This article explores the computational intricacies of H. Rutishauser's quotient-difference (Q-D) algorithm and C programming code, a revolutionary advancement in polynomial analysis. Our specific focus is on cubi...
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This article explores the computational intricacies of H. Rutishauser's quotient-difference (Q-D) algorithm and C programming code, a revolutionary advancement in polynomial analysis. Our specific focus is on cubic polynomials featuring absolute, distinct non-zero real roots. Emphasizing the algorithm's distinctive capability to simultaneously approximate all zeros independently of external data. Notably, it proves invaluable in diverse domains, such as determining continuous fraction representations for meromorphic functions and serving as a powerful tool in complex analysis for the direct localization of poles and zeros. To bring this innovation into practice, the article introduces a meticulously crafted C language program, complete with a comprehensive algorithm and flowchart. Supported by illustrative examples, this implementation underscores the algorithm's robustness and effectiveness across various real-world scenarios.
作者：
Murabayashi, NaokiYoshida, HayatoKansai Univ
Fac Engn Sci Dept Math 3-3-35 Yamate Cho Suita Osaka 5648680 Japan Kansai Univ
Grad Sch Sci & Engn Math Integrated Sci & Engn Major 3-3-35 Yamate Cho Suita Osaka 5648680 Japan
In Buhler et al. (Math Comput 44(170):473-481, 1985), there are given two representations of the function G1(x) (equal to the exponential integral E1(x)) that appear in an expression of the first derivative of the L-f...
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In Buhler et al. (Math Comput 44(170):473-481, 1985), there are given two representations of the function G1(x) (equal to the exponential integral E1(x)) that appear in an expression of the first derivative of the L-function of an elliptic curve defined over Q at 1. One is the Puiseux Series, and the other is the continued fraction representation. In Hitotsumatsu (Introduction to Special Functions, Morikita Publishing Co. Ltd., Chiyoda, 1999), we can see how to construct formally this continued fraction from F(x):=exE1(x) (explained briefly in Introduction), but we have never seen a proof of that it converges to the original function F(x). More precisely an asymptotic expansion of F(x)=exE1(x) is Sigma k=1 infinity(-1)k-1(k-1)!1xk, and this gives the continued fraction by quotient-difference algorithm, which is briefly announced by E. Stiefel and developed by Rutishauser (Zeitschrift f & uuml;r Angewandte Mathematik und Physik 5:233-251, 1954). In this paper we define "a continued fraction expansion of F(x) at infinity", which is analogous to the regular continued fraction expansion of real numbers, and prove that this expansion gives the same continued fraction. Moreover, we give concrete representations of rational functions which are obtained by truncating the continued fraction.
In this paper, novel convergence accelerating techniques are adapted onto the proposed hyperbolic numerical inverse Laplace transform method (hyperbolic-NILT) and analyzed. This Ill NILT method is based on the approxi...
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ISBN:
(纸本)9781509040865
In this paper, novel convergence accelerating techniques are adapted onto the proposed hyperbolic numerical inverse Laplace transform method (hyperbolic-NILT) and analyzed. This Ill NILT method is based on the approximation of the inverse kernel of the Laplace transform Bromwich integral exp(st). It is shown that with the use of the convergence accelerating algorithms onto the essence of the proposed NILT method, an enhancement on the core of the inversion is achieved, with relatively accurate and stable results, while preserving valuable time and memory. The algorithms are tested and their corresponding results are discussed, mainly regarding the accuracy, stability and computational efficiency. The experimental accuracy analysis tests are implemented in the universal MATLAB language with properly chosen Laplace transforms.
This paper proposes an embedded surveillance system for real-time anomaly intrusion detection based on temporal differencealgorithm and theft items detection based on accumulated background subtraction algorithm. Thi...
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ISBN:
(纸本)9781424458653
This paper proposes an embedded surveillance system for real-time anomaly intrusion detection based on temporal differencealgorithm and theft items detection based on accumulated background subtraction algorithm. This design of modified vision algorithm fully utilize the advanced parallelism of Field Programmable Gate Arrays (FPGA) and this hardware implementation realizes time-consumed difference computing with on-chip FIFO and RAM memory. Finally, we fuse these two anomaly detection algorithms in one FPGA and select the algorithm type by user needs. As a result, the detecting validity and robustness of this implementation have been demonstrated through real-time surveillance videos.
The paper deals with a technique for numerical inversion of two-dimensional Laplace transforms based on the FFT & IFFT in conjunction with a quotient-difference algorithm of Rutishauser. In contrast to the existin...
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ISBN:
(纸本)9781424408214
The paper deals with a technique for numerical inversion of two-dimensional Laplace transforms based on the FFT & IFFT in conjunction with a quotient-difference algorithm of Rutishauser. In contrast to the existing FFT-based technique the presented one is based on a repeated application of one-dimensional partial Laplace transform inversions. The method promises a generalization towards multidimensional numerical inverse Laplace transforms because it establishes more effective unified algorithmic approach to a computation. The method was programmed using Matlab language and analyzed as for its accuracy.
作者：
Petrinovic, DUniv Zagreb
Fac Elect Engn & Comp Dept Elect Syst & Informat Proc Zagreb 41000 Croatia
A new, computationally efficient technique for calculation of the Line Spectrum Frequencies (LSF) that can be applied to any order of the LPC analysis is proposed in this paper. It is based on the quotient-difference ...
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ISBN:
(纸本)078036290X
A new, computationally efficient technique for calculation of the Line Spectrum Frequencies (LSF) that can be applied to any order of the LPC analysis is proposed in this paper. It is based on the quotient-difference (Q-D) root-finding algorithm that enables simultaneous solution for all the LSFs. It is an iterative procedure that offers the tradeoff between accuracy and complexity, what is especially important for the real-time applications. To improve the convergence, a nonlinear mapping of the LSFs is also proposed For low accuracy applications, the method is even more effective then the fast converging Newton-Rapshon method, but is at the same time exceptionally simple, has a very regular structure and requires only basic mathematical operations.
Using the framework provided by Clifford algebras, we consider a non-commutative quotient-difference algorithm for obtaining the elements of a continued fraction corresponding to a given vector-valued power series. We...
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Using the framework provided by Clifford algebras, we consider a non-commutative quotient-difference algorithm for obtaining the elements of a continued fraction corresponding to a given vector-valued power series. We demonstrate that these elements are ratios of vectors, which may be calculated with the aid of a cross rule using only vector operations. For vector-valued meromorphic functions we derive the asymptotic behaviour of these vectors, and hence of the continued fraction elements themselves. The behaviour of these elements is similar to that in the scalar case, while the vectors are linked with the residues of the given function. In the particular case of vector power series arising from matrix iteration the new algorithm amounts to a generalisation of the power method to sub-dominant eigenvalues, and their eigenvectors.
We consider polynomials orthogonal relative to a sequence of vectors and derive their recurrence relations within the framework of Clifford algebras. We state sufficient conditions for the existence of a system of suc...
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We consider polynomials orthogonal relative to a sequence of vectors and derive their recurrence relations within the framework of Clifford algebras. We state sufficient conditions for the existence of a system of such polynomials. The coefficients in the above relations may be computed using a cross-rule which is linked to a vector version of the quotient-difference algorithm, both of which are proved here using designants. An alternative route is to employ a vector variant of the Chebyshev algorithm. This algorithm is established and an implementation presented which does not require general Clifford elements. Finally, we comment on the connection with vector Pade approximants.
The recursion relations that were proposed by Ford and Sidi (1988) for implementing vector extrapolation methods are used for devising generalizations of the power method for linear operators. These generalizations ar...
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The recursion relations that were proposed by Ford and Sidi (1988) for implementing vector extrapolation methods are used for devising generalizations of the power method for linear operators. These generalizations are shown to produce approximations to largest eigenvalues of a linear operator under certain conditions. They are similar in form to the quotient-difference algorithm and share similar convergence properties with the latter. These convergence properties resemble also those obtained for the basic LR and QR algorithms. Finally, it is shown that the convergence rate produced by one of these generalizations is twice as fast for normal operators as it is for nonnormal operators.
Certain variants of the Toda flow are continuous analogues of the $QR$ algorithm and other algorithms for calculating eigenvalues of matrices. This was a remarkable discovery of the early eighties. Until very recently...
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Certain variants of the Toda flow are continuous analogues of the $QR$ algorithm and other algorithms for calculating eigenvalues of matrices. This was a remarkable discovery of the early eighties. Until very recently contemporary researchers studying this circle of ideas have been unaware that continuous analogues of the quotient-difference and $LR$ algorithms were already known to Rutishauser in the fifties. Rutishauser’s continuous analogue of the quotient-difference algorithm contains the finite, nonperiodic Toda flow as a special case. A nice feature of Rutishauser’s approach is that it leads from the (discrete) eigenvalue algorithm to the (continuous) flow by a limiting process. Thus the connection between the algorithm and the flow does not come as a surprise. In this paper it is shown how Rutishauser’s approach can be generalized to yield large families of flows in a natural manner. The flows derived include continuous analogues of the $LR$, $QR$, $SR$, and $HR$ algorithms.
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