Chapter 1 SIAM Proceedings Series Macros for Use With LaTeX^{∗}

J. Corey Gray^{†} Tricia Manning^{‡} Vickie Kearn^{§} Nancy Abbott^{¶}
Sue Ciambrano^{∥} Paul Duggan^{∗∗} Robbi Anne Albert^{††} Jean Anderson^{‡‡}

Abstract An equivalence is shown between realizability of input/output (i/o) operators by rational control systems and high-order algebraic differential equations for i/o pairs. This generalizes, to nonlinear systems, the equivalence between autoregressive representations and finite dimensional linear realizability.

1 Problem Specification

In this paper, we consider the solution of the N × N linear system

cos sin Ax = b (1)

where A is large, sparse, symmetric, and positive definite. We consider the direct solution of (1) by means of general sparse Gaussian elimination. In such a procedure, we find a permutation matrix P, and compute the decomposition

P AP^{t} = LDL^{t}

where L is unit lower triangular and D is diagonal.

2 Design Considerations

Several good ordering algorithms (nested dissection and minimum degree) are available for computing P [3], [7]. Since our interest here does not focus directly on the ordering, we assume for convenience that P = I, or that A has been preordered to reflect an appropriate choice of P.

Our purpose here is to examine the nonnumerical complexity of the sparse elimination algorithm given in [1]. As was shown there, a general sparse elimination scheme based on the bordering algorithm requires less storage for pointers and row/column indices than

∗Any information regarding grants should be placed here.

†Production Manager, Society for Industrial and Applied Mathematics, Philadelphia, PA.

‡Publications Specialist, Society for Industrial and Applied Mathematics, Philadelphia, PA.

§Publisher, Society for Industrial and Applied Mathematics, Philadelphia, PA.

¶Design Supervisor, Society for Industrial and Applied Mathematics, Philadelphia, PA ∥Acquisitions Editor, Society for Industrial and Applied Mathematics, Philadelphia, PA ∗∗Composition Specialist, Society for Industrial and Applied Mathematics, Philadelphia, PA ††Production Assistant, Society for Industrial and Applied Mathematics, Philadelphia, PA ‡‡Composition Coordinator, Society for Industrial and Applied Mathematics, Philadelphia, PA

Fig. 1. This is figure 1.

more traditional implementations of general sparse elimination. This is accomplished by exploiting the m-tree, a particular spanning tree for the graph of the filled-in matrix.

Theorem 2.1. The method was extended to three dimensions. For the standard multigrid coarsening (in which, for a given grid, the next coarser grid has 1/8 as many points), anisotropic problems require plane relaxation to obtain a good smoothing factor.

Several good ordering algorithms (nested dissection and minimum degree) are available for computing P [3], [7]. Since our interest here does not focus directly on the ordering, we assume for convenience that P = I, or that A has been preordered to reflect an appropriate choice of P.

Proof. In this paper we consider two methods. The first method is basically the method considered with two differences: first, we perform plane relaxation by a two-dimensional multigrid method, and second, we use a slightly different choice of interpolation operator, which improves performance for nearly singular problems. In the second method coarsening is done by successively coarsening in each of the three independent variables and then ignoring the intermediate grids; this artifice simplifies coding considerably.

Our purpose here is to examine the nonnumerical complexity of the sparse elimination algorithm given in [1]. As was shown there, a general sparse elimination scheme based on the bordering algorithm requires less storage for pointers and row/column indices than more traditional implementations of general sparse elimination. This is accomplished by exploiting the m-tree, a particular spanning tree for the graph of the filled-in matrix.

Definition 2.1. We describe the two methods in §1.2. In § 1.3. we discuss some remaining details.

Our purpose here is to examine the nonnumerical complexity of the sparse elimination algorithm given in [1]. As was shown there, a general sparse elimination scheme based on the bordering algorithm requires less storage for pointers and row/column indices than more traditional implementations of general sparse elimination.

Lemma 2.1. We discuss first the choice for I_{k−1}^{k} which is a generalization. We assume that G^{k−1} is obtained from G^{k} by standard coarsening; that is, if G^{k} is a tensor product grid G_{x}^{k} × G_{y}^{k} × G_{z}^{k}, G^{k−1} = G_{x}^{k−1} × G_{y}^{k−1} × G_{z}^{k−1}, where G_{x}^{k−1} is obtained by deleting every other grid point of G_{x}^{k} and similarly for G_{y}^{k} and G_{z}^{k}.

This is accomplished by exploiting the m-tree, a particular spanning tree for the graph of the filled-in matrix. To our knowledge, the m-tree previously has not been applied in this fashion to the numerical factorization, but it has been used, directly or indirectly, in several optimal order algorithms for computing the fill-in during the symbolic factorization phase [2] - [6], [8], [10].

2.1 Robustness

We do not attempt to present an overview here, but rather attempt to focus on those results that are relevant to our particular algorithm. This section assumes prior knowledge of the role of graph theory in sparse Gaussian elimination; surveys of this role are available in [7] and [3]. More general discussions of elimination trees are given in [4] - [6], [10]. Thus, at the kth stage, the bordering algorithm consists of solving the lower triangular system

L_{k−1}v = c (2)

and setting

(3)  ℓ = D_{k−1}^{−1}v,
(4)  δ = α − ℓ^{t}v.

2.1.1 Versatility. We do not attempt to present an overview here, but rather attempt to focus on those results that are relevant to our particular algorithm.

3 Conclusions

The special structure of this problem allows us to make exact estimates of the complexity. For the old approach, we show that the complexity of the intersection problem is O(n^{3}), the same as the complexity of the numerical computations [3], [9]. For the new approach, the complexity of the second part is reduced to O(n^{2}(log n)^{2}).

References

[1] R. E. Bank and R. K. Smith, General sparse elimination requires no permanent integer storage, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 574–584.

[2] S. C. Eisenstat, M. C. Gursky, M. Schultz, and A. Sherman, Algorithms and data structures for sparse symmetric gaussian elimination, SIAM J. Sci. Stat. Comput., 2 (1982), pp. 225–237.

[3] A. George and J. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice Hall, Englewood Cliffs, NJ, 1981.

[4] K. H. Law and S. J. Fenves, A node addition model for symbolic factorization, ACM TOMS, 12 (1986), pp. 37–50.

[5] J. W. H. Liu, A compact row storage scheme for cholesky factors using elimination trees, ACM TOMS, 12 (1986), pp. 127–148.

[6] , The role of elimination trees in sparse factorization, Tech. Report CS-87-12,Department of Computer Science, York University, Ontario, Canada, 1987.

4 Gray et al.

[7] D. J. Rose, A graph theoretic study of the numeric solution of sparse positive definite systems, in Graph Theory and Computing, Academic Press, New York, 1972.

[8] D. J. Rose, R. E. Tarjan, and G. S. Lueker, Algorithmic aspects of vertex elimination on graphs, SIAM J. Comput., 5 (1976), pp. 226–283.

[9] D. J. Rose and G. F. Whitten, A recursive analysis of disection strategies, in Sparse Matrix Computations, Academic Press, New York, 1976.

[10] R. Schreiber, A new implementation of sparse gaussian elimination, ACM TOMS, 8 (1982), pp. 256–276.