Elimination sequence optimization for SPAR

SPAR is a large-scale computer program for finite element structural analysis. The program allows user specification of the order in which the joints of a structure are to be eliminated since this order can have significant influence over solution performance, in terms of both storage requirements and computer time. An efficient elimination sequence can improve performance by over 50% for some problems. Obtaining such sequences, however, requires the expertise of an experienced user and can take hours of tedious effort to affect. Thus, an automatic elimination sequence optimizer would enhance productivity by reducing the analysts' problem definition time and by lowering computer costs. Two possible methods for automating the elimination sequence specifications were examined. Several algorithms based on the graph theory representations of sparse matrices were studied with mixed results. Significant improvement in the program performance was achieved, but sequencing by an experienced user still yields substantially better results. The initial results provide encouraging evidence that the potential benefits of such an automatic sequencer would be well worth the effort

Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre Polynomials

This article is devoted to compute numerical solutions of some classes and families of fractional order differential equations (FODEs). For the required numerical analysis, we utilize Laguerre polynomials and establish some operational matrices regarding to fractional order derivatives and integrals without discretizing the data. Further corresponding to boundary value problems (BVPs), we establish a new operational matrix which is used to compute numerical solutions of boundary value problems (BVPs) of FODEs. Based on these operational matrices (OMs), we convert the proposed (FODEs) or their system to corresponding algebraic equation of Sylvester type or system of Sylvester type. The resulting algebraic equations are solved by MATLAB® using Gauss elimination method for the unknown coefficient matrix. To demonstrate the suggested scheme for numerical solution, many suitable examples are provided

Completion of partial operator matrices

This work concerns completion problems for partial operator matrices. A partial matrix is an m-by-n array in which some entries are specified and the remaining are unspecified. We allow the entries to be operators acting between corresponding vector spaces (in general, bounded linear operators between Hilbert spaces). Graphs are associated with partial matrices. Chordal graphs and directed graphs with a perfect edge elimination scheme play a key role in our considerations. A specific choice for the unspecified entries is referred to as a completion of the partial matrix. The completion problems studied here involve properties such as: zero-blocks in certain positions of the inverse, positive (semi)definitness, contractivity, or minimum negative inertia for Hermitian operator matrices. Some completion results are generalized to the case of combinatorially nonsymmetric partial matrices. Several applications including a maximum entropy result and determinant formulae for matrices with sparse inverses are given.;In Chapter II we treat completion problems involving zero-blocks in the inverse. Our main result deals with partial operator matrices R, for which the directed graph is associated with an oriented tree. We prove that under invertibility conditions on certain principal minors, R admits a unique invertible completion F such that {dollar}(F\sp{lcub}-1{rcub})\sb{lcub}ij{rcub}{dollar} = 0 whenever {dollar}R\sb{lcub}ij{rcub}{dollar} is unspecified.;Chapter III treats positive semidefinite and Hermitian completions. In the case of partial positive operator matrices with a chordal graph, a maximum entropy principle is presented, generalizing the maximum determinant result in the scalar case. We obtain a linear fractional transform parametrization for the set of all positive semidefinite completions for a generalized banded partial matrix. We also give an inertia formula for Hermitian operator matrices with sparse inverses.;In Chapter IV prior results are applied to obtain facts about contractive and linearly constrained completion problems. The solution to a general n-by-n strong-Parrott type completion problem is the main result. We prove necessary and sufficient conditions for the existence of a solution as well as a cascade transform parametrization for the set of all solutions.;Chapter V extends the results in Chapter II and III to prove determinant formulae for matrices with sparse inverses. Several ideas from graph theory are used. An inheritance principle for chordal graphs is also presented

Solution of linear systems with sparse matrices

For large scale problems in electric circuit simulation as well as in chemical process simulation, the linear solver often needs about 50 - 80 % of the total amount of computing time. For that purpose, we consider direct methods for the numerical solution of linear systems of equations with unsymmetric sparse coefficient matrices. The Gaussian elimination method is applied to solve the linear system. Here, the row permutation is used to provide numerical stability and the column permutation is chosen to control sparsity. In a new approach, implemented in the solver GSPAR2, the determination of the pivot columns is done with a modified algorithm, which has only a complexity of O(n). A partial pivoting technique is used to maintain numerical stability. For solving several linear systems with the same pattern structure of the coefficient matrix efficiently, we generate a list of pseudo code instructions for the factorization of the matrices. With it, the solver GSPAR2 has been proven successful within the simulation of several real life problems. For a number of linear systems arising from different technical problems, the computing times of GSPAR2 are compared to that of some recently released linear solvers

Fast Algorithms on Random Matrices and Structured Matrices

Randomization of matrix computations has become a hot research area in the big data era. Sampling with randomly generated matrices has enabled fast algorithms to perform well for some most fundamental problems of numerical algebra with probability close to 1. The dissertation develops a set of algorithms with random and structured matrices for the following applications: 1) We prove that using random sparse and structured sampling enables rank-r approximation of the average input matrix having numerical rank r. 2) We prove that Gaussian elimination with no pivoting (GENP) is numerically safe for the average nonsingular and well-conditioned matrix preprocessed with a nonsingular and well-conditioned f-Circulant or another structured multiplier. This can be an attractive alternative to the customary Gaussian elimination with partial pivoting (GEPP). 3) By using structured matrices of a large family we compress large-scale neural networks while retaining high accuracy. The results of our extensive are in good accordance with those of our theoretical study

Perfect Elimination Orderings for Symmetric Matrices

We introduce a new class of structured symmetric matrices by extending the notion of perfect elimination ordering from graphs to weighted graphs or matrices. This offers a common framework capturing common vertex elimination orderings of monotone families of chordal graphs, Robinsonian matrices and ultrametrics. We give a structural characterization for matrices that admit perfect elimination orderings in terms of forbidden substructures generalizing chordless cycles in graphs.Comment: 16 pages, 3 figure