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SIAM J. on Matrix Analysis and Applications

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1986

Volume 7, Issue 4, pp. 505-644


The Bandwidth Minimization Problem for Caterpillars with Hair Length 3 is NP-Complete

Burkhard Monien

SIAM. J. on Algebraic and Discrete Methods 7, pp. 505-512 (8 pages) | Cited 19 times

Online Publication Date: July 17, 2006

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It is shown that the Bandwidth Minimization problem remains NP-complete even when restricted to “caterpillars with hairs of length at most three”. “Caterpillars” are special trees; they consist of a simple chain (the “body”) with various simple chains attached to thee vertices of the body (the attached chains are called “hairs”). A previous result in the literature shows that the bandwidth of caterpillars with hairs of length at most 2 can be found in $O( n\log n )$ time (this Journal, 2 (1981), pp. 387–393). We also show that the bandwidth problem is NP-complete when restricted to caterpillars with at most one hair attached to each vertex of the body. The proof is relatively straightforward and thereby also provides an easier proof than found in (SIAM J. Appl. Math., 34 (1978), pp. 477–495) that the bandwidth problem is NP-complete for trees with maximum vertex degree 3.

A Spectrum Enveloping Technique for Iterative Solution of Central Difference Approximations of Convection-Diffusion Equations

Murli M. Gupta

SIAM. J. on Algebraic and Discrete Methods 7, pp. 513-526 (14 pages)

Online Publication Date: July 17, 2006

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When a convection-diffusion equation is discretized using the central difference scheme the resulting coefficient matrix is not diagonally dominant whenever the convection terms are large. If this system of linear equations is solved using the conventional iteration methods, the iterations often fail to converge as some of the eigenvalues of the iteration matrix lie outside the unit circle $C = \{ z: | z |\leqq 1 \}$ in the complex plane.
The eigenvalue spectrum of some of the iteration matrices lies inside the infinite strip $S = \{ z: | \operatorname{Real} ( z ) | < 1, | \operatorname{Imag} ( z ) | < \infty \}$. An example is that of the method of simultaneous displacements or the Jacobi method. In such cases, it is possible to enclose the eigenvalue spectrum inside an ellipse with major axis on the imaginary axis and minor axis in the real interval $( - 1,1 )$. This ellipse is used to define a convergent iteration. A practical computational algorithm is described to obtain such an iteration scheme. Numerical examples show that the spectrum enveloping technique works well when the original iterations diverge. When the original iterations converge the spectrum enveloping technique can converge even faster.

The Null Space Problem I. Complexity

Thomas F. Coleman and Alex Pothen

SIAM. J. on Algebraic and Discrete Methods 7, pp. 527-537 (11 pages) | Cited 20 times

Online Publication Date: July 17, 2006

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The Null Space Problem (NSP) is the following: Given a $t \times n$ matrix $A$ with $t < n$, find a sparsest basis for its null space (a null basis). We show that columns in a sparsest null basis correspond to minimal dependent sets of columns of $A$. Sparsest null bases are characterized by a greedy algorithm that augments a partial basis by a sparsest null vector. Despite this result, (NSP) is NP-hard since finding a sparsest null vector of $A$ is NP-complete. We prove that the related problem of finding a sparsest null basis with an embedded identity matrix is NP-hard too. Finally, we study the zero–nonzero structure of sparsest null bases.

Column LU Factorization with Pivoting on a Message-Passing Multiprocessor

George J. Davis

SIAM. J. on Algebraic and Discrete Methods 7, pp. 538-550 (13 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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A column-oriented algorithm is presented for LU factorization with partial pivoting. Two different mappings of columns to processors are considered. Forward and backsubstitution algorithms to use the factorization for solving linear systems are developed. Timing data, including processor utilization and load balancing, are provided by a hypercube simulator.

Neighborhoods of Dominant Convergence for the SSOR Method

Michael Neumann

SIAM. J. on Algebraic and Discrete Methods 7, pp. 551-559 (9 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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Let $A$ be an $n \times n$ nonsingular irreducible 3-cyclic $H$-matrix and let $J^A$, $L_\omega ^A$ and $S_\omega ^A $ denote, respectively, the Jacobi, the SOR, and the SSOR iteration matrices associated with $A$. In this paper we show that if the spectral radius $\rho ( | J^A | ) \in ( 0,r_0 )$, where $r_0$ is the unique root of the cubic $17r^3 + r^2 - r - 1$ in the interval (0, 1), then there exists a neighborhood $\Omega _{\omega ( A )} $ of $\omega ( A ) : = 2 /( 1 + \rho ( | J^A | )$ such that \[ \rho ( S_\omega ^A ) < | \omega - 1 | \leqq \rho ( L_\omega ^A )\quad \forall \omega \in \Omega _{\omega ( A )} . \]

Fredman–Komlós bounds and information theory

Jénos Körner

SIAM. J. on Algebraic and Discrete Methods 7, pp. 560-570 (11 pages) | Cited 13 times

Online Publication Date: July 17, 2006

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Fredman and Komlós have applied an interesting information-theoretic lemma to two problems in combinatorics. They have derived good lower bounds on the minimum size of a family of partitions of an $n$-element set into at most $b$ classes such that all the subsets (respectively, pairs of subsets) of a certain kind are “separated” by at least one partition in the family.
Our aim is to show that the Fredman–Komlós lemma is a special case of a simple inequality between entropies of graphs. The, general inequality enables us to handle more problems on separating partition systems. Part of the problems relate to hashing.

Optimal Numberings of an $N \times N$ Array

Graeme Mitchison and Richard Durbin

SIAM. J. on Algebraic and Discrete Methods 7, pp. 571-582 (12 pages) | Cited 17 times

Online Publication Date: July 17, 2006

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Given a numbering of the vertices of a graph, one can define the edgesum [6] as the sum of differences between adjacent vertices. The problem of finding numberings which are optimal in the sense of minimizing the edgesum is NP-complete [2] but has been solved in the special case where the graph is the $2^n $ cube [3] and for several instances of graphs with high degrees of symmetry [6]. We find the solutions for numberings of an $N \times N$ array. These have practical application in the problem of representing spatial information in a one-dimensional medium. To find our solutions, we exploit the fact that such numberings can always be taken to be ordered, in the sense that numbers increase along rows and down columns. We also consider a generalization of this problem to the case where the differences are raised to a power $q$. We derive bounds on the edgesum in this case, and show that the optimal numberings for $q < 1$ must be essentially different from those we have found for $q = 1$. While the latter may be assumed to be ordered, and have regions of numbering by rows or columns, neither statement is true for the case $q < 1$. We hypothesize that the solution in this case has a fractal character.

An Approximation to the Stationary Distribution of a Nearly Completely Decomposable Markov Chain and Its Error Bound

Moshe Haviv and Y. Ritov

SIAM. J. on Algebraic and Discrete Methods 7, pp. 583-588 (6 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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In Haviv (Ph.D. dissertation, Yale Univ., New Haven, CT, 1983) an approximation procedure for computing the stationary distribution of a nearly completely decomposable (NCD) Markov chain is suggested. There and in Haviv (this Journal, 7 (1986), pp. 589–593) the incurred error is analyzed. In particular, a series expansion for the error is developed. Courtois and Semal (J. Assoc. Comput. Mach., 31 (1984), pp. 804–825) independently of us, replaced this point approximation with a set of points. Using algebraic methods, they proved that the exact distribution lies in the convex set spanned by this set. We give a probabilistic interpretation for this set and then obtain their results in a more elementary way. We compute the convex combination leading to the exact distribution and develop a bound on it. Finally, we show how approximation to this convex combination leads to an error reduction in a cui`rent approximation. It is the first time that a probabilistic approach is made in order to analyze NCD Markov chains.

An Approximation to the Stationary Distribution of a Nearly Completely Decomposable Markov Chain and Its Error Analysis

Moshe Haviv

SIAM. J. on Algebraic and Discrete Methods 7, pp. 589-593 (5 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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All existing approximations for the stationary distribution of a nearly completely decomposable Markov chain are based on solving systems which are perturbations of the exact system. We develop here an original approximation which is based on probabilistic intuition. A series expansion of the accrued error is given as well.

Computing the Structural Index

I. S. Duff and C. W. Gear

SIAM. J. on Algebraic and Discrete Methods 7, pp. 594-603 (10 pages) | Cited 9 times

Online Publication Date: July 17, 2006

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The index of many differential/algebraic equations (DAEs) is determined by the structure of the system, that is, by the pattern of nonzero entries in the Jacobians. This paper considers an important subclass of DAEs which can be solved by backward differentiation methods if their index does not exceed two. For this reason, it is desirable to determine whether the index exceeds two or not. In this paper we present an algorithm that determines if the index is one, two, or greater, based only on the structure. The algorithm can be exponential in its execution time: we do not know whether it is possible to get an asymptotically faster algorithm. However, in many practical problems, this algorithm will execute in polynomial time.

Sequence Alignments with Matched Sections

Jerrold R. Griggs, Philip J. Hanlon, and Michael S. Waterman

SIAM. J. on Algebraic and Discrete Methods 7, pp. 604-608 (5 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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in an alignment. The number of alignments of two sequences is related to the Stanton–Cowan numbers. This paper gives asymptotics for the number of alignments of two sequences of length $n$ with matching sections of size at least $b$.

Cascade Addition and Subtraction of Matrices

W. N. Anderson, Jr., T. D. Morley, and G. E. Trapp

SIAM. J. on Algebraic and Discrete Methods 7, pp. 609-626 (18 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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The cascade connection of electrical $n$-port networks motivates the cascade sum of matrices. The electrical network situation pertains to Hermitian positive semidefinite matrices, while in this work the cascade addition operation is also considered for arbitrary matrices. Various properties of the cascade sum are presented, including conditions which guarantee the existence of the cascade sum and the associativity of the cascade sum. A related operation, the cascade difference, is also treated. The underlying structure of the cascade operations is developed using the theory of the shorted operator.

Mathematical Aspects of the Relative Gain Array $( A \circ A^{ - T} )$

Charles R. Johnson and Helene M. Shapiro

SIAM. J. on Algebraic and Discrete Methods 7, pp. 627-644 (18 pages) | Cited 9 times

Online Publication Date: July 17, 2006

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For nonsingular $n$-by-$n$ matrices $A$, we investigate the map\[ A \to \Phi ( A ) \equiv A \circ ( A^{ - 1} )^T \] in which $ \circ $ denotes the Hadamard (entry-wise) product. The matrix $\Phi ( A )$ arises in mathematical control theory in chemical engineering design problems, where it is known as the relative gain array, and also in a matrix theoretic problem involving the relation between the diagonal entries and eigenvalues. We first give several elementary properties of $\Phi $ and show that the iterates $\Phi ^k ( A )$ converge to $I$ when $A$ is either positive definite or an $H$-matrix. We then discuss, with examples and partial results, several unsolved problems associated with $\Phi $. These include the range of $\Phi $, inverse images of elements in the range of $\Phi $, fixed points of $\Phi $, etc.
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