date: August 01, 2006

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This work presents an adaptive block Lanczos method for large-scale non-Hermitian Eigenvalue problems (henceforth the ABLE method). The ABLE method is a block version of the non-Hermitian Lanczos algorithm. There are three innovations. First, an adaptive blocksize scheme cures (near) breakdown and adapts the blocksize to the order of multiple or clustered eigenvalues. Second, stopping criteria are developed that exploit the semiquadratic convergence property of the method. Third, a well-known technique from the Hermitian Lanczos algorithm is generalized to monitor the loss of biorthogonality and maintain semibiorthogonality among the computed Lanczos vectors. Each innovation is theoretically justified. Academic model problems and real application problems are solved to demonstrate the numerical behaviors of the method.

The BR Eigenvalue Algorithm

G. A. Geist, G. W. Howell, and D. S. Watkins

SIAM. J. Matrix Anal. & Appl. 20, pp. 1083-1098 (16 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. It is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR algorithm, it is well adapted to computing the eigenvalues of the narrow-band, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the $QR$ algorithm by a factor of 30--60 in computing time and a factor of over 100 in matrix storage space.
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