Showing papers on "Triangular matrix published in 2000"

Invertible completions of 2 x 2 upper triangular operator matrices

Abstract: In this note we prove that if MC = ( A C 0 B ) is a 2×2 upper triangular operator matrix acting on the Banach space X⊕Y , then MC is invertible for some C ∈ L(Y, X) if and only if A ∈ L(X) and B ∈ L(Y ) satisfy the following conditions: (i) A is left invertible; (ii) B is right invertible; (iii) X/A(X) ∼= B−1(0). Furthermore we show that σ(A) ∪ σ(B) = σ(MC ) ∪W , where W is the union of certain of the holes in σ(MC ) which happen to be subsets of σ(A) ∩ σ(B).

Border-Block Triangular Form and Conjunction Schedule in Image Computation

Abstract: Conjunction scheduling in image computation consists of clustering the parts of a transition relation and ordering the clusters, so that the size of the BDDs for the intermediate results of image computation stay small. We present an approach based on the analysis and permutation of the dependence matrix of the transition relation. Our algorithm computes a bordered-block lower triangular form of the matrix that heuristically minimizes the active lifetime of variables, that is, the number of conjunctions in which the variables participate. The ordering procedure guides a clustering algorithm based on the affinity of the transition relation parts. The ordering procedure is then applied again to define the cluster conjunction schedule. Our experimental results show the effectiveness of the new algorithm.

Mixed Integer-Real Valued Adjustment (IRA) Problems: GPS Initial Cycle Ambiguity Resolution by Means of the LLL Algorithm

Abstract: In order to come to GPS solutions of first order accuracy and integrity, carrier phase observations as well as pseudo-ranging observations have to be adjusted with respect to a linear/linearized model. Here the problem of mixed integer-real valued parameter adjustment (IRA) is met. Indeed integer cycle ambiguity unknowns have to be estimated and tested. At first we review the three concepts to deal with IRA: (i) DDD or triple difference observations are produced by a properly chosen difference operator and choice of basis, namely being free of integer-valued unknowns, (ii) The real valued unknown parameters are eliminated by a Gauss elimination step while the remaining integer-valued unknown parameters (initial cycle ambiguities) are determined by Quadratic Programming and (iii) a RA substitute model is firstly implemented (real-valued estimates of initial cycle ambiguities) and secondly a minimum distance map is designed which operates on the real-valued approximation of integers with respect to the integer data in a lattice. This is the place where the integer Gram-Schmidt or-thogonalization by means of the LLL algorithm (modified LLL algorithm) is applied being illustrated by four examples. In particular, we prove that in general it is impossible to transform an oblique base of a lattice to an orthogonal base by Gram-Schmidt orthog-onalization where its matrix entries are integer. The volume preserving Gram-Schmidt orthogonalization operator constraint to integer entries produces “almost orthogonal” bases which, in turn, can be used to produce the integer-valued unknown parameters (initial cycle ambiguities) from the LLL algorithm (modified LLL algorithm). Systematic errors generated by “almost orthogonal” lattice bases are quantified by A.K. Lenstra et al. (1992) as well as M. Pohst (1987). The solution point ẑ of Integer Least Squares generated by the LLL algorithm is ẑ = (L’)-1 [L’x] ∈ Z m where L is the lower triangular Gram-Schmidt matrix rounded to nearest integers, [L], and ẑ = [L’x] are the nearest integers of L’x, x the real valued approximation of z ∈ ℤ m , the m-dimensional lattice space Λ. Indeed due to “almost orthogonality” of the integer Gram-Schmidt procedure, the solution point ẑ is only suboptimal, only close to “least squares”.

QR-factorization method for computing the greatest common divisor of polynomials with inexact coefficients

Abstract: This paper presents a novel means of computing the greatest common divisor (GCD) of two polynomials with real-valued coefficients that have been perturbed by noise. The method involves the QR-factorization of a near-to-Toeplitz matrix derived from the Sylvester matrix of the two polynomials. It turns out that the GCD to within a constant factor is contained in the last nonzero row of the upper triangular matrix R in the QR-factorization of the near-to-Toeplitz matrix. The QR-factorization is efficiently performed by an algorithm due to Chun et al. (1987). A condition number estimator due to Bischof (1990) and an algorithm for rank estimation due to Zarowski (1998) are employed to account for the effects of noise.

Balanced Parametrizations of Stable SISO All-Pass Systems in Discrete Time

Abstract: A balanced canonical form for discrete-time stable SISO all-pass systems is obtained by requiring the realization to be balanced and such that the reachability matrix is upper triangular with positive diagonal entries, in analogy to the continuous-time balanced canonical form of Ober [O1]. It is shown that the resulting balanced canonical form can be parametrized by Schur parameters. The relation with the Schur parameters for stable AR systems is established. Using the structure of the canonical form it is shown that, for the space of stable all-pass systems of order less than or equal to a fixed number n, the topology of pointwise convergence and the topology induced by H 2 coincide. The topological space thus obtained has the structure of a hypersphere. Model reduction procedures based on truncation, which respect the canonical form, are discussed.

Degree of Approximation to Functions in a Normed Space

Abstract: We compute the degree of approximation of functions, in the H$ouml;lder metric,using lower triangular nonnegative matrices. Ourwork generalizes that of Singh [5], and other authors, since we do notassume any monotonicity condition on the rows of the matrix.

On constructing unit triangular matrices with prescribed singular values

Abstract: We propose an efficient algorithm for computing a unit lower triangular n×n matrix with prescribed singular values of O(n2) cost. This is a solution of the question raised by N. J. Higham in [4, Problem 26.3, p. 528].

Regular Basis and R-Matrices for the ^(su)(n) k Knizhnik–Zamolodchikov Equation

Abstract: Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations of the U q (sl n ) covariant quantum matrix algebra derived previously by solving the dynamical Yang–Baxter equation. As a byproduct, we extend the regular basis introduced earlier for SU(2) chiral fields to SU(n) step operators and display the corresponding triangular matrix representation of the braid group.

Meet irreducible ideals in direct limit algebras

Abstract: These ideals have a description in terms of the coordinates, or spectrum, that is a natural extension of one description of meet irreducible ideals in the upper triangular matrices. Additional information is available if the limit algebra is an analytic subalgebra of its C-envelope or if the analytic algebra is trivially analytic with an injective 0-cocycle. In the latter case, we obtain a complete description of the meet irreducible ideals, modeled on the description in the algebra of upper triangular matrices. This applies, in particular, to all full nest algebras.

Regular basis and R-matrices for the su(n)_k Knizhnik-Zamolodchikov equation

Abstract: Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations for the U_q(sl_n) covariant quantum matrix derived previously by solving the dynamical Yang-Baxter equation. As a byproduct, we extend the regular basis introduced earlier for SU(2) chiral fields to SU(n) step operators and display the corresponding triangular matrix representation of the braid group.

Parallel givens sequences for solving the general linear model on a erew pram

Abstract: Parallel Givens sequences for solving the General Linear Model (GLM) are developed and analyzed. The block updating GLM estimation problem is also considered. The solution of the GLM employs as a main computational device the Generalized QR Decomposition, where one of the two matrices is initially upper triangular. The proposed Givens sequences efficiently exploit the initial triangular structure of the matrix and special properties of the solution method. The complexity analysis of the sequences is based on a Exclusive Read-Exclusive Write (EREW) Parallel Random Access Machine (PRAM) model with limited parallelism. Furthermore, the number of operations performed by a Givens rotation is determined by the size of the vectors used in the rotation. With these assumptions one conclusion drawn is that a sequence which applies the smallest number of compound disjoint Givens rotations to solve the GLM estimation problem does not necessarily have the lowest computational complexity. The various Givens se...

Random walk on upper triangular matrices mixes rapidly

Abstract: We present an upper bound O(n 2 ) for the mixing time of a simple random walk on upper triangular matrices. We show that this bound is sharp up to a constant, and find tight bounds on the eigenvalue gap. We conclude by applying our results to indicate that the asymmetric exclusion process on a circle indeed mixes more rapidly than the corresponding symmetric process.

Two Random Walks on Upper Triangular Matrices

Abstract: We study two random walks on a group of upper triangular matrices. In each case, we give upper bound on the mixing time by using a stopping time technique.

Parallel Two-Stage Reduction of a Regular Matrix Pair to Hessenberg-Triangular Form

Abstract: A parallel two-stage algorithm for reduction of a regular matrix pair (A,B) to Hessenberg-triangular form (H, T) is presented Stage one reduces the matrix pair to a block upper Hessenberg-triangular form (Hr, T), where Hr is upper r-Hessenberg with r > 1 subdiagonals and T is upper triangular In stage two, the desired upper Hessenberg-triangular form is computed using two-sided Givens rotations Performance results for the ScaLAPACK-style implementations show that the parallel algorithms can be used to solve large scale problems effectively

An improved simulation method and apparatus

Abstract: A Linear Solver method and apparatus, embodied in a Simulator, (70), and adapted for solving systems of non-linear partial differential equations and systems of linear equations representing physical characteristics of an oil and/or gas reservoir, (72), includes receiving a first signal representing physical characteristics of a reservoir, obtaining a residual vector, (Ro) from the first signal (representing errors associated with a system of nonlinear equations describing the reservoir), and a first matrix, (Ao), (78), (representing the sensitivity of the residual vector to changes in a system of nonlinear equations), recursively decomposing matrix (Ao), into a lower block triangular matrix, an upper block triangular matrix, and a diagonal matrix, and generating, (80), a second matrix (Mo) that is an approximation to matrix, (Ao). A solution to the systems of non-linear partial differential equations may then be found by using certain values that were used to produce the matrix, (Mo), and that solution, (86), does not require the direct computation of (Ao) z = b, (representing the system of linear equations) as required by conventional methods.

Extensions of simple-injective rings

Abstract: A ring Ris called right simple-injective if every itMinear map with simple image from a right ideal to Rcan be extended to R. We characterize when matrix rings, upper triangular matrix rings and trivial extensions are right simple-injective. We also study split null extensions of simple-injective rings.

Perturbation method with triangular propagators and anharmonicities of intermediate strength

Abstract: We propose a new, very flexible version of the Rayleigh–Schrodinger perturbation method which admits a lower triangular matrix in place of the usual diagonal unperturbed propagator. The technique and its enhanced efficiency are illustrated on rational anharmonicities V(1)(x)=β×polynomial(x)/polynomial(x). They are shown tractable, in the intermediate coupling regime, as \(\mathcal{O}(\beta {\text{ - }}\beta ^{{\text{(0)}}} )\) perturbations of exact states at non-vanishing β(0)≠0. In this sense our method bridges the gap between the current weak- and strong-coupling expansions.

On designing optimal parallel triangular solvers

Abstract: This paper explores the problem of solving triangular linear systems on parallel distributed-memory machines. Working within the LogP model, tight asymptotic bounds for solving these systems using forward/backward substitution are presented. Specifically, lower bounds on execution time independent of the data layout, lower bounds for data layouts in which the number of data items per processor is bounded, and lower bounds for specific data layouts commonly used in designing parallel algorithms for this problem are presented in this paper. Furthermore, algorithms are provided which have running times within a constant factor of the lower bounds described. One interesting result is that the popular two-dimensional block matrix layout necessarily results in significantly longer running times than simpler one-dimensional schemes. Finally, a generalization of the lower bounds to banded triangular linear systems is presented.

Fast and Scalable Parallel Matrix Computations with Optical Buses

Abstract: We present fast and highly scalable parallel computations for a number of important and fundamental matrix problems on linear arrays with reconfigurable pipelined optical bus systems. These problems include computing the Nth power, the in verse, the characteristic polynomial, the determinant, the rank, and an LU- and a QR-factorization of a matrix, and solving linear systems of equations. These computations are based on efficient implementation of the fastest sequential matrix multiplication algorithm, and are highly scalable over a wide range of system size. Such fast and scalable parallel matrix computations were not seen before on distributed memory parallel computing systems.

Realizations for Schur upper triangular operators centered at an arbitrary point

Abstract: Reproducing kernel spaces introduced by L. de Branges and J. Rovnyak provide isometric, coisometric and unitary realizations for Schur functions, i.e. for matrix-valued functions analytic and contractive in the open unit disk. In our previous paper [12] we showed that similar realizations exist in the “nonstationary setting”, i.e. when one considers upper triangular contractions (which appear in time-variant system theory as “transfer functions” of dissipative systems) rather than Schur functions and diagonal operators rather than complex numbers. We considered in [12] realizations centered at the origin. In the present paper we study realizations of a more general kind, centered at an arbitrary diagonal operator. Analogous realizations (centered at a point α of the open unit disk) for Schur functions were introduced and studied in [3] and [4].

Exponents of Subvarieties of Upper Triangular Matrices over Arbitrary Fields are Integral

Abstract: Let Uc be the variety of associative algebras generated by the algebra of all upper triangular matrices, the field being arbitrary. We prove that the upper exponent of any subvariety V ⊂ Uc coincides with the lower exponent and is an integer.

Minimum mean-squared error block-decision feedback sequence estimation in digital communication systems

Abstract: A method of estimating the sequence of transmitted symbols in a digital communication system is provided. By assuming that the transmitted symbols have a zero mean and a covariance matrix given by an identity matrix, as is the case in an EDGE communication system, the solution to the minimization of the expectation value of ∥{circumflex over (s)}−s∥2 always allows a lower triangular matrix to be found through Cholesky decomposition. The lower triangular matrix allows an efficient block decision-feedback sequence estimation to be carried out. The method is no more complex than the zero-forcing block decision-feedback sequence estimation technique, yet a solution always exists and performance is improved over the zero-forcing method.