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Numerical solution of singular regular boundary value problems by pole detection with qd-algorithm

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2009年第2期233卷 420-436页

作者： Allouche, H. Marhraoui, N. Fac Sci Dept Math & Comp Sci Zitoune Meknes Morocco

In this paper, we present a method using qd-algorithm combined with finite difference method for numerically solving singular boundary value problems for certain ordinary differential equation having singular coefficients. (We can also use qd-algorithm combined with other numerical methods.) These problems arise when reducing partial differential equation to ordinary differential equation by physical symmetry. Some illustrative examples are given to demonstrate the efficiency and high accuracy of the proposed method and compare it to the numerical results made with other methods. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Boundary value problems Degenerate problems Pole detection Singularity Formally orthogonal Hadamard polynomials qd-algorithm Finite difference scheme Frobenius method

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VECTOR ORTHOGONAL RELATIONS - VECTOR qd-algorithm

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 1987年第1期19卷 141-150页

作者： VANISEGHEM, J UNIV LILLE 1 UFR IEEAF-59655 VILLENEUVE DASCQFRANCE

Vector-Padé approximants to a function F = (ƒ 1 ; … ƒ d ) from C to C d have been defined, uniquely, without any auxiliary choice than the degrees of the numerator and the denominator (the same for all the components ƒ i ), as in the scalar case [1,5]. The denominators are associated to polynomials P s r , which are given by vector orthogonal properties (R) and which satisfy for each s , recurrence relations of order d + 1 (i.e. with d + 2 terms), called relations (D). We study here consequences of (R) and (D): first we prove an algorithm similar to the generalized MNA-algorithm; then we define a vector qd-algorithm which links two diagonals ( P s r ) r and ( P s + 1 r ) r . Conversely if a family ( P r ) r ⩾ 0 verifying (D) is given, it is possible to build ( P s r ) r ⩾ 0, s ⩾ 0 , and d linear functionals C α , α = 1,…, d , such that P 0 r = P r and ( P s r ) verify the orthogonal relations (R), with respect to the C α .

关键词： Padé approximants orthogonal polynomials qd-algorithm

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CONVERGENCE OF THE VECTOR qd-algorithm - ZEROS OF VECTOR ORTHOGONAL POLYNOMIALS

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 1989年第1期25卷 33-46页

作者： VANISEGHEM, J UNIV SCI & TECH LILLE FLANDRES ARTOIS UFR MATH PURES & APPLF-59655 VILLENEUVE DASCQFRANCE

The convergence of the vector qd-algorithm, associated to d meromorphic functions, is established. As a consequence a De Montessus–De Ballore theorem for vector Padé approximants is proved. A short numerical stu... 详细信息

关键词： Vector Padé approximants qd-algorithm

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On the calculation of the poles of multivariate meromorphic functions using the symbolic-numeric two-point qd-algorithm

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NUMERICAL algorithmS 2020年第4期84卷 1443-1458页

作者： Elidrissi, A. Abouir, J. Benouahmane, B. Hassan II Univ Casablanca Fac Sci & Technol Lab LMCMAN BP 146 Mohammadia 20650 Morocco

The so-called quotient-difference algorithm, or qd-algorithm, is used for determining the poles of a meromorphic function from its Taylor coefficients. A generalization of this algorithm to the univariate and multivariate two-point cases applied to a power series (positive or negative exponents) is presented. We describe also the symbolic-numeric two-point qd-algorithm to compute the poles of multivariate meromorphic functions in a given domain from its series expansion coefficients. This algorithm can be regarded as computing the parametrized eigenvalues for a tridiagonal matrix. Numerical examples are furnished to illustrate our results.

关键词： Two-point Pade approximants qd-algorithm Series expansions Multivariate approximation Poles Eigenvalues

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Reliable root detection with the qd-algorithm: When Bernoulli, Hadamard and Rutishauser cooperate

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1188-1208页

作者： Allouche, Hassane Cuyt, Annie Fac Sci Dept Math & Informat Beni Mhamed Meknes 4010 Morocco Univ Antwerp Dept Math & Comp Sci B-2020 Antwerp Belgium

When using Rutishauser's qd-algorithm for the determination of the roots of a polynomial (originally the poles of a meromorphic function), or for related problems, conditions have been formulated for the interpretation of the computed q- and e-values. For a correct interpretation, the so-called critical indices play a crucial role. They index a column of e-values that tends to zero because of a jump in modulus among the poles. For more than 50 years the qd-algorithm in exact arithmetic was considered to be fully understood. In this presentation we push the detailed theoretical investigation of the qd-algorithm even further and we present a new aspect that seems to have been overlooked. We indicate a new element that makes a column of e-values tend to zero, namely a jump in multiplicity among equidistant poles. This result is obtained by combining the qd-algorithm with a deflation technique, and hence mainly relying on Bernoulli's method and Hadamard's formally orthogonal polynomials. Our results round up the theoretical analysis of the qd-algorithm as formulated in its original form, and are of importance in a variety of practical applications as outlined in the introduction. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Meromorphic function Pole detection Formally orthogonal Hadamard polynomials Method of Bernoulli qd-algorithm

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ON THE CONVERGENCE OF THE MULTIVARIATE HOMOGENEOUS qd-algorithm

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BIT 1994年第4期34卷 535-545页

作者： CUYT, A UNIV INSTELLING ANTWERP DEPT MATH & COMP SCIB-2610 WILRIJKBELGIUM

The convergence of columns in the univariate qd-algorithm to reciprocals of polar singularities of meromorphic functions has often proved to be very useful. A multivariate qd-algorithm was discovered in 1982 for the construction of the so-called homogeneous Pade approximants. In the first section we repeat the univariate convergence results. In the second section we summarize the ''homogeneous'' multivariate qd-algorithm. In the third section a multivariate convergence result is proved by combining results from the previous sections. This convergence result is compared with another theorem for the general order multivariate qdg-algorithm. The main difference lies in the fact that the homogeneous form detects the polar singularities ''pointwise'' while the general form detects them ''curvewise''.

关键词： qd-algorithm MULTIVARIATE RATIONAL APPROXIMATION

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Reliable root detection with the qd-algorithm: When Bernoulli, Hadamard and Rutishauser cooperate

Reliable root detection with the qd-algorithm: When Bernoull...

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Conference on Approximation and Extrapolation of Convergent and Divergent Sequences and Series

作者： Allouche, Hassane Cuyt, Annie Fac Sci Dept Math & Informat Beni Mhamed Meknes 4010 Morocco Univ Antwerp Dept Math & Comp Sci B-2020 Antwerp Belgium

关键词： Meromorphic function Pole detection Formally orthogonal Hadamard polynomials Method of Bernoulli qd-algorithm

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Detecting discontinuity points from spectral data with the quotient-difference (qd) algorithm

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2012年第9期236卷 2406-2424页

作者： Allouche, Hassane Ghanou, Noura Tigma, Khalid Fac Sci Dept Math & Comp Sci Meknes Morocco

This paper introduces a new technique for the localization of discontinuity points from spectral data. Through this work, we will be able to detect discontinuity points of a 2 pi-periodic piecewise smooth function from its Fourier coefficients. This could be useful in detecting edges and reducing the effects of the Gibbs phenomenon which appears near discontinuities and affects signal restitution. Our approach consists in moving from a discontinuity point detection problem to a pole detection problem, then adapting the quotient-difference (qd) algorithm in order to detect those discontinuity points. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Location of discontinuity points Edge detection Fourier series Piecewise smooth function Gibbs phenomenon qd-algorithm

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qd-algorithms and recurrence relations for biorthogonal polynomials

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 1999年第1期107卷 53-72页

作者： da Rocha, Z Univ Porto Fac Ciencias Dept Matemat Aplicada P-4050 Porto Portugal

Biorthogonal polynomials P-n((i, j)) include as particular cases vector orthogonal polynomials of dimension d and - d(d is an element of N). We pay special attention to the cases of dimension 1 and -1. We discuss the problem of computing P-n((i, j)) using only one or several recurrence relations. Furthermore, we deduce all recurrence relations of a certain type that give P-n((i, j)) from two other biorthogonal polynomials. The coefficients that appear in any two independent relations satisfy some identities from which it is possible to establish qd-like algorithms. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： biorthogonal polynomial vector orthogonal polynomial of dimension d and -d (d epsilon N) recurrence relation qd-algorithm

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Convergence acceleration algorithms related to a generalized E-transformation and its particular cases

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JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 2013年第2期30卷 263-285页

作者： He, Yi Hu, Xing-Biao Tam, Hon-Wah Tsujimoto, Satoshi Chinese Acad Sci Wuhan Inst Phys & Math Wuhan 430071 Peoples R China Chinese Acad Sci AMSS Inst Computat Math & Sci Engn Comp LSEC Beijing 100190 Peoples R China Hong Kong Baptist Univ Dept Comp Sci Kowloon Tong Hong Kong Peoples R China Kyoto Univ Grad Sch Informat Dept Appl Math & Phys Kyoto 6068501 Japan

In this paper, a generalized E-transformation arising from the study of a generalization of sequence transformations and triangular recursion schemes is proposed. Three new algorithms, namely, the generalized E-algorithm, the generalized FS-algorithm and the generalized hungry type E-algorithm, are constructed for implementing the generalization of the E-transformation. Some convergence results of the generalized E-algorithm are obtained. In addition, some particular cases of the generalized E-transformation and the recursive algorithms for their computation are also studied.

关键词： E-transformation epsilon-algorithm G-algorithm rs-algorithm qd-algorithm

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