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Relevant bibliographies by topics / Polynomials. Algorithms

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Academic literature on the topic 'Polynomials. Algorithms'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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Journal articles on the topic "Polynomials. Algorithms"

1

Kritzer, Peter, and Friedrich Pillichshammer. "Constructions of general polynomial lattices for multivariate integration." Bulletin of the Australian Mathematical Society 76, no. 1 (2007): 93–110. http://dx.doi.org/10.1017/s0004972700039496.

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We study a construction algorithm for certain polynomial lattice rules modulo arbitrary polynomials. The underlying polynomial lattices are special types of digital nets as introduced by Niederreiter. Dick, Kuo, Pillichshammer and Sloan recently introduced construction algorithms for polynomial lattice rules modulo irreducible polynomials which yield a small worst-case error for integration of functions in certain weighted Hilbert spaces. Here, we generalize these results to the case where the polynomial lattice rules are constructed moduloarbitrarypolynomials.

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AL-Utaibi, Khaled A., Sadiq H. Abdulhussain, Basheera M. Mahmmod, Marwah Abdulrazzaq Naser, Muntadher Alsabah, and Sadiq M. Sait. "Reliable Recurrence Algorithm for High-Order Krawtchouk Polynomials." Entropy 23, no. 9 (2021): 1162. http://dx.doi.org/10.3390/e23091162.

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Krawtchouk polynomials (KPs) and their moments are promising techniques for applications of information theory, coding theory, and signal processing. This is due to the special capabilities of KPs in feature extraction and classification processes. The main challenge in existing KPs recurrence algorithms is that of numerical errors, which occur during the computation of the coefficients in large polynomial sizes, particularly when the KP parameter (p) values deviate away from 0.5 to 0 and 1. To this end, this paper proposes a new recurrence relation in order to compute the coefficients of KPs

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Bulmer, M., D. Fearnley-Sander, and T. Stokes. "Towards a calculus of algorithms." Bulletin of the Australian Mathematical Society 50, no. 1 (1994): 81–89. http://dx.doi.org/10.1017/s000497270000959x.

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We develop a generalised polynomial formalism which captures the concept of an algebra of piece-wise denned polynomials. The formalism is based on the Boolean power construction of universal algebra. A generalisation of the theory of substitution homomorphisms is developed. The abstract operation of composition of generalised polynomials in one variable is denned and shown to correspond to function composition.

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Astapov, N. S. "Algorithms for Factorization of Polynomials of Low Degree." Programmnaya Ingeneria 12, no. 4 (2021): 200–208. http://dx.doi.org/10.17587/prin.12.200-208.

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For polynomials of the third degree of a special type, expansions into linear factors are found. Various methods of factorization of fourth-degree polynomials of general and particular types are proposed. For polynomials of the sixth degree of a special kind, representations are given in the form of a product of polynomials of lower degrees. Special attention is paid to representations through square trinomials. The decomposition of the generalized reciprocal polynomial of the sixth degree into square trinomials is given.

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Neunhöffer, Max, and Cheryl E. Praeger. "Computing Minimal Polynomials of Matrices." LMS Journal of Computation and Mathematics 11 (2008): 252–79. http://dx.doi.org/10.1112/s1461157000000590.

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AbstractWe present and analyse a Monte-Carlo algorithm to compute the minimal polynomial of ann × nmatrix over a finite field that requiresO(n3) field operations andO(n) random vectors, and is well suited for successful practical implementation. The algorithm, and its complexity analysis, use standard algorithms for polynomial and matrix operations. We compare features of the algorithm with several other algorithms in the literature. In addition we present a deterministic verification procedure which is similarly efficient in most cases but has a worst-case complexity ofO(n4). Finally, we repo

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Asadi, Mohammadali, Alexander Brandt, Robert H. C. Moir, and Marc Moreno Maza. "Algorithms and Data Structures for Sparse Polynomial Arithmetic." Mathematics 7, no. 5 (2019): 441. http://dx.doi.org/10.3390/math7050441.

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We provide a comprehensive presentation of algorithms, data structures, and implementation techniques for high-performance sparse multivariate polynomial arithmetic over the integers and rational numbers as implemented in the freely available Basic Polynomial Algebra Subprograms (BPAS) library. We report on an algorithm for sparse pseudo-division, based on the algorithms for division with remainder, multiplication, and addition, which are also examined herein. The pseudo-division and division with remainder operations are extended to multi-divisor pseudo-division and normal form algorithms, re

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Coulter, Robert S., George Havas, and Marie Henderson. "On decomposition of sub-linearised-polynomials." Journal of the Australian Mathematical Society 76, no. 3 (2004): 317–28. http://dx.doi.org/10.1017/s1446788700009885.

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AbstractWe give a detailed exposition of the theory of decompositions of linearised polynomials, using a well-known connection with skew-polynomial rings with zero derivative. It is known that there is a one-to-one correspondence between decompositions of linearised polynomials and sub-linearised polynomials. This correspondence leads to a formula for the number of indecomposable sub-linearised polynomials of given degree over a finite field. We also show how to extend existing factorisation algorithms over skew-polynomial rings to decompose sub-linearised polynomials without asymptotic cost.

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van der Hoeven, Joris, and Michael Monagan. "Computing one billion roots using the tangent Graeffe method." ACM Communications in Computer Algebra 54, no. 3 (2020): 65–85. http://dx.doi.org/10.1145/3457341.3457342.

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Let p be a prime of the form p = σ2 k + 1 with σ small and let F p denote the finite field with p elements. Let P ( z ) be a polynomial of degree d in F p [ z ] with d distinct roots in F p . For p =5 · 2 55 + 1 we can compute the roots of such polynomials of degree 10 9 . We believe we are the first to factor such polynomials of size one billion. We used a multi-core computer with two 10 core Intel Xeon E5 2680 v2 CPUs and 128 gigabytes of RAM. The factorization takes just under 4,000 seconds on 10 cores and uses 121 gigabytes of RAM. We used the tangent Graeffe root finding algorithm from [2

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Gutin, Gregory, Felix Reidl, Magnus Wahlström, and Meirav Zehavi. "Designing deterministic polynomial-space algorithms by color-coding multivariate polynomials." Journal of Computer and System Sciences 95 (August 2018): 69–85. http://dx.doi.org/10.1016/j.jcss.2018.01.004.

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Khirbeet, Ahmed Salih, and Ravie Chandren Muniyandi. "New Heuristic Model for Optimal CRC Polynomial." International Journal of Electrical and Computer Engineering (IJECE) 7, no. 1 (2017): 521. http://dx.doi.org/10.11591/ijece.v7i1.pp521-525.

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Cyclic Redundancy Codes (CRCs) are important for maintaining integrity in data transmissions. CRC performance is mainly affected by the polynomial chosen. Recent increases in data throughput require a foray into determining optimal polynomials through software or hardware implementations. Most CRC implementations in use, offer less than optimal performance or are inferior to their newer published counterparts. Classical approaches to determining optimal polynomials involve brute force based searching a population set of all possible polynomials in that set. This paper evaluates performance of

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Dissertations / Theses on the topic "Polynomials. Algorithms"

1

Cheng, Howard. "Algorithms for Normal Forms for Matrices of Polynomials and Ore Polynomials." Thesis, University of Waterloo, 2003. http://hdl.handle.net/10012/1088.

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In this thesis we study algorithms for computing normal forms for matrices of Ore polynomials while controlling coefficient growth. By formulating row reduction as a linear algebra problem, we obtain a fraction-free algorithm for row reduction for matrices of Ore polynomials. The algorithm allows us to compute the rank and a basis of the left nullspace of the input matrix. When the input is restricted to matrices of shift polynomials and ordinary polynomials, we obtain fraction-free algorithms for computing row-reduced forms and weak Popov forms. These algorithms can be used to com

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Fernandez, Carlos K. "Pascal polynomials over GF(2)." Thesis, Monterey, Calif. : Naval Postgraduate School, 2008. http://handle.dtic.mil/100.2/ADA483773.

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Thesis (M.S. in Applied Mathematics)--Naval Postgraduate School, June 2008.<br>Thesis Advisor(s): Fredricksen, Harold M. ; Stanica, Pantelimon. "June 2008." Description based on title screen as viewed on August 25, 2008. Includes bibliographical references (p. 51). Also available in print.

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Yuan, Chenyang. "Focused polynomials, random projections and approximation algorithms for polynomial optimization over the sphere." Thesis, Massachusetts Institute of Technology, 2018. http://hdl.handle.net/1721.1/120396.

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Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 73-75).<br>In this thesis, we study approximation algorithms for polynomial optimization over the sphere, concentrating on classes of polynomials whose optimum on the sphere can be efficiently approximated to a factor that only depends on the degree of the polynomial, and not on the dimension of the problem. We extend and generalize an existing class of polynomials known as focused polynomial

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Hanif, Sajid, and Muhammad Imran. "Factorization Algorithms for Polynomials over Finite Fields." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-11553.

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Integer factorization is a dicult task. Some cryptosystem such asRSA (which stands for Rivest, Shamir and Adleman ) are in fact designedaround the diculty of integer factorization.For factorization of polynomials in a given nite eld Fp we can useBerlekamp's and Zassenhaus algorithms. In this project we will see howBerlekamp's and Zassenhaus algorithms work for factorization of polyno-mials in a nite eld Fp. This project is aimed toward those with interestsin computational algebra, nite elds, and linear algebra.

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Webb, Marcus David. "Isospectral algorithms, Toeplitz matrices and orthogonal polynomials." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/264149.

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An isospectral algorithm is one which manipulates a matrix without changing its spectrum. In this thesis we study three interrelated examples of isospectral algorithms, all pertaining to Toeplitz matrices in some fashion, and one directly involving orthogonal polynomials. The first set of algorithms we study come from discretising a continuous isospectral flow designed to converge to a symmetric Toeplitz matrix with prescribed eigenvalues. We analyse constrained, isospectral gradient flow approaches and an isospectral flow studied by Chu in 1993. The second set of algorithms compute the spectr

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Munro, Christopher James. "Algorithms for matrix polynomials and structured matrix problems." Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/algorithms-for-matrix-polynomials-and-structured-matrix-problems(9154f9f0-8b86-46f8-8066-40c5139fcc51).html.

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McKee, James. "Some elliptic curve algorithms." Thesis, University of Cambridge, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.319564.

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Anderson, Robert Lawrence. "An Exposition of the Deterministic Polynomial-Time Primality Testing Algorithm of Agrawal-Kayal-Saxena." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd869.pdf.

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Abu, Salem Fatima Khaled. "Factorisation algorithms for univariate and bivariate polynomials over finite fields." Thesis, University of Oxford, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403928.

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Pachon, Ricardo. "Algorithms for polynomial and rational approximation." Thesis, University of Oxford, 2010. http://ora.ox.ac.uk/objects/uuid:f268a835-46ef-45ea-8610-77bf654b9442.

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Robust algorithms for the approximation of functions are studied and developed in this thesis. Novel results and algorithms on piecewise polynomial interpolation, rational interpolation and best polynomial and rational approximations are presented. Algorithms for the extension of Chebfun, a software system for the numerical computation with functions, are described. These algorithms allow the construction and manipulation of piecewise smooth functions numerically with machine precision. Breakpoints delimiting subintervals are introduced explicitly, implicitly or automatically, the latter metho

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Books on the topic "Polynomials. Algorithms"

1

Gore, Vivek. A quasi-polynomial-time algorithm for sampling words from a context-free language. LFCS, Dept. of Computer Science, University of Edinburgh, 1995.

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2

Gruevski, Trpe. Algorithms for solving the polynomial algebraic equations of any power. Company Samojlik, 2000.

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3

Gragg, William B. Downdating of Szego polynomials and data fitting applications. Naval Postgraduate School, 1992.

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Tovey, Craig A. A polynomial-time algorithm for computing the yolk in fixed dimension. Naval Postgraduate School, 1991.

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5

Hirshfeld, Yoram. A polynomial algorithm for deciding bisimularity of normed context-free processes. LFCS, Dept. of Computer Science, University of Edinburgh, 1994.

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Ho, Chung-jen. Fast parallel GCD algorithms for several polynomials over integral domain. Courant Institute of Mathematical Sciences, New York University, 1988.

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Hirshfeld, Yoram. A polynomial-time algorithm for deciding bisimulation equivalence of normed Basic Parallel Processes. LFCS, Dept. of Computer Science, University of Edinburgh, 1994.

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Crouch, P. E. The explicit computation of integration algorithms and first integrals for ordinary differential equations with polynomials coefficients using trees. National Aeronautics and Space Administration, 1992.

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9

Simai, He, Zhang Shuzhong, and SpringerLink (Online service), eds. Approximation Methods for Polynomial Optimization: Models, Algorithms, and Applications. Springer New York, 2012.

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Freund, Roland W. Quasi-kernal polynomials and convergance results for quasi-minimal residual iterations. Research Institute for Advanced Computer Science, NASA Ames Research Center, 1992.

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Book chapters on the topic "Polynomials. Algorithms"

1

Cohen, Henri. "Algorithms on Polynomials." In A Course in Computational Algebraic Number Theory. Springer Berlin Heidelberg, 1993. http://dx.doi.org/10.1007/978-3-662-02945-9_3.

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Engeln-Müllges, Gisela, and Frank Uhlig. "Roots of Polynomials." In Numerical Algorithms with C. Springer Berlin Heidelberg, 1996. http://dx.doi.org/10.1007/978-3-642-61074-5_3.

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Engeln-Müllges, Gisela, and Frank Uhlig. "Roots of Polynomials." In Numerical Algorithms with Fortran. Springer Berlin Heidelberg, 1996. http://dx.doi.org/10.1007/978-3-642-80043-6_3.

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Schinzel, A. "Arithmetical Properties of Polynomials." In Algorithms and Combinatorics. Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/978-3-642-60408-9_12.

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Winkler, Franz. "Decomposition of polynomials." In Polynomial Algorithms in Computer Algebra. Springer Vienna, 1996. http://dx.doi.org/10.1007/978-3-7091-6571-3_6.

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Watt, Stephen M. "Algorithms for Symbolic Polynomials." In Computer Algebra in Scientific Computing. Springer Berlin Heidelberg, 2006. http://dx.doi.org/10.1007/11870814_26.

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Cygan, Marek, Fedor V. Fomin, Łukasz Kowalik, et al. "Algebraic techniques: sieves, convolutions, and polynomials." In Parameterized Algorithms. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-21275-3_10.

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Giesbrecht, Mark, Daniel S. Roche, and Hrushikesh Tilak. "Computing Sparse Multiples of Polynomials." In Algorithms and Computation. Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-17517-6_25.

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Winkler, Franz. "5 Factorization of polynomials." In Polynomial Algorithms in Computer Algebra. Springer Vienna, 1996. http://dx.doi.org/10.1007/978-3-7091-6571-3_5.

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Nüsken, Michael, and Martin Ziegler. "Fast Multipoint Evaluation of Bivariate Polynomials." In Algorithms – ESA 2004. Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-540-30140-0_49.

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Conference papers on the topic "Polynomials. Algorithms"

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Mulholland, Jamie, and Michael Monagan. "Algorithms for trigonometric polynomials." In the 2001 international symposium. ACM Press, 2001. http://dx.doi.org/10.1145/384101.384135.

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2

Fischer, Cyril, and Jiří Náprstek. "Remarks on inverse of matrix polynomials." In Programs and Algorithms of Numerical Mathematics 18. Institute of Mathematics, Czech Academy of Sciences, 2017. http://dx.doi.org/10.21136/panm.2016.03.

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Ishai, Yuval, Eyal Kushilevitz, and Anat Paskin-Cherniavsky. "From randomizing polynomials to parallel algorithms." In the 3rd Innovations in Theoretical Computer Science Conference. ACM Press, 2012. http://dx.doi.org/10.1145/2090236.2090244.

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Toledo, Sivan, and Amit Waisel. "Parallel Algorithms for Evaluating Matrix Polynomials." In ICPP 2019: 48th International Conference on Parallel Processing. ACM, 2019. http://dx.doi.org/10.1145/3337821.3337871.

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Kauffman, Louis H., and Samuel J. Lomonaco, Jr. "Quantum algorithms for colored Jones polynomials." In SPIE Defense, Security, and Sensing, edited by Eric J. Donkor, Andrew R. Pirich, and Howard E. Brandt. SPIE, 2009. http://dx.doi.org/10.1117/12.821974.

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Bhattacharyya, Arnab, Pooya Hatami, and Madhur Tulsiani. "Algorithmic regularity for polynomials and applications." In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, 2014. http://dx.doi.org/10.1137/1.9781611973730.125.

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WATT, STEPHEN M. "TWO FAMILIES OF ALGORITHMS FOR SYMBOLIC POLYNOMIALS." In Proceedings of the Waterloo Workshop. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812778857_0012.

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Cheng, Qi, Shuhong Gao, and Daqing Wan. "Constructing high order elements through subspace polynomials." In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, 2012. http://dx.doi.org/10.1137/1.9781611973099.115.

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Dolecek, Gordana Jovanovic, and Lara Dolecek. "Exploiting features of symmetric polynomials for improved comb filter design." In 2016 Signal Processing: Algorithms, Architectures, Arrangements and Applications (SPA). IEEE, 2016. http://dx.doi.org/10.1109/spa.2016.7763581.

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Castillo, K., and F. R. Rafaeli. "A short note on interlacing and monotonicity of zeros of orthogonal polynomials." In NUMERICAL COMPUTATIONS: THEORY AND ALGORITHMS (NUMTA–2016): Proceedings of the 2nd International Conference “Numerical Computations: Theory and Algorithms”. Author(s), 2016. http://dx.doi.org/10.1063/1.4965368.

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Reports on the topic "Polynomials. Algorithms"

1

Orlin, James B., Serge A. Plotkin, and Eva Tardos. Polynomial Dual Network Simplex Algorithms. Defense Technical Information Center, 1991. http://dx.doi.org/10.21236/ada254340.

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Brown, Christopher W. Algorithmic Reformulation of Polynomial Problems. Defense Technical Information Center, 2007. http://dx.doi.org/10.21236/ada469251.

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Dutt, A., M. Gu, and V. Rokhlin. Fast Algorithms for Polynomial Interpolation, Integration and Differentiation. Defense Technical Information Center, 1993. http://dx.doi.org/10.21236/ada267505.

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Dantzig, George B. Converting a Converging Algorithm into a Polynomially Bounded Algorithm. Defense Technical Information Center, 1991. http://dx.doi.org/10.21236/ada234961.

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Orlin, James B. A Faster Strongly Polynomial Minimum Cost Flow Algorithm. Defense Technical Information Center, 1988. http://dx.doi.org/10.21236/ada457044.

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Herman, Martin, and Rama Chellappa. A reliable optical flow algorithm using 3-D hermite polynomials. National Institute of Standards and Technology, 1993. http://dx.doi.org/10.6028/nist.ir.5333.

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Tseng, Paul. A Very Simple Polynomial-Time Algorithm for Linear Programming. Defense Technical Information Center, 1988. http://dx.doi.org/10.21236/ada202502.

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Conforti, Michele, Gerard Cornuejols, and M. R. Rao. Decomposition of Balanced Matrices. Part 7. A Polynomial Recognition Algorithm. Defense Technical Information Center, 1991. http://dx.doi.org/10.21236/ada247400.

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Tovey, Craig A. A Polynomial-Time Algorithm for Computing the Yolk in Fixed Dimension. Defense Technical Information Center, 1991. http://dx.doi.org/10.21236/ada240060.

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Joyce, Kevin, and Aaron Luttman. Dewarping Algorithm Comparison: Local vs. Global 2D Polynomial Interpolation for Dewarping. Office of Scientific and Technical Information (OSTI), 2016. http://dx.doi.org/10.2172/1755854.

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