Showing papers on "Matrix (mathematics) published in 1997"

LINCS : A linear constraint solver for molecular simulations

Abstract: In this article, we present a new LINear Constraint Solver (LINCS) for molecular simulations with bond constraints. The algorithm is inherently stable, as the constraints themselves are reset instead of derivatives of the constraints, thereby eliminating drift. Although the derivation of the algorithm is presented in terms of matrices, no matrix matrix multiplications are needed and only the nonzero matrix elements have to be stored, making the method useful for very large molecules. At the same accuracy, the LINCS algorithm is three to four times faster than the SHAKE algorithm. Parallelization of the algorithm is straightforward. (C) 1997 John Wiley & Sons, Inc.

A blind source separation technique using second-order statistics

Abstract: Separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed. In many situations, no a priori information on the mixing matrix is available: The linear mixture should be "blindly" processed. This typically occurs in narrowband array processing applications when the array manifold is unknown or distorted. This paper introduces a new source separation technique exploiting the time coherence of the source signals. In contrast with other previously reported techniques, the proposed approach relies only on stationary second-order statistics that are based on a joint diagonalization of a set of covariance matrices. Asymptotic performance analysis of this method is carried out; some numerical simulations are provided to illustrate the effectiveness of the proposed method.

Density-functional method for very large systems with LCAO basis sets

Abstract: We have implemented a linear scaling, fully self-consistent density-functional method for performing first-principles calculations on systems with a large number of atoms, using standard norm-conserving pseudopotentials and flexible linear combinations of atomic orbitals (LCAO) basis sets. Exchange and correlation are treated within the local-spin-density or gradient-corrected approximations. The basis functions and the electron density are projected on a real-space grid in order to calculate the Hartree and exchange–correlation potentials and matrix elements. We substitute the customary diagonalization procedure by the minimization of a modified energy functional, which gives orthogonal wave functions and the same energy and density as the Kohn–Sham energy functional, without the need of an explicit orthogonalization. The additional restriction to a finite range for the electron wave functions allows the computational effort (time and memory) to increase only linearly with the size of the system. Forces and stresses are also calculated efficiently and accurately, allowing structural relaxation and molecular dynamics simulations. We present test calculations beginning with small molecules and ending with a piece of DNA. Using double-z, polarized bases, geometries within 1% of experiments are obtained. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 453–461, 1997

On Krylov Subspace Approximations to the Matrix Exponential Operator

Abstract: Krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper. We derive error bounds via a functional calculus of Arnoldi and Lanczos methods that reduces the study of Krylov subspace approximations of functions of matrices to that of linear systems of equations. As a side result, we obtain error bounds for Galerkin-type Krylov methods for linear equations, namely, the biconjugate gradient method and the full orthogonalization method. For Krylov approximations to matrix exponentials, we show superlinear error decay from relatively small iteration numbers onwards, depending on the geometry of the numerical range, the spectrum, or the pseudospectrum. The convergence to exp$(\tau A)v$ is faster than that of corresponding Krylov methods for the solution of linear equations $(I-\tau A)x=v$, which usually arise in the numerical solution of stiff ordinary differential equations (ODEs). We therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use Krylov approximations to the exponential function of the Jacobian instead of solving linear or nonlinear systems of equations in every time step.

Krylov Projection Methods for Model Reduction

Abstract: This dissertation focuses on efficiently forming reduced-order models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reduced-order models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczos-based methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also develop ed to form a complete model-reduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multiple-input multiple-output systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees

Nonnegative Matrices and Applications

Abstract: This book provides an integrated treatment of the theory of nonnegative matrices (matrices with only positive numbers or zero as entries) and some related classes of positive matrices, concentrating on connections with game theory, combinatorics, inequalities, optimisation and mathematical economics. The wide variety of applications, which include price fixing, scheduling and the fair division problem, have been carefully chosen both for their elegant mathematical content and for their accessibility to students with minimal preparation. Many results in matrix theory are also presented. The treatment is rigorous and almost all results are proved completely. These results and applications will be of great interest to researchers in linear programming, statistics and operations research. The minimal prerequisites also make the book accessible to first-year graduate students.

Haar wavelet method for solving lumped and distributed-parameter systems

Abstract: An operational matrix of integration based on Haar wavelets is established, and a procedure for applying the matrix to analyse lumped and distributed-parameters dynamic systems is formulated. The technique can be interpreted from the incremental and multiresolution viewpoint. Crude as well as accurate solutions can be obtained by changing the parameter m; in the mean time, the main features of the solution are preserved. Several nontrivial examples are included for demonstrating the fast, flexible and convenient capabilities of the new method.

Pseudospectra of linear operators

Abstract: If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ||An|| or ||exp(tA)||. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ||(zI - A)-1||. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.

An Unsymmetric-Pattern Multifrontal Method for Sparse LU Factorization

Abstract: Sparse matrix factorization algorithms for general problems are typically characterized by irregular memory access patterns that limit their performance on parallel-vector supercomputers. For symmetric problems, methods such as the multifrontal method avoid indirect addressing in the innermost loops by using dense matrix kernels. However, no efficient LU factorization algorithm based primarily on dense matrix kernels exists for matrices whose pattern is very unsymmetric. We address this deficiency and present a new unsymmetric-pattern multifrontal method based on dense matrix kernels. As in the classical multifrontal method, advantage is taken of repetitive structure in the matrix by factorizing more than one pivot in each frontal matrix, thus enabling the use of Level 2 and Level 3 BLAS. The performance is compared with the classical multifrontal method and other unsymmetric solvers on a CRAY C-98.

Multi-Terminal Binary Decision Diagrams: An Efficient DataStructure for Matrix Representation

Abstract: In this paper, we discuss the use of binary decision diagrams to represent general matrices. We demonstrate that binary decision diagrams are an efficient representation for every special-case matrix in common use, notably sparse matrices. In particular, we demonstrate that for any matrix, the BDD representation can be no larger than the corresponding sparse-matrix representation. Further, the BDD representation is often smaller than any other conventional special-case representation: for the n×n Walsh matrix, for example, the BDD representation is of size O(log n). No other special-case representation in common use represents this matrix in space less than O(n²). We describe termwise, row, column, block, and diagonal selection over these matrices, standard an Strassen matrix multiplication, and LU factorization. We demonstrate that the complexity of each of these operations over the BDD representation is no greater than that over any standard representation. Further, we demonstrate that complete pivoting is no more difficult over these matrices than partial pivoting. Finally, we consider an example, the Walsh Spectrum of a Boolean function.

NP-Hardness of Some Linear Control Design Problems

Abstract: We show that some basic linear control design problems are NP-hard, implying that, unless P=NP, they cannot be solved by polynomial time algorithms. The problems that we consider include simultaneous stabilization by output feedback, stabilization by state or output feedback in the presence of bounds on the elements of the gain matrix, and decentralized control. These results are obtained by first showing that checking the existence of a stable matrix in an interval family of matrices is NP-hard.

Trafficking of matrix collagens through bone-resorbing osteoclasts [see comments]

Abstract: An intracellular pathway for proteins liberated from mineralized matrix during resorption was identified in osteoclasts. Analysis by confocal microscopy of sites of active bone resorption showed that released matrix proteins, including degraded type I collagen, were endocytosed along the ruffled border within the resorption compartment and transcytosed through the osteoclast to the basolateral membrane. Intracellular trafficking of degraded collagen, as typified by the resorbing osteoclast, may provide the cell with a regulatory mechanism for the control of tissue degradation.

Matrix Description of M-theory on $T^5$ and $T^5/Z_2$

Abstract: We present four infinite series of new quantum theories with super-Poincare symmetry in six dimensions, which are not local quantum field theories. They have string like excitations but the string coupling is of order one. Compactifying these theories on $T^5$ we find a Matrix theory description of M theory on $T^5$ and on $T^5/\IZ_2$, which is well defined and is manifestly U-duality invariant.

Interfacing relativistic and nonrelativistic methods. i. normalized elimination of the small component in the modified dirac equation

Abstract: The introduction of relativistic terms into the nonrelativistic all-electron Schrodinger equation is achieved by the method of normalized elimination of the small component (ESC) within the matrix representation of the modified Dirac equation. In contrast to the usual method of ESC, the method presented retains the correct relativistic normalization, and permits the construction of a single matrix relating the large and small component coefficient matrices for an entire set of positive energy one-particle states, thus enabling the whole set to be obtained with a single diagonalization. This matrix is used to define a modified set of one- and two-electron integrals which have the same appearance as the nonrelativistic integrals, and to which they reduce in the limit α→0. The normalized method corresponds to a projection of the Dirac–Fock matrix onto the positive energy states. Inclusion of the normalization reduces the discrepancy between the eigenvalues of the ESC approach and the Dirac eigenvalues for a ...

A combined unifrontal/multifrontal method for unsymmetric sparse matrices

Abstract: We discuss the organization of frontal matrices in multifrontal methods for the solution of large sparse sets of unsymmetric linear equations. In the multifrontal method, work on a frontal matrix can be suspended, the frontal matrix can be stored for later reuse, and a new frontal matrix can be generated. There are thus several frontal matrices stored during the factorization, and one or more of these are assembled (summed) when creating a new frontal matrix. Although this means that arbitrary sparsity patterns can be handled efficiently, extra work is required to sum the frontal matrices together and can be costly because indirect addressing is requred. The (uni)frontal method avoids this extra work by factorizing the matrix with a single frontal matrix. Rows and columns are added to the frontal matrix, and pivot rows and columns are removed. Data movement is simpler, but higher fill-in can result if the matrix cannot be permuted into a variable-band form with small profile. We consider a combined unifrontal/multifrontal algorithm to enable general fill-in reduction orderings to be applied without the data movement of previous multifrontal approaches. We discuss this technique in the context of a code designed for the solution of sparse systems with unsymmetric pattern.

H ∞ -control for Markovian jumping linear systems with parametric uncertainty

Abstract: This paper studies the problem of H∞-control for linear systems with Markovian jumping parameters The jumping parameters considered here are two separable continuous-time, discrete-state Markov processes, one appearing in the system matrices and one appearing in the control variable Our attention is focused on the design of linear state feedback controllers such that both stochastic stability and a prescribed H∞-performance are achieved We also deal with the robust H∞-control problem for linear systems with both Markovian jumping parameters and parameter uncertainties The parameter uncertainties are assumed to be real, time-varying, norm-bounded, appearing in the state matrix Both the finite-horizon and infinite-horizon cases are analyzed We show that the control problems for linear Markovian jumping systems with and without parameter uncertainties can be solved in terms of the solutions to a set of coupled differential Riccati equations for the finite-horizon case or algebraic Riccati equations for the infinite-horizon case Particularly, robust H∞-controllers are also designed when the jumping rates have parameter uncertainties

Matrix description of interacting theories in six-dimensions

Abstract: We propose descriptions of interacting (2,0) supersymmetric theories without gravity in six dimensions in the infinite momentum frame. They are based on the large $N$ limit of quantum mechanics or 1+1 dimensional field theories on the moduli space of $N$ instantons in $\IR^4$.

SuperLU Users'' Guide

Abstract: This document describes a collection of three related ANSI C subroutine libraries for solving sparse linear systems of equations AX = B: Here A is a square, nonsingular, n x n sparse matrix, and X and B are dense n x nrhs matrices, where nrhs is the number of right-hand sides and solution vectors. Matrix A need not be symmetric or definite; indeed, SuperLU is particularly appropriate for matrices with very unsymmetric structure. All three libraries use variations of Gaussian elimination optimized to take advantage both of sparsity and the computer architecture, in particular memory hierarchies (caches) and parallelism.

Jacobians of matrix transformations and functions of matrix argument

Abstract: Jacobians of matrix transformations Jacobians in orthogonal and related transformations Jacobians in the complex case transformations involving Eigenvalues and unitary matrices some special functions of matrix argument functions of matrix argument in the complex case.

Linear scaling conjugate gradient density matrix search as an alternative to diagonalization for first principles electronic structure calculations

Abstract: Advances in the computation of the Coulomb, exchange, and correlation contributions to Gaussian-based Hartree–Fock and density functional theory Hamiltonians have demonstrated near-linear scaling with molecular size for these steps. These advances leave the O(N3) diagonalization bottleneck as the rate determining step for very large systems. In this work, a conjugate gradient density matrix search (CG-DMS) method has been successfully extended and computationally implemented for use with first principles calculations. A Cholesky decomposition of the overlap matrix and its inverse is used to transform to and back from an orthonormal basis, which can be formed in near-linear time for sparse systems. Linear scaling of CPU time for the density matrix search and crossover of CPU time with diagonalization is demonstrated for polyglycine chains containing up to 493 atoms and water clusters up to 900 atoms.

Highly scalable parallel algorithms for sparse matrix factorization

Abstract: In this paper, we describe scalable parallel algorithms for symmetric sparse matrix factorization, analyze their performance and scalability, and present experimental results for up to 1,024 processors on a Gray T3D parallel computer. Through our analysis and experimental results, we demonstrate that our algorithms substantially improve the state of the art in parallel direct solution of sparse linear systems-both in terms of scalability and overall performance. It is a well known fact that dense matrix factorization scales well and can be implemented efficiently on parallel computers. In this paper, we present the first algorithms to factor a wide class of sparse matrices (including those arising from two- and three-dimensional finite element problems) that are asymptotically as scalable as dense matrix factorization algorithms on a variety of parallel architectures. Our algorithms incur less communication overhead and are more scalable than any previously known parallel formulation of sparse matrix factorization. Although, in this paper, we discuss Cholesky factorization of symmetric positive definite matrices, the algorithms can be adapted for solving sparse linear least squares problems and for Gaussian elimination of diagonally dominant matrices that are almost symmetric in structure. An implementation of one of our sparse Cholesky factorization algorithms delivers up to 20 GFlops on a Gray T3D for medium-size structural engineering and linear programming problems. To the best of our knowledge, this is the highest performance ever obtained for sparse Cholesky factorization on any supercomputer.

D0 Branes on T^n and Matrix Theory

Abstract: The Hamiltonian describing Matrix theory on T^n is identified with the Hamiltonian describing the dynamics of D0-branes on T^n in an appropriate weak coupling limit for all n up to 5. New subtleties arise in taking this weak coupling limit for n=6, since the transverse size of the D0 brane system blows up in this limit. This can be attributed to the appearance of extra light states in the theory from wrapped D6 branes. This subtlety is related to the difficulty in finding a Matrix formulation of M-theory on T^6.

Matrix market: a web resource for test matrix collections

Abstract: We describe a repository of data for the testing of numerical algorithms and mathematical software for matrix computations. The repository is designed to accommodate both dense and sparse matrices, as well as software to generate matrices. It has been seeded with the well-known Harwell-Boeing sparse matrix collection. The raw data files have been augmented with an integrated World Wide Web interface which describes the matrices in the collection quantitatively and visually. For example, each matrix has a Web page which details its attributes, graphically depicts its sparsity pattern, and provides access to the matrix itself in several formats. In addition, a search mechanism is included which allows retrieval of matrices based on a variety of attributes, such as type and size, as well as through free-text search in abstracts. The URL is http://math.nist.gov/MatrizMarket/.

The design and use of algorithms for permuting large entries to the diagonal of sparse matrices

Abstract: 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.