Structured Matrix Based Methods for Approximate Polynomial GCD
Structured Matrix Based Methods for Approximate Polynomial GCD

Author: Paola Boito

Publisher: Springer Science & Business Media

Published: 2012-03-13

Total Pages: 208

ISBN-13: 8876423818

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Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree.

Advanced Numerical Methods in Applied Sciences
Advanced Numerical Methods in Applied Sciences

Author: Luigi Brugnano

Publisher: MDPI

Published: 2019-06-20

Total Pages: 306

ISBN-13: 3038976660

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The use of scientific computing tools is currently customary for solving problems at several complexity levels in Applied Sciences. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and better performing numerical methods that are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.

Computer Algebra in Scientific Computing
Computer Algebra in Scientific Computing

Author: Vladimir P. Gerdt

Publisher: Springer

Published: 2013-08-15

Total Pages: 457

ISBN-13: 3319022970

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This book constitutes the proceedings of the 14th International Workshop on Computer Algebra in Scientific Computing, CASC 2013, held in Berlin, Germany, in September 2013. The 33 full papers presented were carefully reviewed and selected for inclusion in this book. The papers address issues such as polynomial algebra; the solution of tropical linear systems and tropical polynomial systems; the theory of matrices; the use of computer algebra for the investigation of various mathematical and applied topics related to ordinary differential equations (ODEs); applications of symbolic computations for solving partial differential equations (PDEs) in mathematical physics; problems arising at the application of computer algebra methods for finding infinitesimal symmetries; applications of symbolic and symbolic-numeric algorithms in mechanics and physics; automatic differentiation; the application of the CAS Mathematica for the simulation of quantum error correction in quantum computing; the application of the CAS GAP for the enumeration of Schur rings over the group A5; constructive computation of zero separation bounds for arithmetic expressions; the parallel implementation of fast Fourier transforms with the aid of the Spiral library generation system; the use of object-oriented languages such as Java or Scala for implementation of categories as type classes; a survey of industrial applications of approximate computer algebra.

Computer Algebra in Scientific Computing
Computer Algebra in Scientific Computing

Author: François Boulier

Publisher: Springer Nature

Published: 2021-08-16

Total Pages: 485

ISBN-13: 3030851656

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This book constitutes the proceedings of the 23rd International Workshop on Computer Algebra in Scientific Computing, CASC 2021, held in Sochi, Russia, in September 2021. The 24 full papers presented together with 1 invited talk were carefully reviewed and selected from 40 submissions. The papers cover theoretical computer algebra and its applications in scientific computing.

Numerical Methods for Structured Matrices and Applications
Numerical Methods for Structured Matrices and Applications

Author: Dario Andrea Bini

Publisher: Springer Science & Business Media

Published: 2011-02-09

Total Pages: 439

ISBN-13: 3764389966

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This cross-disciplinary volume brings together theoretical mathematicians, engineers and numerical analysts and publishes surveys and research articles related to topics such as fast algorithms, in which the late Georg Heinig made outstanding achievements.

Latent Variable Analysis and Signal Separation
Latent Variable Analysis and Signal Separation

Author: Yannick Deville

Publisher: Springer

Published: 2018-06-05

Total Pages: 583

ISBN-13: 3319937642

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This book constitutes the proceedings of the 14th International Conference on Latent Variable Analysis and Signal Separation, LVA/ICA 2018, held in Guildford, UK, in July 2018.The 52 full papers were carefully reviewed and selected from 62 initial submissions. As research topics the papers encompass a wide range of general mixtures of latent variables models but also theories and tools drawn from a great variety of disciplines such as structured tensor decompositions and applications; matrix and tensor factorizations; ICA methods; nonlinear mixtures; audio data and methods; signal separation evaluation campaign; deep learning and data-driven methods; advances in phase retrieval and applications; sparsity-related methods; and biomedical data and methods.

Numerical Methods for Roots of Polynomials - Part II
Numerical Methods for Roots of Polynomials - Part II

Author: J.M. McNamee

Publisher: Elsevier Inc. Chapters

Published: 2013-07-19

Total Pages: 94

ISBN-13: 0128077050

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The zeros of a polynomial can be readily recovered from its linear factors. The linear factors can be approximated by first splitting a polynomial numerically into the product of its two nonconstant factors and then recursively splitting every computed nonlinear factor in similar fashion. For both the worst and average case inputs the resulting algorithms solve the polynomial factorization and root-finding problems within fixed sufficiently small error bounds by using nearly optimal arithmetic and Boolean time, that is using nearly optimal numbers of arithmetic and bitwise operations; in the case of a polynomial with integer coefficients and simple roots we can immediately extend factorization to root isolation, that is to computing disjoint covering discs, one for every root on the complex plane. The presented algorithms compute highly accurate approximations to all roots nearly as fast as one reads the input coefficients. Furthermore, our algorithms allow processor efficient parallel acceleration, which enables root-finding, factorization, and root isolation in polylogarithmic arithmetic and Boolean time. The chapter thoroughly covers the design and analysis of these algorithms, including auxiliary techniques of independent interest. At the end we compare the presented polynomial root-finders with alternative ones, in particular with the popular algorithms adopted by users based on supporting empirical information. We also comment on some promising directions to further progress.

LATIN 2008: Theoretical Informatics
LATIN 2008: Theoretical Informatics

Author: Eduardo Sany Laber

Publisher: Springer

Published: 2008-04-04

Total Pages: 809

ISBN-13: 3540787739

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This proceedings volume examines a range of topics in theoretical computer science, including automata theory, data compression, logic, machine learning, mathematical programming, parallel and distributed computing, quantum computing and random structures.

A Graduate Introduction to Numerical Methods
A Graduate Introduction to Numerical Methods

Author: Robert M. Corless

Publisher: Springer Science & Business Media

Published: 2013-12-12

Total Pages: 896

ISBN-13: 1461484537

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This book provides an extensive introduction to numerical computing from the viewpoint of backward error analysis. The intended audience includes students and researchers in science, engineering and mathematics. The approach taken is somewhat informal owing to the wide variety of backgrounds of the readers, but the central ideas of backward error and sensitivity (conditioning) are systematically emphasized. The book is divided into four parts: Part I provides the background preliminaries including floating-point arithmetic, polynomials and computer evaluation of functions; Part II covers numerical linear algebra; Part III covers interpolation, the FFT and quadrature; and Part IV covers numerical solutions of differential equations including initial-value problems, boundary-value problems, delay differential equations and a brief chapter on partial differential equations. The book contains detailed illustrations, chapter summaries and a variety of exercises as well some Matlab codes provided online as supplementary material. “I really like the focus on backward error analysis and condition. This is novel in a textbook and a practical approach that will bring welcome attention." Lawrence F. Shampine A Graduate Introduction to Numerical Methods and Backward Error Analysis” has been selected by Computing Reviews as a notable book in computing in 2013. Computing Reviews Best of 2013 list consists of book and article nominations from reviewers, CR category editors, the editors-in-chief of journals, and others in the computing community.

Structured Matrices and Polynomials
Structured Matrices and Polynomials

Author: Victor Y. Pan

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 299

ISBN-13: 1461201292

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This user-friendly, engaging textbook makes the material accessible to graduate students and new researchers who wish to study the rapidly exploding area of computations with structured matrices and polynomials. The book goes beyond research frontiers and, apart from very recent research articles, includes previously unpublished results.