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On the recursive and explicit form of the general JCP Miller formula with applications

ADVANCES IN APPLIED MATHEMATICS 2024年 156卷

作者： Bugajewski, Dariusz Bugajewski, Dawid Gan, Xiao-Xiong Mackowiak, Piotr Adam Mickiewicz Univ Fac Math & Comp Sci Dept Nonlinear Anal & Appl Topol Uniwersytetu Poznanskiego 4 PL-61614 Poznan Poland Univ Warsaw Fac Phys Pasteura 5 PL-02093 Warsaw Poland Morgan State Univ Dept Math Baltimore MD 21251 USA

The famous J.C.P. Miller formula provides a recurrence algorithm for the composition B a degrees f , where B a is the formal binomial series and f is a formal power series, however it requires that f has to be a nonunit. In this paper we provide the general J.C.P. Miller formula which eliminates the requirement of nonunitness of f and, instead, we establish a necessary and sufficient condition for the existence of the composition B a degrees f . We also provide the general J.C.P. Miller recurrence algorithm for computing the coefficients of that composition, if B a degrees f is well defined, obviously. Our generalizations cover both the case in which f is a one-variable formal power series and the case in which f is a multivariable formal power series. In the central part of this article we state, using some combinatorial techniques, the explicit form of the general J.C.P. Miller formula for one -variable case. As applications of these results we provide an explicit formula for the inverses of polynomials and formal power series for which the inverses exist, obviously. We also use our results to investigation of approximate solution to a differential equation which cannot be solved in an explicit way. (c) 2024 Elsevier Inc. All rights reserved.

关键词： Approximate solution Composition of formal power series Determinants Formal power series Hessenberg matrices Infinite matrices Initial value problem Inverses JCP Miller formula recurrence algorithm Separable equation Trudi formula

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Efficient Hermite Spectral-Galerkin Methods for Nonlocal Diffusion Equations in Unbounded Domains

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Numerical Mathematics(Theory,Methods and Applications) 2022年第4期15卷 1009-1040页

作者： Huiyuan Li Ruiqing Liu Li-Lian Wang State Key Laboratory of Computer Science/Laboratory of Parallel Computing Institute of SoftwareChinese Academy of SciencesBeijing 100190China University of Chinese Academy of Sciences Beijing 100190China Division of Mathematical Sciences School of Physical and Mathematical SciencesNanyang Technological University637371Singapore

In this paper,we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded *** show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal *** a result,the“stiffness”matrix can be fast computed and assembled via the four-point stable recursive algorithm with O(N^(2))arithmetic ***,the singular factor in a typical kernel function can be fully absorbed by the *** the aid of Fourier analysis,we can prove the convergence of the *** demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis *** provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.

关键词： Nonlocal diffusion equation spectral-Galerkin Hermite functions correlation/convolution recurrence algorithm

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Multithreading-Based algorithm for High-Performance Tchebichef Polynomials with Higher Orders

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algorithmS 2024年第9期17卷 381页

作者： Al-sudani, Ahlam Hanoon Mahmmod, Basheera M. Sabir, Firas A. Abdulhussain, Sadiq H. Alsabah, Muntadher Flayyih, Wameedh Nazar Univ Baghdad Dept Comp Engn Baghdad Iraq Al Farahidi Univ Med Tech Coll Baghdad Iraq

Tchebichef polynomials (TPs) play a crucial role in various fields of mathematics and applied sciences, including numerical analysis, image and signal processing, and computer vision. This is due to the unique properties of the TPs and their remarkable performance. Nowadays, the demand for high-quality images (2D signals) is increasing and is expected to continue growing. The processing of these signals requires the generation of accurate and fast polynomials. The existing algorithms generate the TPs sequentially, and this is considered as computationally costly for high-order and larger-sized polynomials. To this end, we present a new efficient solution to overcome the limitation of sequential algorithms. The presented algorithm uses the parallel processing paradigm to leverage the computation cost. This is performed by utilizing the multicore and multithreading features of a CPU. The implementation of multithreaded algorithms for computing TP coefficients segments the computations into sub-tasks. These sub-tasks are executed concurrently on several threads across the available cores. The performance of the multithreaded algorithm is evaluated on various TP sizes, which demonstrates a significant improvement in computation time. Furthermore, a selection for the appropriate number of threads for the proposed algorithm is introduced. The results reveal that the proposed algorithm enhances the computation performance to provide a quick, steady, and accurate computation of the TP coefficients, making it a practical solution for different applications.

关键词： Tchebichef polynomials Tchebichef moments multithread recurrence algorithm

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Parallel Framework for Memory-Efficient Computation of Image Descriptors for Megapixel Images

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BIG DATA RESEARCH 2023年 33卷

作者： Abdeltif, Amr M. Hosny, Khalid M. Darwish, Mohamed M. Salah, Ahmad Li, Kenli Zagazig Univ Fac Comp & Informat Dept Comp Sci Zagazig Egypt Zagazig Univ Fac Comp & Informat Dept Informat Technol Zagazig Egypt Assiut Univ Fac Comp & Informat Dept Comp Sci Asyut Egypt Univ Technol & Appl Sci Coll Comp & Informat Sci Informat Technol Dept Ibri Oman Hunan Univ Coll Comp Sci & Elect Engn Changsha 410082 Peoples R China Zagazig Univ Fac Comp & Informat Zagazig Egypt

Image moments are image descriptors widely utilized in several image processing, pattern recognition, computer vision, and multimedia security applications. In the era of big data, the computation of image moments yields a huge memory demand, especially for large moment order and/or high-resolution images (i.e., megapixel images). The state-of-the-art moment computation methods successfully accelerate the image moment computation for digital images of a resolution smaller than 1K x 1K pixels. For digital images of higher resolutions, image moment computation is problematic. Researchers utilized GPU-based parallel processing to overcome this problem. In practice, the parallel computation of image moments using GPUs encounters the non-extended memory problem, which is the main challenge. This paper proposed a recurrent-based method for computing the Polar Complex Exponent Transform (PCET) moments of fractional orders. The proposed method utilized the symmetry of the image kernel to reduce kernel computation. In the proposed method, once a kernel value is computed in one quaternion, the other three corresponding values in the remaining three quaternions can be trivially computed. Moreover, the proposed method utilized recurrence equations to compute kernels. Thus, the required memory to store the pre-computed memory is saved. Finally, we implemented the proposed method on the GPU parallel architecture. The proposed method overcomes the memory limit due to saving the kernel's memory. The experiments show that the proposed parallel-friendly and memory-efficient method is superior to the state-of-the-art moment computation methods in memory consumption and runtimes. The proposed method computes the PCET moment of order 50 for an image of size 2K x 2K pixels in 3.5 seconds while the state-of-the-art method of comparison needs 7.0 seconds to process the same image, the memory requirements for the proposed method and the method of comparison for the were 67.0 MB and 3.4 GB,

关键词： Fractional-order moments Parallel implementation Memory-efficient recurrence algorithm Megapixel images

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On the computational aspects of Charlier polynomials

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COGENT ENGINEERING 2020年第1期7卷 Article: 1763553页

作者： Abdul-Hadi, Alaa M. Abdulhussain, Sadiq H. Mahmmod, Basheera M. Univ Baghdad Dept Comp Engn Baghdad 10071 Iraq

Charlier polynomials (CHPs) and their moments are commonly used in image processing due to their salient performance in the analysis of signals and their capability in signal representation. The major issue of CHPs is the numerical instability of coefficients for high-order polynomials. In this study, a new recurrence algorithm is proposed to generate CHPs for high-order polynomials. First, sufficient initial values are obtained mathematically. Second, the reduced form of the recurrence algorithm is determined. Finally, a new symmetry relation for CHPs is realized to reduce the number of recurrence times. The symmetry relation is applied to calculate,50% of the polynomial coefficients. The performance of the proposed recurrence algorithm is evaluated in terms of computational cost and reconstruction error. The evaluation involves a comparison with existing recurrence algorithms. Moreover, the maximum size that can be generated using the proposed recurrence algorithm is investigated and compared with those of existing recurrence algorithms. Comparison results;indicate that the proposed algorithm exhibits better performance because it can generate a polynomial 44 times faster than existing recurrence algorithms. In addition, the improvement of the proposed

关键词： orthogonal polynomials Charlier polynomials Charlier moments recurrence algorithm

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Stationary Channel Factors and Signal Disturbances in a Direct Converter Receiver in a System with MIMO Joint Estimation algorithm

Stationary Channel Factors and Signal Disturbances in a Dire...

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Conference on Systems of Signal Synchronization, Generating and Processing in Telecommunications (SYNCHROINFO)

作者： Poborchaya, N. E. Moscow Tech Univ Commun & Informat Moscow Russia

ISBN: (纸本)9781728160726

The article discusses the algorithm for the joint estimation of the stationary channel factors and signal distortions, such as DC drift, the frequency shift remaining from the demodulation procedure, amplitude and phase imbalance (IQ-imbalance). The problem of estimating unknown parameters is solved in two stages, by combining two procedures. The first, based on the polynomial approximation of the generalized communication channel and the linear least squares (MLS) method, estimates the constant components of the signal quadrature, amplitude and phase imbalance, as well as a rough estimate of the channel frequency and factors. The second procedure is synthesized using the Taylor approximation, a modified method of least squares in the form of a functional A.N. Tikhonov and regularization method. It is a nonlinear recursive algorithm for obtaining a more accurate estimate of the frequency and channel factors. This approach allows a sufficiently high estimation accuracy to reduce the complexity of the algorithm compared to using only the second procedure. The resulting algorithm works under conditions of uncorrelated Rayleigh fading in MIMO systems with spatial multiplexing with a priori uncertainty regarding the statistical characteristics of the communication channel (except for the dispersion of additive noise) and the laws of noise distribution, both phase and additive.

关键词： MIMO estimation least squares method recurrence algorithm regularization theory direct conversion receiver signal distortion constant component drift frequency shift amplitude and phase imbalance

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Some computational aspects of Tchebichef moments for higher orders

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PATTERN RECOGNITION LETTERS 2018年第Sep.1期112卷 332-339页

作者： Camacho-Bello, Cesar Rivera-Lopez, Jose S. Univ Politecn Tulancingo Tulancingo Hidalgo Mexico

In this work, we propose a new algorithm for the computation of Tchebichef moments by means of a recurrence relation with respect to order and the Gram-Schmidt process, which reduces the numerical instability and the carry error caused by the computation of high-order moments. Results and comparison with other methods are presented. (c) 2018 Elsevier B.V. All rights reserved.

关键词： Discrete orthogonal polynomials Tchebichef polynomials Tchebichef moments recurrence algorithm Numerical propagations errors

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Performance Analysis of M2M Traffic in LTE Network Using Queuing Systems with Random Resource Requirements

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AUTOMATIC CONTROL AND COMPUTER SCIENCES 2018年第5期52卷 345-353页

作者： Sopin, E. S. Ageev, K. A. Markova, E. V. Vikhrova, O. G. Gaidamaka, Yu. V. RUDN Univ Peoples Friendship Univ Russia Appl Probabil & Informat Dept Moscow 117198 Russia

We analyze a multiserver queuing system, in which customers require a server and a certain amount of limited resources for the duration of their service. For the case of discrete resources, we develop a recurrence algorithm to evaluate the model's stationary probability distribution and its various stationary characteristics, such as the blocking probability and the average amount of occupied resources. The algorithm is applied to analysis of M2M traffic characteristics in a LTE network cell. We derive the cumulative distribution function of radio resource requirements of M2M devices and propose a sampling approach in order to apply the recurrence algorithm to the case of continuous resources.

关键词： Queueing system limited resources random requirements recurrence algorithm normalization constant CDF approximation M2M LTE

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Frequency Analysis of recurrence Variational P-Splines

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OPTOELECTRONICS INSTRUMENTATION AND DATA PROCESSING 2017年第6期53卷 591-598页

作者： Kochegurova, E. A. Kochegurov, A. I. Rozhkova, N. E. Tomsk Polytech Univ Pr Lenina 30 Tomsk 634050 Russia

Frequency responses of the procedure of spline-smoothing of information coming in real time are obtained. A recurrence spline is studied from the standpoint of the theory of linear dynamical systems. The estimation of quality and sustainability of the recurrence spline filter are described. The spline conversion laws identified in a frequency analysis are confirmed by the indicators of smoothing quality in the time domain.

关键词： recurrence algorithm spline filter free spline variational spline system function hardware function

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On Computational Aspects of Tchebichef Polynomials for Higher Polynomial Order

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IEEE ACCESS 2017年 5卷 2470-2478页

作者： Abdulhussan, Sadiz H. Ramli, Abd Rahman Al-Haddad, Syed Abdul Rahman Mahmmod, Basheera M. Jassim, Wissam A. Univ Putra Malaysia Dept Comp & Commun Syst Engn Serdang 43400 Malaysia Univ Dublin Trinity Coll Dublin ADAPT Ctr Sch Engn Dublin 2 Ireland Univ Baghdad Dept Comp Engn Baghdad 10071 Iraq

Tchebichef polynomials (TPs) and their moments are widely used in signal processing due to their remarkable performance in signal analysis, feature extraction, and compression capability. The common problem of the TP is that the coefficients computation is prone to numerical instabilities when the polynomial order becomes large. In this paper, a new algorithm is proposed to compute the TP coefficients (TPCs) for higher polynomial order by combining two existing recurrence algorithms: the three-term recurrence relations in the n-direction and x-direction. First, the TPCs are computed for x, n = 0, 1, ..., (N/2) - 1 using the recurrence in the x-direction. Second, the TPCs for x = 0, 1, ..., (N/2) - 1 and n = (N/2), (N/2) + 1,..., N - 1 based on n and x directions are calculated. Finally, the symmetry condition is applied to calculate the rest of the coefficients for x = (N/2), (N/2) + 1, ..., N - 1. In addition to the ability of the proposed algorithm to reduce the numerical propagation errors, it also accelerates the computational speed of the TPCs. The performance of the proposed algorithm was compared to that of existing algorithms for the reconstruction of speech and image signals taken from different databases. The performance of the TPCs computed by the proposed algorithm was also compared with the performance of the discrete cosine transform coefficients for speech compression systems. Different types of speech quality measures were used for evaluation. According to the results of the comparative analysis, the proposed algorithm makes the computation of the TP superior to that of conventional recurrence algorithms when the polynomial order is large.

关键词： Orthogonal polynomials recurrence algorithm Tchebichef polynomials numerical propagations errors

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