31 Matching Results

Search Results

Advanced search parameters have been applied.

Iterative Solution of Linear Programs

Description: By perturbing a linear program to a quadratic program, it is possible to solve the latter in its dual variable space by iterative techniques such as successive over-relaxation (SOR) methods. This provides a solution to the original linear program.
Date: July 1979
Creator: Mangasarian, Olvi L.
Partner: UNT Libraries Government Documents Department

Improving the Accuracy of Computed Eigenvalues and Eigenvectors

Description: This paper describes a computational method for improving the accuracy of a given eigenvalue and its associated eigenvector. The method is analogous to iterative improvement for the solution of linear systems. An iterative algorithm using working precision arithmetic is applied to increase the accuracy of the eigenpair. The only extended precision computation is the residual calculation. The method is related to inverse iteration and to Newton's method applied to the eigenvalue problem.
Date: July 1981
Creator: Dongarra, J. J.; Moler, Cleve B. & Wilkinson, J. H.
Partner: UNT Libraries Government Documents Department

Examples and Applications of Infinite Iterated Function Systems

Description: The aim of this work is the study of infinite conformal iterated function systems. More specifically, we investigate some properties of a limit set J associated to such system, its Hausdorff and packing measure and Hausdorff dimension. We provide necessary and sufficient conditions for such systems to be bi-Lipschitz equivalent. We use the concept of scaling functions to obtain some result about 1-dimensional systems. We discuss particular examples of infinite iterated function systems derived from complex continued fraction expansions with restricted entries. Each system is obtained from an infinite number of contractions. We show that under certain conditions the limit sets of such systems possess zero Hausdorff measure and positive finite packing measure. We include an algorithm for an approximation of the Hausdorff dimension of limit sets. One numerical result is presented. In this thesis we also explore the concept of positively recurrent function. We use iterated function systems to construct a natural, wide class of such functions that have strong ergodic properties.
Date: August 2000
Creator: Hanus, Pawel Grzegorz
Partner: UNT Libraries

Users Manual for KSP Data-Structure-Neutral Codes Implementing Krylov Space Methods

Description: The combination of a Krylov space method and a pre-conditioner is at the heart of most modern numerical codes for the iterative solution of linear systems. This document contains both a users manual and a description of the implementation for the Krylov space methods package KSP included as part of the Portable, Extensible Tools for Scientific computation package (PETSc). PETSc is a large suite of data-structure-neutral libraries for the solution of large-scale problems in scientific computation, in particular on massively parallel computers. The methods in KSP are conjugate gradient method, GMRES, BiCG-Stab, two versions of transpose-free QMR, and others. All of the methods are coded using a common, data-structure-neutral framework and are compatible with the sequential, parallel, and out-of-core solution of linear systems. The codes make no assumptions about the representation of the linear operator; implicitly defined operators (say, calculated using differencing) are fully supported. In addition, unlike all other iterative packages we are aware of, the vector operations are also data-structure neutral. Once certain vector primitives are provided, the same KSP software runs unchanged using any vector storage format. It is not restricted to a few common vector representations. The codes described are actual working codes that run on a large variety of machines including the IBM SP1, Intel DELTA, workstations, networks of workstations, the TMC CM-5, and the CRAY C90. New Krylov space methods may be easily added to the package and used immediately with any application code that has been written using KSP; no changes to the application code are needed.
Date: August 1994
Creator: Gropp, William & Smith, Barry
Partner: UNT Libraries Government Documents Department

Newton's Method with a Model Trust-Region Modification

Description: A modified Newton method for unconstrained minimization is presented and analyzed. The modification is based upon the model trust region approach. This report contains a thorough analysis of the locally constrained quadratic minimizations that arise as sub-problems in the modified Newton iteration. Several promising alternatives are presented for solving these sub-problems in ways that overcome certain theoretical difficulties exposed by this analysis. Very strong convergence results are presented concerning the minimization algorithm. In particular, the explicit use of second-order information is justified by demonstrating that the iterates converge to a point that satisfies the second-order necessary conditions for minimization. With the exception of very pathological cases this convergence occurs whenever the algorithm is applied to problems with continuous second partial derivatives.
Date: September 1980
Creator: Sorensen, D. C.
Partner: UNT Libraries Government Documents Department

Newton's Method

Description: Newton's method plays a central role in the development of numerical techniques for optimization. In fact, most of the current practical methods for optimization can be viewed as variations on Newton's method. It is therefore important to understand Newton's method as an algorithm in its own right and as a key introduction to the most recent ideas in this area. One of the aims of this expository paper is to present and analyze two main approaches to Newton's method for unconstrained minimization: the line search approach and the trust region approach. The other aim is to present some of the recent developments in the optimization field which are related to Newton's method. In particular, we explore several variations on Newton's method which are appropriate for large scale problems, and we also show how quasi-Newton methods can be derived quite naturally from Newton's method.
Date: February 1982
Creator: Moré, Jorge J. & Sorensen, D. C.
Partner: UNT Libraries Government Documents Department

Simplified Linear Equation Solvers : Users Manual

Description: The solution of large sparse systems of linear equations is at the heart of many algorithms in scientific computing. The SLES package is a set of easy-to-use yet powerful and extensible routines for solving large sparse linear systems. The design of the package allows new techniques to be used in existing applications without any source code changes in the applications.
Date: February 1993
Creator: Gropp, William & Smith, Barry
Partner: UNT Libraries Government Documents Department

Iterative Solution of Linear Boundary Value Problems

Description: The investigation is initially a continuation of Neuberger's work on linear boundary value problems. A very general iterative procedure for solution of these problems is described. The alternating-projection theorem of von Neumann is the mathematical starting point for this study. Later theorems demonstrate the validity of numerical approximation for Neuberger's method under certain conditions. A sampling of differential equations within the scope of our iterative method is given. The numerical evidence is that the procedure works well on neutral-state equations, for which no software is written now.
Date: August 1983
Creator: Walsh, John Breslin
Partner: UNT Libraries

Application of a Multigrid Method to a Buoyancy-Induced Flow Problem

Description: The numerical prediction of buoyancy-induced flows provides special difficulties for standard numerical techniques associated with velocity-buoyancy coupling. We present a multigrid algorithm based upon a novel relaxation scheme that handles this coupling correctly. Numerical experiments have been performed that show that this approach is reasonably efficient and robust for a range of Rayleigh numbers and a variety of cycling strategies.
Date: October 1987
Creator: Thompson, C. P.; Leaf, G. K. & Vanka, S. P.
Partner: UNT Libraries Government Documents Department

The Application of Automated Reasoning to Proof Translation and to Finding Proofs with Specified Properties: a Case Study in Many-Valued Sentential Calculus

Description: In both mathematics and logic, many theorems exist such that each can be proved in entirely different ways. For a striking example, there exist theorems from group theory that can be proved by relying solely on equality and (from the viewpoint of automated reasoning) the use of paramodulation, but can also be proved in a notation in which equality is totally absent and the inference rule is condensed detachment (captured with a single clause and the rule hyper-resolution). A study of such examples immediately shows how far from obvious is the problem of producing a proof in one system even in the presence of a proof in another; such problems can be viewed as ones of translation, where the rules of translation and the translation itself are frequently difficult to obtain. In this report, we discuss in detail various techniques that can be applied by the automated reasoning program OTTER to address the translation problem to obtain a proof in one notation and inference system given a proof in a completely different notation and inference system. To illustrate the techniques, we present a full treatment culminating in a successful translation'' of a proof of a theorem from many-valued sentential calculus. To our delight and amazement, instead of the expected translation consisting of approximately 175 applications of condensed detachment, OTTER obtained a far shorter proof. We also touch on techniques for finding shorter proofs and techniques for finding proofs satisfying some given property.
Date: August 1991
Creator: Wos, Larry & McCune, William W.
Partner: UNT Libraries Government Documents Department

A Preconditioned Conjugate Gradient Method for Solving a Class of Non-Symmetric Linear Systems

Description: This report describes a conjugate gradient preconditioning scheme for solving a certain system of equations which arises in the solution of a three dimensional partial differential equation. The problem involves solving systems of equations where the matrices are large, sparse, and non-symmetric.
Date: October 1981
Creator: Dongarra, J. J.; Leaf, G. K. & Minkoff, Michael
Partner: UNT Libraries Government Documents Department

Dispersive Approximations for Hyperbolic Conservation Laws

Description: Necessary and sufficient conditions are given so that the Sobolev-type partial differential equations generate a contraction semigroup. It is shown that any nonlinear contraction from L/sup 1/(R) to itself that preserves the integral and commutes with translations satisfies maximum and minimum principles. This lemma is applied to the solution operator S/sub t/ to give necessary and sufficient conditions that S/t/ satisfy a maximum principle, despite the dispersive nature. Sufficient conditions are given so that the solutions converge, as nu and beta tend to zero, to the entropy solution of the conservation law. A larger class of monotone finite-difference schemes for the numerical solution of the conservation law motivated by finite-difference discretizations of the Sobolev equations, is introduced, and convergence results are proved for methods in this class. The methods analyzed include some that were previously used to approximate the solution of a linear waterflood problem in petroleum engineering.
Date: December 1981
Creator: Lucier, Bradley J.
Partner: UNT Libraries Government Documents Department

Computing a Trust Region Step

Description: An algorithm is proposed for the problem of minimizing a quadratic function subject to an ellipsoidal constraint which is guaranteed to produce a nearly optimal solution in a finite number of iterations. A robust and efficient algorithm for this problem is required to compute the step between iterates in trust region methods for optimization problems. We also consider the use of our algorithm in a trust region Newton's method. In particular, we prove that under reasonable assumptions the sequence (X/sub k/) generated by Newton's method has a limit point X* which satisfies the first and second order necessary conditions for a minimizer of the objective function f. Numerical results for GQTPAR, which is a Fortran implementation of our algorithm, show that GQTPAR is quite successful in a trust region method. In our tests a call to GQTPAR only required 1.6 iterations on the average.
Date: December 1981
Creator: Moré, Jorge J. & Sorensen, D. C.
Partner: UNT Libraries Government Documents Department

Improving the Accuracy of Computed Singular Values

Description: This paper describes a computational method for improving the accuracy of a given singular value and its associated left and right singular vectors. The method is analogous to iterative improvement for the solution of linear systems. That is, by means of a low-precision computation, an iterative algorithm is applied to increase the accuracy of the singular value and vectors; extended precision computations are used in the residual calculation. The method is related to Newton's Method applied to the singular value problem and inverse iteration for the eigenvalue problem.
Date: January 1982
Creator: Dongarra, J. J.
Partner: UNT Libraries Government Documents Department

Updating the Symmetric Indefinite Factorization with Applications in a Modified Newton's Method: A Dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosphy in Mathematics

Description: In recent years the use of quasi-Newton methods in optimization algorithms has inspired much of the research in an area of numerical linear algebra called updating matrix factorizations. Previous research in this area has been concerned with updating the factorization of a symmetric positive definite matrix. Here, a numerical algorithm is presented for updating the Symmetric Indefinite Factorization of Bunch and Parlett. The algorithm requires only O(n²) arithmetic operations to update the factorization of a symmetric matrix when modified by a rank-one matrix. An error analysis of this algorithm is given. Computational results are presented that investigate the timing and accuracy of this algorithm. Another algorithm is presented for the unconstrained minimization of a nonlinear functional. The algorithm is a modification of Newton's method. At points where the Hessian is indefinite the search for the next iterate is conducted along a quadratic curve in the plane spanned by a direction of negative curvature and a gradient related descent direction. The stopping criteria for this search takes into account the second-order derivative information. The result is that the iterates are shown to converge globally to a critical point at which the Hessian is positively semidefinite. Computational results are presented which indicate that the method is promising.
Date: 1977
Creator: Sorensen, Danny C.
Partner: UNT Libraries Government Documents Department

BlockSolve v1. 1: Scalable Library Software for the Parallel Solution of Sparse Linear Systems

Description: BlockSolve is a software library for solving large, sparse systems of linear equations on massively parallel computers. The matrices must be symmetric, but may have an arbitrary sparsity structure. BlockSolve is a portable package that is compatible with several different message-passing pardigms. This report gives detailed instructions on the use of BlockSolve in applications programs.
Date: March 1993
Creator: Jones, Mark T. & Plassmann, Paul E.
Partner: UNT Libraries Government Documents Department

Analysis of Nonlinear Fluid Structure Interaction Transient in Fast Reactors

Description: A generalized Eulerian method is described for analyzing the fluid transients and the structural response in nuclear reactors under the postulated accident conditions. The phenomena considered are the wave propagation, slug impact, sodium spillage, bubble migration, and the fluid-structure interaction. The basic equations and numerical formulation are presented in detail. Sample calculations are given to illustrate the analysis. It is shown from the results that the implicit, iterative method used is unconditionally stable and is especially suitable for problems involving large material distortions.
Date: 1978
Creator: Yang, C. Y.
Partner: UNT Libraries Government Documents Department

The Global Structure of Iterated Function Systems

Description: I study sets of attractors and non-attractors of finite iterated function systems. I provide examples of compact sets which are attractors of iterated function systems as well as compact sets which are not attractors of any iterated function system. I show that the set of all attractors is a dense Fs set and the space of all non-attractors is a dense Gd set it the space of all non-empty compact subsets of a space X. I also investigate the small trans-finite inductive dimension of the space of all attractors of iterated function systems generated by similarity maps on [0,1].
Date: May 2009
Creator: Snyder, Jason Edward
Partner: UNT Libraries

DISPL: a Software Package for One and Two Spatially Dimensioned Kinetics-Diffusion Problems

Description: DISPL is a software package for solving some second-order nonlinear systems of partial differential equations including parabolic, elliptic, hyperbolic, and some mixed types such as parabolic-elliptic equations. Fairly general nonlinear boundary conditions are allowed as well as interface conditions for problems in an inhomogeneous media. The spatial domain is one- or two-dimensional with Cartesian, cylindrical, or spherical (in one dimension only) geometry. The numerical method is based on the use of Galerkin's procedure combined with the use of B-splines in order to reduce the system of PDE's to a system of ODE's. The latter system is then solved with a sophisticated ODE software package. Software features include extensive dump/restart facilities, free format input, moderate printed output capability, dynamic storage allocation, and three graphics packages.
Date: November 1978
Creator: Leaf, G. K.; Minkoff, M.; Byrne, G. D.; Sorensen, D.; Bleakney, T. & Saltzman, J.
Partner: UNT Libraries Government Documents Department

COMMIX-SA-1: a Three-Dimensional Thermohydrodynamic Computer Program for Solar Applications

Description: COMMIX-SA-1 is a three-dimensional, transient, single-phase, compressible-flow, component computer program for thermohydrodynamic analysis. It was developed for solar applications in general, and for analysis of thermocline storage tanks in particular. The conservation equations (in cylindrical coordinates) for mass, momentum, and energy are solved as an initial-boundary-value problem. The detailed numerical-solution procedure based on a modified ICE (Implicit Continuous-Fluid Eulerian) technique is described. A method for treating the singularity problem arising at the origin of a cylindrical-coordinate system is presented. In addition, the thermal interactions between fluid and structures (tank walls, baffles, etc.) are explicitly accounted for. Finally, the COMMIX-SA-1 code structure is delineated, and an input description and sample problems are presented.
Date: November 1980
Creator: Sha, W. T.; Lin, E. I. H.; Schmitt, R. C.; Liu, K. V.; Hull, J. R.; Oras, J. J. et al.
Partner: UNT Libraries Government Documents Department

A Survey of Numerical Methods for Hydraulic Transients

Description: The finite difference methods of Lax, Lax-Wendroff (two-step), donor cell type and finite-element method using Galerkin procedure with B-splines as approximating functions are compared with the method of characteristics for the solution of water-hammer transients as typified by valve closure problems. From the point of view of accuracy, the two-step Lax-Wendroff method and the method of characteristics are comparable and produce the best results. The Lax methods fair worst. The donor cell type and the Galerkin procedure with quadratic B-spline basis as approximating functions display roughly the same accuracy. From the comparison presented, it appears that Galerkin technique offers no substantial advantage over the other finite-difference methods except that of ease in handling boundary conditions as compared to finite-difference methods.
Date: October 1977
Creator: Leaf, G. K. & Chawla, T. C.
Partner: UNT Libraries Government Documents Department

Computations of Turbulent Recirculating Flows with Fully Coupled Solution of Momentum and Continuity Equations

Description: A fully coupled solution algorithm for pressure-linked fluid flow equations earlier found to be rapidly convergent in laminar flows has been extended to calculate turbulent flows. The governing mean flow equations are solved in conjunction with a two-equation (k - epsilon) turbulence model. A number of two-dimensional recirculating flows have been computed and it is shown that the calculation procedure is rapidly convergent in all the cases. The calculations have been compared with published experimental data; their agreement is in accord with other published experiences with the (k - epsilon) model in similar flows.
Date: August 1983
Creator: Vanka, S. P.
Partner: UNT Libraries Government Documents Department

Theoretical Studies of the Atmospheric Triatomic Molecules H₂O, N₂O, NO₂, CO₂, O₃, and their Ions

Description: Definitive location of excited energy levels and the delineation of reaction mechanisms were provided by ab initio calculations of the triatomic atmospheric molecules and their ions. Linear geometry to bent geometry correlation diagrams were also determined. These provide simple ways of predicting reaction possibilities. Potential energy surfaces were computed for the ground ²B₁ and ²B₂ states of H2O⁺. Previous theoretical spectra of carbon dioxide were shown to be of limited accuracy and therefore experimental assignments based on these are not definitive. Correlated results for the vertical excitation energies of ozone are in very good agreement with experiments and with other calculations. The ground state potential energy surface of N2O is characterized and related to the ground state potential energy surface of the neutral N2O molecule by combining theoretical and experimental information. A definitive theoretical study provided energy levels for NO2⁺ and NO2⁻, and extension of previous studies on NO2. Agreement between the two studies was good.
Date: 1977?
Creator: Wahl, Arnold C.; England, Walter B.; Rosenberg, Bruce J.; Hopper, Darrel G. & Fortune, Patrick J.
Partner: UNT Libraries Government Documents Department

Fully-Coupled Solution of Pressure-Linked Fluid-Flow Equations

Description: A robust and efficient numerical scheme has been developed for the solution of the finite-differenced pressure linked fluid flow equations. The algorithm solves the set of nonlinear simultaneous equations by a combination of Newton's method and efficient sparse matrix techniques. In tests on typical recirculating flows the method is rapidly convergent. The method does not require any under-relaxation or other convergence-enhancing techniques employed in iterative schemes. It is currently described for two-dimensional steady state flows but is extendible to three dimensions and mildly time-varying flows. The method is robust to changes in Reynolds number, grid aspect ratio, and mesh size. This paper reports the algorithm and the results of calculations performed.
Date: August 1983
Creator: Vanka, S. P. & Leaf, G. K.
Partner: UNT Libraries Government Documents Department