The interplay between recent results and methodologies in numerical linear algebra and mathematical software and their application to problems arising in systems, control, and estimation theory is discussed. The impac...
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The interplay between recent results and methodologies in numerical linear algebra and mathematical software and their application to problems arising in systems, control, and estimation theory is discussed. The impact of finite precision, finite range arithmetic [including the implications of the proposed IEEE floating point standard(s)] on control design computations is illustrated with numerous examples as are pertinent remarks concerning numerical stability and conditioning. Basic tools from numerical linear algebra such as linear equations, linear least squares, eigenproblems, generalized eigenproblems, and singular value decomposition are then outlined. A selected list of applications of the basic tools then follows including algorithms for solution of problems such as matrix exponentials, frequency response, system balancing, and matrix Riccati equations. The implementation of such algorithms as robust mathematical software is then discussed. A number of issues are addressed including characteristics of reliable mathematical software, availability and evaluation, language implications (Fortran, Ada, etc.), and the overall role of mathematical software as a component of computer-aided control system design.
This note provides a short, self-contained treatment, using linear algebra and matrix theory, for establishing maximal periods, underlying structure, and choice of starting values for shift-register and lagged-Fibonac...
This note provides a short, self-contained treatment, using linear algebra and matrix theory, for establishing maximal periods, underlying structure, and choice of starting values for shift-register and lagged-Fibonacci random number generators.
The first structure theory in abstract algebra was that of finite dimensional Lie algebras (Cartan-Killing), followed by the structure theory of associative algebras (Wedderburn-Artin). These theories determine, in a ...
ISBN:
(纸本)9780897911511
The first structure theory in abstract algebra was that of finite dimensional Lie algebras (Cartan-Killing), followed by the structure theory of associative algebras (Wedderburn-Artin). These theories determine, in a non-constructive way, the basic building blocks of the respective algebras (the radical and the simple components of the factor by the radical). In view of the extensive computations done in such algebras, it seems important to design efficient algorithms to find these building *** find polynomial time solutions to a substantial part of these problems. We restrict our attention to algebras over finite fields and over algebraic number fields. We succeed in determining the radical (the “bad part” of the algebra) in polynomial time, using (in the case of prime characteristic) some new algebraic results developed in this paper. For associative algebras we are able to determine the simple components as well. This latter result generalizes factorization of polynomials over the given field. Correspondingly, our algorithm over finite fields is Las *** of the results generalize to fields given by *** fundamental problems remain open. An example: decide whether or not a given rational algebra is a noncommutative field.
We develop fast parallel solutions to a number of basic problems involving solvable and nilpotent permutation groups. Testing solvability is in NC, and RNC includes, for solvable groups, finding order, testing members...
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We develop fast parallel solutions to a number of basic problems involving solvable and nilpotent permutation groups. Testing solvability is in NC, and RNC includes, for solvable groups, finding order, testing membership, finding the derived series and finding a composition series. Additionally, for nilpotent groups, one can, in RNC, find the center, a central composition series, and point-wise stabilizers of sets. There are applications to graph isomorphism. In fact, we exhibit a class of vertex-colored graphs for which determining isomorphism is NC-equivalent to computing ranks of matrices Over small fields. A useful tool is the observation that the problem of finding the smallest subspace containing a given set of vectors and closed under a given set of linear transformations (all over a small field) belongs to RNC.
applications of computeralgebra to linear partial differential equations of first order and to nonlinear control theory are presented. It is shown how symbolic systems can compute automatically: (i) the dimension of ...
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The condition number (relative to the Frobenius norm) of the n × n matrix P n = [ p i −1 ( x j )] i , j = 1 n is investigated, where p r (·) = p r (·; dλ ) are orthogonal polynomials with respect to so...
The condition number (relative to the Frobenius norm) of the n × n matrix P n = [ p i −1 ( x j )] i , j = 1 n is investigated, where p r (·) = p r (·; dλ ) are orthogonal polynomials with respect to some weight distribution dλ and x j are pairwise distinct real numbers. If the nodes x j are the zeros of p n , the condition number is either expressed, or estimated from below and above, in terms of the Christoffel numbers for dλ , depending on whether the p r are normalized or not. For arbitrary real x j and normalized p r a lower bound of the condition number is obtained in terms of the Christoffel function evaluated at the nodes. Numerical results are given for minimizing the condition number as a function of the nodes for selected classical distributions dλ .
The proceedings contain 28 papers. The special focus in this conference is on computeralgebra. The topics include: The design of Maple: A compact, portable, and powerful computeralgebra system;LISP compilation viewe...
ISBN:
(纸本)9783540128687
The proceedings contain 28 papers. The special focus in this conference is on computeralgebra. The topics include: The design of Maple: A compact, portable, and powerful computeralgebra system;LISP compilation viewed as provable semantics preserving program transformation;Implementing REDUCE on a microcomputer;a note on the complexity of constructing Gröbner-bases;gröbner bases, Gaussian elimination and resolution of systems of algebraic equations;the computation of the Hilbert function;an algorithm for constructing detaching bases in the ring of polynomials over a field;on the problem of Behā Eddīn 'Amūlī and the computation of height functions;a procedure for determining algebraic integers of given norm;integration — What do we want from the theory?;computation of integral solutions of a special type of systems of quadratic equations;factorisation of sparse polynomials;early detection of true factors in univariate polynomial factorization;on the complexity of finding short vectors in integer lattices;factoring polynomials over algebraic number fields;the construction of a complete minimal set of contextual normal forms;a knowledge-based approach to user-friendliness in symbolic computing;computeralgebra and VLSI, prospects for cross fertilization;code optimization of multivariate polynomial schemes: A pragmatic approach;the Euclidean algorithm for Gaussian integers;multi polynomial remainder sequence and its application to linear diophantine equations;towards mechanical solution of the Kahan ellipse problem I;automatically determining symmetries of ordinary differential equations;algebraic computation of the statistics of the solution of some nonlinear stochastic differential equations;characterization of a linear differential system with a regular singularity.
Let S,T be n×n integral matrices and one of them positive definite. We develop a method to decide whether a solution X∈n×n of XtrSX=T exists and, if the answer is affirmative, an algorithm for the computati...
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The fuzzy relational database model originated by the authors permits fuzzy domain values from a discrete, finite universe. The model is extended here by demonstrating that fuzzy numbers may be employed as domain valu...
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The fuzzy relational database model originated by the authors permits fuzzy domain values from a discrete, finite universe. The model is extended here by demonstrating that fuzzy numbers may be employed as domain values without loss of consistency with respect to representation or the relational algebra. Where equivalence is required in an ordinary relational database, similarity is employed in a fuzzy relational database. For discrete, finite universes, similarity between atomic elements is described via a fuzzy similarity relation with max-min transitivity. Two or more fuzzy numbers are defined to be α-similar if their union forms a continuous α-level set over the real line. This convention effects the partitioning of fuzzy number domains that is necessary to assure the well-definedness of the fuzzy relational algebra.
We use a simple example (the projective plane on seven points) to give an introductory survey on the problems and methods in finite geometries - an area of mathematics related to geometry, combinatorial theory, algebr...
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