Polynomial Evaluation Research Articles

Fast Matrix Multiplication

We give an overview of the history of fast algorithms for matrix multiplication. Along the way, we look at some other fundamental problems in algebraic complexity like polynomial evaluation. This exposition is self-contained. To make it accessible to a broad audience, we only assume a minimal mathematical background: basic linear algebra, familiarity with polynomials in several variables over rings, and rudimentary knowledge in combinatorics should be sufficient to read (and understand) this article. This means that we have to treat tensors in a very concrete way (which might annoy people coming from mathematics), occasionally prove basic results from combinatorics, and solve recursive inequalities explicitly (because we want to annoy people with a background in theoretical computer science, too).

A New Upper Bound on the Query Complexity of Testing Generalized Reed-Muller Codes.

Over a finite field \({\mathbb{F}}_q\) the (n,d,q)-Reed-Muller code is the code given by evaluations of n-variate polynomials of total degree at most d on all points (of \({\mathbb{F}}_q^n\)). The task of testing if a function \(f:{\mathbb{F}}_q^n \to {\mathbb{F}}_q\) is close to a codeword of an (n,d,q)-Reed-Muller code has been of central interest in complexity theory and property testing. The query complexity of this task is the minimal number of queries that a tester can make (minimum over all testers of the maximum number of queries over all random choices) while accepting all Reed-Muller codewords and rejecting words that are δ-far from the code with probability Ω(δ). (In this work we allow the constant in the Ω to depend on d.)

Reconstructing Hyperbolic Cross Trigonometric Polynomials by Sampling along Rank-1 Lattices

With given Fourier coefficients the evaluation of multivariate trigonometric polynomials at the nodes of a rank-1 lattice leads to a one-dimensional discrete Fourier transform. In many applications one is also interested in the reconstruction of the Fourier coefficients from samples in the spatial domain. We present necessary and sufficient conditions on rank-1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic crosses. In addition, we suggest approaches for determining suitable rank-1 lattices using a component-by-component algorithm. We present numerical results for reconstructing trigonometric polynomials up to spatial dimension 100.

A General Condition Number for Polynomials

This paper presents a generic condition number for polynomials that is useful for polynomial evaluation of a finite series of polynomial basis defined by means of a linear recurrence. This expression extends the classical one for the power and Bernstein bases, but it also provides us a general framework for all the families of orthogonal polynomials like Chebyshev, Legendre, Gegenbauer, Jacobi, and Sobolev orthogonal polynomial bases. The standard algorithm for the evaluation of finite series in any of these polynomial bases is the extended Clenshaw algorithm. The use of this new condition number permits us to give a general theorem about the forward error for that evaluation algorithm. A running-error bound of the extended algorithm is also presented and all the bounds are compared in several numerical examples.

Numerical Solution of a Reaction-Diffusion System with Fast Reversible Reaction by Using Adomian’s Decomposition Method and He’s Variational Iteration Method

In this paper, the approximate solution of a reaction-diffusion system with fast reversible reaction is obtained by using Adomian decomposition method (ADM) and variational iteration method (VIM) which are two powerful methods that were recently developed. The VIM requires the evaluation of the Lagrange multiplier, whereas ADM requires the evaluation of the Adomian polynomials. The behavior of the approximate solutions and the effects of different values of t are shown graphically.

On the evaluation of multivariate polynomials over finite fields

A method to evaluate multivariate polynomials over a finite field is described and its multiplicative complexity is discussed.

A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices

We propose a new fast algorithm for computing the spectrum of an N×N symmetric tridiagonal matrix in O(NlnN) operations. Such an algorithm may be combined with any of the existing methods for the determination of eigenvectors of a symmetric tridiagonal matrix with known eigenvalues. The underlying technique is a divide-and-conquer approach which determines eigenvalues of a larger tridiagonal matrix from those of constituent matrices by the use of relations of their characteristic polynomials. The evaluation of characteristic polynomials is accelerated by the use of a technique known as the fast multipole method. An implementation of the algorithm has been developed in Fortran, providing for a comparison with existing techniques in terms of running time and accuracy. We present numerical results which demonstrate the effectiveness of the method.

High-speed and parallel approach for decoding of binary BCH codes with application to Flash memory devices

In this article, we propose a high-speed decoding algorithm for binary BCH codes that can correct up to 7 bits in error. Evaluation of the error-locator polynomial is the most complicated and time-consuming step in the decoding of a BCH code. We have derived equations for specifying the coefficients of the error-locator polynomial, which can form the basis for the development of a parallel architecture for the decoder. This approach has the advantage that all the coefficients of the error locator polynomial are computed in parallel (in one step). The roots of error-locator polynomial can be obtained by Chien's search and inverting these roots gives the error locations. This algorithm can be employed in any application where high-speed decoding of data encoded by a binary BCH code is required. One important application is in Flash memories where data integrity is preserved using a long, high-rate binary BCH code. We have synthesized generator polynomials for binary BCH codes (error-correcting capability, s) that can be employed in Flash memory devices to improve the integrity of information storage. The proposed decoding algorithm can be used as an efficient, high-speed decoder in this important application.

Accurate summation, dot product and polynomial evaluation in complex floating point arithmetic

Several different techniques and softwares intend to improve the accuracy of results computed in a fixed finite precision. Here we focus on methods to improve the accuracy of summation, dot product and polynomial evaluation. Such algorithms exist real floating point numbers. In this paper, we provide new algorithms which deal with complex floating point numbers. We show that the computed results are as accurate as if computed in twice the working precision. The algorithms are simple since they only require addition, subtraction and multiplication of floating point numbers in the same working precision as the given data.

Good Linear Codes from Polynomial Evaluations

In the present paper, we generalize the ideas of code constructions from our previous papers . It turns out that the codes in the previous papers can be viewed as special cases of those in this paper. Moreover, our constructions produce some good codes in terms of their parameters. In particular, some best-known codes can be obtained through our methods. Furthermore, our constructions are explicit and the codes can be easily implemented as shown in the tables of Appendix. Besides, one new code, i.e., a 4-ary [64,15,31]-linear code, is found through our constructions.

Comment on “On the optimal symmetric purification scheme of the one-particle density matrix” [Chem. Phys. Lett. 511 (2011) 159–160

Recently, Kim and Jung reported on the efficiency of certain density matrix purification polynomials for calculation of the one-particle density matrix. In this comment, it is shown that a different conclusion is reached if a more efficient method is used for polynomial evaluation. The impact of the choice of purification polynomial on sparsity and memory usage is also discussed. We comment on how the use of nonmonotonic polynomials can make the convergence even faster.

SYNTHESIS OF MICROWAVE RESONATOR DIPLEXERS USING LINEAR FREQUENCY TRANSFORMATION AND OPTIMIZATION

This paper presents a method for synthesizing coupled resonator diplexers composed of TX and RX fllters (two types of junctions connecting the TX and RX fllters are considered). For the flrst junction type, the common port is directly coupled to the flrst resonator of the TX and RX fllters, respectively. For the second junction type, common node is realized by adding an extra resonator besides those of the TX and RX fllters. The method is based on the evaluation of the characteristic polynomials of the diplexer using the proposed linear frequency transformation and well-established method, and then the \N + 3 coupling matrix of overall diplexer can be obtained using hybrid optimization methods. Two diplexers have been designed and fabricated to validate the proposed method.

Large-Integer Multiplication Based on Homogeneous Polynomials

Several algorithms based on homogeneous polynomials for multiplication of large integers are described in the paper. The homogeneity of polynomials provides several simplifications: reduction of system of equations and elimination of necessity to evaluate polynomials in points with larger coordinates. It is demonstrated that a two-stage implementation of the proposed and Toom-Cook algorithms asymptotically require twice as many standard multiplications than their direct implementation. A multistage implementation of these algorithms is also less efficient than their direct implementation. Although the proposed algorithms as well as the corresponding Toom-Cook algorithms require numerous algebraic additions, the Generalized Horner rule for evaluation of homogeneous polynomials, provided in the paper, decrease this number twice.

A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods

In this work, we present a class of fast first order finite volume solvers, called PVM (polynomial viscosity matrix), for balance laws or, more generally, for nonconservative hyperbolic systems. They are defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a Roe matrix. These methods have the advantage that they only need some information about the eigenvalues of the system to be defined, and no spectral decomposition of a Roe matrix is needed. As a consequence, they are faster than the Roe method. These methods can be seen as a generalization of the schemes introduced by Degond et al. in [C. R. Acad. Sci. Paris Sér. l Math., 328 (1999), pp. 479--483] for balance laws and nonconservative systems. The first order path conservative methods to be designed here are intended to be used as the basis for higher order methods for multidimensional problems. In this work, some well known solvers, such as Rusanov, Lax--Friedrichs, FORCE (see [E. F. Toro and S. J. Billett, IMA J. Numer. Anal., 20 (2000), pp. 47--79; M. J. Castro, A. Pardo, C. Parés, and E. F. Toro, Math. Comp., 79 (2010), pp. 1427--1472]), GFORCE (see [E. F. Toro and V. A. Titarev, J. Comput. Phys., 216 (2006), pp. 403--429]), and HLL (see [A. Harten, P. D. Lax, and B. van Leer, SIAM Rev., 25 (1983), pp. 35--61]), are redefined under this form, and then some new solvers are proposed. Finally, some numerical tests are presented, and the performances of the numerical schemes are compared with each others and with Roe scheme for the 1D and 2D two-fluid flow model of Pitman and Le.

On the Bringer–Chabanne EPIR protocol for polynomial evaluation

Extended private information retrieval (EPIR) was defined by \cite{BCPT07} at CANS'07 and generalized by \cite{BC09} at AFRICACRYPT'09. In the generalized setting, EPIR allows a user to evaluate a function on a database block such that the database can learn neither which function has been evaluated nor on which block the function has been evaluated and the user learns no more information on the database blocks except for the expected result. An EPIR protocol for evaluating polynomials over a finite field $L$ was proposed by Bringer and Chabanne in \cite{BC09}. We show that the protocol does not satisfy the correctness requirement as they have claimed. In particular, we show that it does not give the user the expected result with large probability if one of the coefficients of the polynomial to be evaluated is primitive in $L$ and the others belong to the prime subfield of $L$.

Polynomial evaluation over finite fields: new algorithms and complexity bounds

An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques, when the degree of the polynomial is large enough compared to the field characteristic. Specifically, if n is the degree of the polynomiaI, the asymptotic complexity is shown to be $${O(\sqrt{n})}$$ , versus O(n) of classical algorithms. Applications to the syndrome computation in the decoding of Reed-Solomon codes are highlighted.

Numerical calculation of Bessel, Hankel and Airy functions

The numerical evaluation of an individual Bessel or Hankel function of large order and large argument is a notoriously problematic issue in physics. Recurrence relations are inefficient when an individual function of high order and argument is to be evaluated. The coefficients in the well-known uniform asymptotic expansions have a complex mathematical structure which involves Airy functions. For Bessel and Hankel functions, we present an adapted algorithm which relies on a combination of three methods: (i) numerical evaluation of Debye polynomials, (ii) calculation of Airy functions with special emphasis on their Stokes lines, and (iii) resummation of the entire uniform asymptotic expansion of the Bessel and Hankel functions by nonlinear sequence transformations. In general, for an evaluation of a special function, we advocate the use of nonlinear sequence transformations in order to bridge the gap between the asymptotic expansion for large argument and the Taylor expansion for small argument (“principle of asymptotic overlap”). This general principle needs to be strongly adapted to the current case, taking into account the complex phase of the argument. Combining the indicated techniques, we observe that it possible to extend the range of applicability of existing algorithms. Numerical examples and reference values are given.

The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

This paper was motivated by a conjecture of Brändén [P. Brändén, Actions on permutations and unimodality of descent polynomials, European J. Combin. 29 (2) (2008) 514–531] about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the symmetric and unimodal property of the Eulerian numbers. We show that such a formula with the conjectured property can be derived from the combinatorial theory of continued fractions. We also discuss an analogous expansion for the corresponding formula for derangements and prove a ( p , q ) -analogue of the fact that the (-1)-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). The ( p , q ) -analogue unifies and generalizes our recent results [H. Shin, J. Zeng, The q -tangent and q -secant numbers via continued fractions, European J. Combin. 31 (7) (2010) 1689–1705] and that of Josuat-Vergès [M. Josuat-Vergés, A q -enumeration of alternating permutations, European J. Combin. 31 (7) (2010) 1892–1906].

RF Power Consumption Emulation Optimized with Interval Valued Homotopies

This paper presents a methodology towards the emulation of the electrical power consumption of the RF device during the cellular phone/handset transmission mode using the LTE technology. The emulation methodology takes the physical environmental variables and the logical interface between the baseband and the RF system as inputs to compute the emulated power dissipation of the RF device. The emulated power, in between the measured points corresponding to the discrete values of the logical interface parameters is computed as a polynomial interpolation using polynomial basis functions. The evaluation of polynomial and spline curve fitting models showed a respective divergence (test error) of 8% and 0.02% from the physically measured power consumption. The precisions of the instruments used for the physical measurements have been modeled as intervals. We have been able to model the power consumption of the RF device operating at 5MHz using homotopy between 2 continuous power consumptions of the RF device operating at the bandwidths 3MHz and 10MHz. Keywords—Radio frequency, High Power Amplifier, Baseband, LTE, Power, Emulation, Homotopy, Interval analysis, Tx power, Register-Transfer Level.

On Certain Divisibility Property of Polynomials over Integral Domains

An integral domain $R$ is a \textit{degree-domain} if for given two polynomials $f(x)$ and $g(x)$ in $R[x]$ such that for all $k\in R$ $(g(k)\not=0\Rightarrow g(k)|f(k))$, then $f(x)=0$ or $\deg f\geq \deg g$. We prove that the ring of integers $\mathcal{O}_L$ is a degree-domain, where $\mathbb{Q}\subseteq L$ is a finite Galois extension. Then we study degree-domains that are also unique factorization domains to determine divisibility of polynomials using polynomial evaluations