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SIAM J. on Matrix Analysis and Applications

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1986

Volume 7, Issue 2, pp. 167-336


On the Eigenvalue Problem for a Class of Band Matrices Including Those with Toeplitz Inverses

William F. Trench

SIAM. J. on Algebraic and Discrete Methods 7, pp. 167-179 (13 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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We study the eigenvalue problem for a class $\mathcal{H}$ of band matrices which includes as a proper subclass all band matrices with Toeplitz inverses. Toeplitz matrices of this kind occur, for example, as autocorrelation matrices of purely autoregressive stationary time series. A formula is given for the characteristic polynomial $p_n ( \lambda )$ of an $n$th order matrix $H_n $ in $\mathcal{H}$, with bandwidth $k + 1\leqq n$, as the ratio of $k \times k$ determinants whose entries are polynomials in the zeros of a certain $k$th degree polynomial which is independent of $n$ and has one coefficient which depends upon $\lambda $. The formula permits the evaluation of $p_n ( \lambda )$ by means of a computation with complexity independent of $n$. Also given is a formula for the eigenvectors in terms of these zeros and $k$ coefficients which can be obtained by solving a $k \times k$ homogeneous system.

$LU$-Decompositions of Tridiagonal Irreducible $H$-Matrices

W. J. Harrod

SIAM. J. on Algebraic and Discrete Methods 7, pp. 180-187 (8 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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In this paper bounds are developed and investigated on the growth factors and the multipliers resulting from Gaussian elimination applied to an irreducible tridiagonal $H$-matrix. These results extend the study of the stability of Gaussian elimination without pivoting on certain tridiagonal matrices by Gunzburger and Nicolaides.

Inverse Problems for Means of Matrices

William N. Anderson, Jr. and George E. Trapp

SIAM. J. on Algebraic and Discrete Methods 7, pp. 188-192 (5 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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Given two positive semidefinite Hermitian matrices $A$ and $B$ there are natural definitions for their arithmetic and harmonic means. In this work we consider the following question: Given positive semidefinite matrices $C$ and $D$, when do there exist positive semidefinite matrices $A$ and $B$ such that $C$ is the arithmetic mean of $A$ and $B$ and $D$ is the harmonic mean of $A$ and $B$. Uniqueness questions are also answered. Similar questions are answered concerning the geometric mean.

Incomplete Factorization of Singular $M$-Matrices

J. J. Buoni

SIAM. J. on Algebraic and Discrete Methods 7, pp. 193-198 (6 pages) | Cited 5 times

Online Publication Date: July 17, 2006

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In 1981, Varga and Cai characterized those $M$-matrices $A$ (perhaps singular) which admit a factorization into $M$-matrices $L$ and $U ( A = LU )$ where $L$ is required to be a nonsingular and lower triangular $M$-matrix and $U$ is required to be an upper triangular $M$-matrix, a result that was first proved by Fiedler and Ptak (1962) in the case when $A$ is nonsingular. Because this factorization may, as a result of fill-in, produce a lower triangular matrix which is considerably less sparse than $A$, one attempts to control the fill-in of the factorization of $A$ by means of a graph. This method leads to the concept of incomplete factorizations of $A$. Meijerink and van der Vorst (1977).who have shown that incomplete factorizations of nonsingular $M$-matrices are possible, while Manteufiel (1980) has extended this result to the $H$-matrix case. The purpose of this paper is to give a condition on a singular $M$-matrix which guarantees the incomplete factorization of a singular $M$-matrix.

Packings by Complete Bipartite Graphs

P. Hell and D. G. Kirkpatrick

SIAM. J. on Algebraic and Discrete Methods 7, pp. 199-209 (11 pages) | Cited 5 times

Online Publication Date: July 17, 2006

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Given any set $\mathcal{B}$ of complete bipartite graphs, we ask whether a graph $H$ admits a $\mathcal{B}$-factor, i.e., a spanning subgraph, each of whose components is a member of $\mathcal{B}$. More generally, we seek in $H$ a maximum $\mathcal{B}$-packing, i.e., a $\mathcal{B}$-factor of a maximum size subgraph of $H$. We first treat the interesting special case when $\mathcal{B}$ is a set of stars. The results are generalized to arbitrary $\mathcal{B}$ in the last section. We prove for most of these problems that they are $\mathcal{N}\mathcal{P}$-hard; we also show that the remaining problems admit polynomial algorithms based on augmenting configurations. The simplicity of these algorithms, as well as the implied min-max theorems, resemble the theory of matchings in bipartite, rather than general, graphs.

Values of Graph-Restricted Games

Guillermo Owen

SIAM. J. on Algebraic and Discrete Methods 7, pp. 210-220 (11 pages) | Cited 31 times

Online Publication Date: July 17, 2006

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We consider the problem of modifying $n$-person games so as to take account of the difficulties imposed by lack of communications, and the opportunities this might accord to intermediaries.
In this model, the members of a finite set are simultaneously players in a game and vertices of a graph. A combination of these two structures gives rise to a new, modified game in which the only effective coalitions are those corresponding to connected partial graphs. We study the relationship between the power indices of the original game and the restricted game; for the special case where the graph is a tree, this relationship is especially easy to analyze.
Several examples are studied in detail.

The Cyclic Coloring Problem and Estimation of Sparse Hessian Matrices

Thomas F. Coleman and Jin-Yi Cai

SIAM. J. on Algebraic and Discrete Methods 7, pp. 221-235 (15 pages) | Cited 13 times

Online Publication Date: July 17, 2006

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Numerical optimization algorithms often require the (symmetric) matrix of second derivatives, $\nabla ^2 f( x )$. If the Hessian matrix is large and sparse, then estimation by finite differences can be quite attractive since several schemes allow for estimation in much fewer than $n$ gradient evaluations.
The purpose of this paper is to analyze, from a combinatorial point of view, a class of methods known as substitution methods. We present a concise characterization of such methods in graph-theoretic terms. Using this characterization, we develop a complexity analysis of the general problem and derive a roundoff error bound on the Hessian approximation. Moreover, the graph model immediately reveals procedures to effect the substitution process optimally (i.e. using fewest possible substitutions given the differencing directions) in space proportional to the number of nonzeros in the Hessian matrix.

Difference Methods for the Numerical Solution of Time-Varying Singular Systems of Differential Equations

Kenneth D. Clark

SIAM. J. on Algebraic and Discrete Methods 7, pp. 236-246 (11 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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In this note, we introduce a class of difference methods for the numerical solution of differential equations of the form\[ A ( t ) x' + B ( t ) x ( t ) = f ( t ) \] where $A$, $B$, and $f$ are assumed sufficiently smooth in $t$ in the interval $I = [ 0,T ]$ and $A ( t )$ is identically singular on $I$. These methods are straightforward extensions of the well-known Gear’s backward difference methods (BDF’s) and correspond to BDF’s whenever $A$ is constant. It is shown that the modified methods (MBDF’s) work whenever the system can be transformed to a constant coefficient problem by a change of variable $x = Ly$, and also whenever a related system can be transformed into a certain canonical form.
We also investigate the relationship between the convergence of BDF’s and the continuous regularization of the system by its pencil perturbation. In particular, we show the existence of examples where the BDF’s converge but the pencil perturbation is not a continuous regularization.

A Packing Problem You Can Almost Solve by Sitting on Your Suitcase

Dorit S. Hochbaum and David B. Shmoys

SIAM. J. on Algebraic and Discrete Methods 7, pp. 247-257 (11 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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In this paper, we present a novel approach for approximating solutions to the bin-packing and machine scheduling problems. In obtaining our results, we exploit a certain dual relationship that exists between these two problems.
We introduce the notion of a dual approximation algorithm, where for the bin-packing problem, the aim is to find approximate packings where at most the optimal number of bins are used, but the bins are allowed to be filled beyond their capacity. For this approach, the objective is to minimize the tardiness of the machine that finishes last. For bin-packing instances where the size of each piece is at least $( 1 /3 - \varepsilon )$ times the capacity of the bin, we give an approximation algorithm $A_\varepsilon $ that is guaranteed to produce a solution where no bin contains more than $( 1 + 3\varepsilon /2 )$ times the bin capacity. Thus we have a family of dual approximation algorithms, dependent on the problem instance, where the “closer” the instance is to belonging to a class that can be solved in polynomial-time, the better performance is guaranteed.
Using this result, we construct an approximation algorithm for the minimum makespan scheduling problem, that always finds a schedule where all jobs are completed by $\frac{5} {4}$ times the best completion time.

Simplified Reliabilities for Consecutive-$k$-out-of-$n$ Systems

F. K. Hwang

SIAM. J. on Algebraic and Discrete Methods 7, pp. 258-264 (7 pages) | Cited 10 times

Online Publication Date: July 17, 2006

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Reliabilities for consecutive-$k$-out-of-$n$ systems are typically given in the form of recursive equations. Some attempts have been made to use combinatorics to obtain closed-form solutions, but the solutions contain $k - 1$ summations. In this paper we obtain closed form solutions with one summation over $n/ k$ terms. For $k = 2$ we are able to eliminate all summations. We apply our result to compute the reliability of a $k$-loop computer network.

On the Spectral Radius of Complementary Acyclic Matrices of Zeros and Ones

Richard A. Brualdi and Ernie S. Solheid

SIAM. J. on Algebraic and Discrete Methods 7, pp. 265-272 (8 pages) | Cited 7 times

Online Publication Date: July 17, 2006

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For an $n \times n$ complementary acyclic matrix $A$ of 0’s and l’s we show that the spectral radius $\rho ( A )$ of $A$ satisfies $\rho ( A )\geqq n - 2$ and determine those matrices $A$ for which equality holds. When $A$ is an $n \times n$ irreducible, complementary tree matrix, we also obtain that $\rho ( A )\leqq \rho _n $, where $\rho _n $ is the largest root of the polynomial $\lambda^3 - (n - 2 )\lambda^2 - ( n - 3 )\lambda - 1$.

Using the QR Factorization and Group Inversion to Compute, Differentiate, and Estimate the Sensitivity of Stationary Probabilities for Markov Chains

Gene H. Golub and Carl D. Meyer, Jr.

SIAM. J. on Algebraic and Discrete Methods 7, pp. 273-281 (9 pages) | Cited 18 times

Online Publication Date: July 17, 2006

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For an $n$-state finite, homogeneous, ergodic Markov chain, with transition matrix ${\bf P}$ and stationary distribution ${\boldsymbol \pi} $ we assume that the entries of ${\bf P}$ are differentiable functions of a parameter $t$ and we obtain an expression for $d{\boldsymbol \pi} /dt$. This expression is given in terms of the group inverse of ${\bf I} - {\bf P}$ and is used in a sensitivity analysis of ${\boldsymbol \pi}$. Finally, it is demonstrated how a ${\boldsymbol QR}$ factorization can be used to simultaneously compute the stationary distribution of an ergodic chain along with estimates which gauge the sensitivity of the stationary distribution to perturbations in the transition probabilities.

Expanders and Diffusers

Marshall W. Buck

SIAM. J. on Algebraic and Discrete Methods 7, pp. 282-304 (23 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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Expander graphs are ingredients for making concentrating, switching, and sorting networks, and are closely related to sparse, doubly-stochastic matrices called diffusers. The best explicit examples of diffusers are defined by means of the action of elements of the matrix group $SL (2,{\bf Z} )$ on certain finite mathematical objects. Some corresponding, explicit expanders were introduced by Margulis. However, Gabber and Galil were the first to obtain good estimates for the expanders and produce from them a family of directed acyclic superconcentrators having density 271.8. In this paper we review various techniques for making expanders from diffusers. We also demonstrate asymptotic upper bounds on the strength of algebraically defined classes of degree $k$ diffusers. Each upper bound is given as the norm of a diffusion operator on an infinite discrete group, and bounds for several examples are calculated. Numerical evidence is supplied in support of our conjecture that these bounds can be achieved by certain algebraically defined examples. The conjecture, if true, would lead to superconcentrators of density less than 58.

Characterization and Recognition of Partial 3-Trees

Stefan Arnborg and Andrzej Proskurowski

SIAM. J. on Algebraic and Discrete Methods 7, pp. 305-314 (10 pages) | Cited 17 times

Online Publication Date: July 17, 2006

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Our interest in the class of $k$-trees and their partial graphs and subgraphs is motivated by some practical questions about the reliability of communication networks in the presence of constrained line- and site-failures, and about the complexity of queries in a data base system. We have found a set of confluent graph reductions such that any graph can be reduced to the empty graph if and only if it is a subgraph of a 3-tree. This set of reductions yields a polynomial time algorithm for deciding if a given graph is a partial 3-tree and for finding one of its embeddings in a 3-tree when such an embedding exists. Our result generalizes a previously known recognition algorithm for partial 2-trees (series-parallel graphs).

An Application of the Singular Value Decomposition to Manipulability and Sensitivity of Industrial Robots

Masaki Togai

SIAM. J. on Algebraic and Discrete Methods 7, pp. 315-320 (6 pages)

Online Publication Date: July 17, 2006

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In designing and evaluating industrial robots, it is important to find optimal configurations and locate optimum points in the workspace for the anticipated tasks. In the current paper the singular value decomposition and perturbation analysis are applied to the Jacobian of robot kinematics; the condition number of the Jacobian is then proposed to be a measure of the “nearness” to degeneracy. Then qualitative measures called kinematic “manipulability” and “sensitivity” are proposed. Some properties of proposed measures are investigated and the relation between these measures are discussed. Optimal postures of various types of industrial robots are obtained.

A Multiplier Method for Identifying Keyblocks in Excavations Through Jointed Rock

J. L. Delport and D. H. Martin

SIAM. J. on Algebraic and Discrete Methods 7, pp. 321-330 (10 pages)

Online Publication Date: July 17, 2006

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G. H. Shi and R. E. Goodman have recently pointed out the practical importance of the keyblock principle in supporting underground and surface excavations in jointed rock and have given an elegant procedure involving stereographic projection and graphics for identifying the shapes of keyblocks. This paper presents the mathematical analysis of an entirely different keyblock characterisation and identification algorithm. The characterisation rests upon Tucker’s Theorem of the Alternative, and the algorithm requires only the execution of a few linear programming type pivot operations and sign tests.

Hard Enumeration Problems in Geometry and Combinatorics

Nathan Linial

SIAM. J. on Algebraic and Discrete Methods 7, pp. 331-335 (5 pages) | Cited 22 times

Online Publication Date: July 17, 2006

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A number of natural enumeration problems in geometry and combinatorics are shown to be complete in the class # P introduced by Valiant. Among others this is established for the numeration of vertices and of facets of a polytope, acyclic orientations of a graph and satisfying assignments of implicative boolean formulas.

Erratum: Volterra Multipliers II

Ray Redheffer

SIAM. J. on Algebraic and Discrete Methods 7, pp. 336-336 (1 page)

Online Publication Date: July 17, 2006

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