Search results for: Sparse approximation.

Authors: V. T. S. Dao, T. G. Etoh, C. Vo Le, H. D. Nguyen, K. Takehara, T. Akino, K. Nishi

Abstract:

Since 2004, we have been developing an in-situ storage image sensor (ISIS) that captures more than 100 consecutive images at a frame rate of 10 Mfps with ultra-high sensitivity as well as the video camera for use with this ISIS. Currently, basic research is continuing in an attempt to increase the frame rate up to 100 Mfps and above. In order to suppress electro-magnetic noise at such high frequency, a digital-noiseless imaging transfer scheme has been developed utilizing solely sinusoidal driving voltages. This paper presents highly efficient-yet-accurate expressions to estimate attenuation as well as phase delay of driving voltages through RC networks of an ultra-high-speed image sensor. Elmore metric for a fundamental RC chain is employed as the first-order approximation. By application of dimensional analysis to SPICE data, we found a simple expression that significantly improves the accuracy of the approximation. Similarly, another simple closed-form model to estimate phase delay through fundamental RC networks is also obtained. Estimation error of both expressions is much less than previous works, only less 2% for most of the cases . The framework of this analysis can be extended to address similar issues of other VLSI structures.

Keywords: Dimensional Analysis, ISIS, Digital-noiseless, RC network, Attenuation, Phase Delay, Elmore model

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Authors: Jack R. McKenzie, Peter A. Appleby, Thomas House, Neil Walton

Abstract:

Cold-start is a notoriously difficult problem which can occur in recommendation systems, and arises when there is insufficient information to draw inferences for users or items. To address this challenge, a contextual bandit algorithm – the Fast Approximate Bayesian Contextual Cold Start Learning algorithm (FAB-COST) – is proposed, which is designed to provide improved accuracy compared to the traditionally used Laplace approximation in the logistic contextual bandit, while controlling both algorithmic complexity and computational cost. To this end, FAB-COST uses a combination of two moment projection variational methods: Expectation Propagation (EP), which performs well at the cold start, but becomes slow as the amount of data increases; and Assumed Density Filtering (ADF), which has slower growth of computational cost with data size but requires more data to obtain an acceptable level of accuracy. By switching from EP to ADF when the dataset becomes large, it is able to exploit their complementary strengths. The empirical justification for FAB-COST is presented, and systematically compared to other approaches on simulated data. In a benchmark against the Laplace approximation on real data consisting of over 670, 000 impressions from autotrader.co.uk, FAB-COST demonstrates at one point increase of over 16% in user clicks. On the basis of these results, it is argued that FAB-COST is likely to be an attractive approach to cold-start recommendation systems in a variety of contexts.

Keywords: Cold-start, expectation propagation, multi-armed bandits, Thompson sampling, variational inference.

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Authors: Wanhyun Cho, Soonja Kang, Sangkyoon Kim, Soonyoung Park

Abstract:

In this paper, we propose the variational EM inference algorithm for the multi-class Gaussian process classification model that can be used in the field of human behavior recognition. This algorithm can drive simultaneously both a posterior distribution of a latent function and estimators of hyper-parameters in a Gaussian process classification model with multiclass. Our algorithm is based on the Laplace approximation (LA) technique and variational EM framework. This is performed in two steps: called expectation and maximization steps. First, in the expectation step, using the Bayesian formula and LA technique, we derive approximately the posterior distribution of the latent function indicating the possibility that each observation belongs to a certain class in the Gaussian process classification model. Second, in the maximization step, using a derived posterior distribution of latent function, we compute the maximum likelihood estimator for hyper-parameters of a covariance matrix necessary to define prior distribution for latent function. These two steps iteratively repeat until a convergence condition satisfies. Moreover, we apply the proposed algorithm with human action classification problem using a public database, namely, the KTH human action data set. Experimental results reveal that the proposed algorithm shows good performance on this data set.

Keywords: Bayesian rule, Gaussian process classification model with multiclass, Gaussian process prior, human action classification, laplace approximation, variational EM algorithm.

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Authors: K. A. D. N. K Wimalawarne

Abstract:

In this paper we introduce a novel kernel classifier based on a iterative shrinkage algorithm developed for compressive sensing. We have adopted Bregman iteration with soft and hard shrinkage functions and generalized hinge loss for solving l1 norm minimization problem for classification. Our experimental results with face recognition and digit classification using SVM as the benchmark have shown that our method has a close error rate compared to SVM but do not perform better than SVM. We have found that the soft shrinkage method give more accuracy and in some situations more sparseness than hard shrinkage methods.

Keywords: Compressive sensing, Bregman iteration, Generalisedhinge loss, sparse, kernels, shrinkage functions

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Authors: P. Ashok, G. M. Kadhar Nawaz

Abstract:

Rough set theory is used to handle uncertainty and incomplete information by applying two accurate sets, Lower approximation and Upper approximation. In this paper, the rough clustering algorithms are improved by adopting the Similarity, Dissimilarity–Similarity and Entropy based initial centroids selection method on three different clustering algorithms namely Entropy based Rough K-Means (ERKM), Similarity based Rough K-Means (SRKM) and Dissimilarity-Similarity based Rough K-Means (DSRKM) were developed and executed by yeast dataset. The rough clustering algorithms are validated by cluster validity indexes namely Rand and Adjusted Rand indexes. An experimental result shows that the ERKM clustering algorithm perform effectively and delivers better results than other clustering methods. Outlier detection is an important task in data mining and very much different from the rest of the objects in the clusters. Entropy based Rough Outlier Factor (EROF) method is seemly to detect outlier effectively for yeast dataset. In rough K-Means method, by tuning the epsilon (ᶓ) value from 0.8 to 1.08 can detect outliers on boundary region and the RKM algorithm delivers better results, when choosing the value of epsilon (ᶓ) in the specified range. An experimental result shows that the EROF method on clustering algorithm performed very well and suitable for detecting outlier effectively for all datasets. Further, experimental readings show that the ERKM clustering method outperformed the other methods.

Keywords: Clustering, Entropy, Outlier, Rough K-Means, validity index.

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Authors: Navdeep Goel, Salvador Gabarda

Abstract:

Digital images are widely used in computer applications. To store or transmit the uncompressed images requires considerable storage capacity and transmission bandwidth. Image compression is a means to perform transmission or storage of visual data in the most economical way. This paper explains about how images can be encoded to be transmitted in a multiplexing time-frequency domain channel. Multiplexing involves packing signals together whose representations are compact in the working domain. In order to optimize transmission resources each 4 × 4 pixel block of the image is transformed by a suitable polynomial approximation, into a minimal number of coefficients. Less than 4 × 4 coefficients in one block spares a significant amount of transmitted information, but some information is lost. Different approximations for image transformation have been evaluated as polynomial representation (Vandermonde matrix), least squares + gradient descent, 1-D Chebyshev polynomials, 2-D Chebyshev polynomials or singular value decomposition (SVD). Results have been compared in terms of nominal compression rate (NCR), compression ratio (CR) and peak signal-to-noise ratio (PSNR) in order to minimize the error function defined as the difference between the original pixel gray levels and the approximated polynomial output. Polynomial coefficients have been later encoded and handled for generating chirps in a target rate of about two chirps per 4 × 4 pixel block and then submitted to a transmission multiplexing operation in the time-frequency domain.

Keywords: Chirp signals, Image multiplexing, Image transformation, Linear canonical transform, Polynomial approximation.

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Authors: Arshia Azam, J. Amarnath, Ch. D. V. Paradesi Rao

Abstract:

The advantage of solving the complex nonlinear problems by utilizing fuzzy logic methodologies is that the experience or expert-s knowledge described as a fuzzy rule base can be directly embedded into the systems for dealing with the problems. The current limitation of appropriate and automated designing of fuzzy controllers are focused in this paper. The structure discovery and parameter adjustment of the Branched T-S fuzzy model is addressed by a hybrid technique of type constrained sparse tree algorithms. The simulation result for different system model is evaluated and the identification error is observed to be minimum.

Keywords: Fuzzy logic, branch T-S fuzzy model, tree modeling, complex nonlinear system.

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Authors: Sundararajan Sangeetha, Joseph Jesu Christopher, Swaminathan Ramakrishnan

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In this work, the primary compressive strength components of human femur trabecular bone are qualitatively assessed using image processing and wavelet analysis. The Primary Compressive (PC) component in planar radiographic femur trabecular images (N=50) is delineated by semi-automatic image processing procedure. Auto threshold binarization algorithm is employed to recognize the presence of mineralization in the digitized images. The qualitative parameters such as apparent mineralization and total area associated with the PC region are derived for normal and abnormal images.The two-dimensional discrete wavelet transforms are utilized to obtain appropriate features that quantify texture changes in medical images .The normal and abnormal samples of the human femur are comprehensively analyzed using Harr wavelet.The six statistical parameters such as mean, median, mode, standard deviation, mean absolute deviation and median absolute deviation are derived at level 4 decomposition for both approximation and horizontal wavelet coefficients. The correlation coefficient of various wavelet derived parameters with normal and abnormal for both approximated and horizontal coefficients are estimated. It is seen that in almost all cases the abnormal show higher degree of correlation than normals. Further the parameters derived from approximation coefficient show more correlation than those derived from the horizontal coefficients. The parameters mean and median computed at the output of level 4 Harr wavelet channel was found to be a useful predictor to delineate the normal and the abnormal groups.

Keywords: Image processing, planar radiographs, trabecular bone and wavelet analysis.

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Authors: E. N. Mathioudakis, E. P. Papadopoulou

Abstract:

The problem addressed herein is the efficient management of the Grid/Cluster intense computation involved, when the preconditioned Bi-CGSTAB Krylov method is employed for the iterative solution of the large and sparse linear system arising from the discretization of the Modified Helmholtz-Dirichlet problem by the Hermite Collocation method. Taking advantage of the Collocation ma-trix's red-black ordered structure we organize efficiently the whole computation and map it on a pipeline architecture with master-slave communication. Implementation, through MPI programming tools, is realized on a SUN V240 cluster, inter-connected through a 100Mbps and 1Gbps ethernet network,and its performance is presented by speedup measurements included.

Keywords: Collocation, Preconditioned Bi-CGSTAB, MPI, Grid and DSM Systems.

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Authors: Kankeyanathan Kannan

Abstract:

The uniform Roe C*-algebra (also called uniform translation)C^*- algebra provides a link between coarse geometry and C^*- algebra theory. The uniform Roe algebra has a great importance in geometry, topology and analysis. We consider some of the elementary concepts associated with coarse spaces. 

Keywords: Invariant Approximation Property, Uniform Roe algebras.

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Authors: J. Kavikumar, N. S. Manian, M. B. K. Moorthy

Abstract:

The main purpose of this paper is to consider the new kind of approximation which is called as t-best coapproximation in fuzzy n-normed spaces. The set of all t-best coapproximation define the t-coproximinal, t-co-Chebyshev and F-best coapproximation and then prove several theorems pertaining to this sets. 

Keywords: Fuzzy-n-normed space, best coapproximation, co-proximinal, co-Chebyshev, F-best coapproximation, orthogonality

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Authors: Mohammed A. Abutheraa, David Lester

Abstract:

We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions.

Keywords: Approximation Theory, Chebyshev Polynomial, Computable Functions, Computable Real Arithmetic, Integration, Numerical Analysis.

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Authors: Kemalettin Erbatur, Özer Koca, Evrim Taşkıran, Metin Yılmaz, Utku Seven

Abstract:

Recent fifteen years witnessed fast improvements in the field of humanoid robotics. The human-like robot structure is more suitable to human environment with its supreme obstacle avoidance properties when compared with wheeled service robots. However, the walking control for bipedal robots is a challenging task due to their complex dynamics. Stable reference generation plays a very important role in control. Linear Inverted Pendulum Model (LIPM) and the Zero Moment Point (ZMP) criterion are applied in a number of studies for stable walking reference generation of biped walking robots. This paper follows this main approach too. We propose a natural and continuous ZMP reference trajectory for a stable and human-like walk. The ZMP reference trajectories move forward under the sole of the support foot when the robot body is supported by a single leg. Robot center of mass trajectory is obtained from predefined ZMP reference trajectories by a Fourier series approximation method. The Gibbs phenomenon problem common with Fourier approximations of discontinuous functions is avoided by employing continuous ZMP references. Also, these ZMP reference trajectories possess pre-assigned single and double support phases, which are very useful in experimental tuning work. The ZMP based reference generation strategy is tested via threedimensional full-dynamics simulations of a 12-degrees-of-freedom biped robot model. Simulation results indicate that the proposed reference trajectory generation technique is successful.

Keywords: Biped robot, Linear Inverted Pendulum Model, Zero Moment Point, Fourier series approximation.

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Authors: El-Said Badr, Mahmoud Moussa, Konstantinos Paparrizos, Nikolaos Samaras, Angelo Sifaleras

Abstract:

There are two major variants of the Simplex Algorithm: the revised method and the standard, or tableau method. Today, all serious implementations are based on the revised method because it is more efficient for sparse linear programming problems. Moreover, there are a number of applications that lead to dense linear problems so our aim in this paper is to present some computational results on parallel implementation of dense Simplex Method. Our implementation is implemented on a SMP cluster using C programming language and the Message Passing Interface MPI. Preliminary computational results on randomly generated dense linear programs support our results.

Keywords: Linear Programming, MPI, Parallel Implementation, Simplex Algorithm.

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Authors: Isao Taguchi, Yasuo Sugai

Abstract:

This paper proposes an efficient learning method for the layered neural networks based on the selection of training data and input characteristics of an output layer unit. Comparing to recent neural networks; pulse neural networks, quantum neuro computation, etc, the multilayer network is widely used due to its simple structure. When learning objects are complicated, the problems, such as unsuccessful learning or a significant time required in learning, remain unsolved. Focusing on the input data during the learning stage, we undertook an experiment to identify the data that makes large errors and interferes with the learning process. Our method devides the learning process into several stages. In general, input characteristics to an output layer unit show oscillation during learning process for complicated problems. The multi-stage learning method proposes by the authors for the function approximation problems of classifying learning data in a phased manner, focusing on their learnabilities prior to learning in the multi layered neural network, and demonstrates validity of the multi-stage learning method. Specifically, this paper verifies by computer experiments that both of learning accuracy and learning time are improved of the BP method as a learning rule of the multi-stage learning method. In learning, oscillatory phenomena of a learning curve serve an important role in learning performance. The authors also discuss the occurrence mechanisms of oscillatory phenomena in learning. Furthermore, the authors discuss the reasons that errors of some data remain large value even after learning, observing behaviors during learning.

Keywords: data selection, function approximation problem, multistage leaning, neural network, voluntary oscillation.

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Authors: Wenshuai Zhai, Yunhua Zhang

Abstract:

Synthetic aperture radar (SAR) imaging usually requires echo data collected continuously pulse by pulse with certain bandwidth. However in real situation, data collection or part of signal spectrum can be interrupted due to various reasons, i.e. there will be gaps in spatial spectrum. In this case we need to find ways to fill out the resulted gaps and get image with defined resolution. In this paper we introduce our work on how to apply iterative spatially variant apodization (Super-SVA) technique to extrapolate the spatial spectrum in both azimuthal and range directions so as to fill out the gaps and get correct radar image.

Keywords: SAR imaging, Sparse aperture, Stepped frequencychirp signal, high resolution, Super-SVA

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Authors: Qiang Niu, Linzhang Lu

Abstract:

Restarted GMRES methods augmented with approximate eigenvectors are widely used for solving large sparse linear systems. Recently a new scheme of augmenting with error approximations is proposed. The main aim of this paper is to develop a restarted GMRES method augmented with the combination of harmonic Ritz vectors and error approximations. We demonstrate that the resulted combination method can gain the advantages of two approaches: (i) effectively deflate the small eigenvalues in magnitude that may hamper the convergence of the method and (ii) partially recover the global optimality lost due to restarting. The effectiveness and efficiency of the new method are demonstrated through various numerical examples.

Keywords: Arnoldi process, GMRES, Krylov subspace, systems of linear equations.

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Authors: Ali Al-Ataby , Fawzi Al-Naima

Abstract:

Symbolic Circuit Analysis (SCA) is a technique used to generate the symbolic expression of a network. It has become a well-established technique in circuit analysis and design. The symbolic expression of networks offers excellent way to perform frequency response analysis, sensitivity computation, stability measurements, performance optimization, and fault diagnosis. Many approaches have been proposed in the area of SCA offering different features and capabilities. Numerical Interpolation methods are very common in this context, especially by using the Fast Fourier Transform (FFT). The aim of this paper is to present a method for SCA that depends on the use of Wavelet Transform (WT) as a mathematical tool to generate the symbolic expression for large circuits with minimizing the analysis time by reducing the number of computations.

Keywords: Numerical Interpolation, Sparse Matrices, SymbolicAnalysis, Wavelet Transform.

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Authors: Hasmayadi Abdul Majid, Yuzman Yusoff, Noor Shelida Salleh

Abstract:

This paper presents the development of a single-ended 38.5 kS/s 10-bit programmable reference SAR ADC which is realized in MIMOS’s 0.35 µm CMOS process. The design uses a resistive DAC, a dynamic comparator with pre-amplifier and a SAR digital logic to create 10 effective bits ADC. A programmable reference circuitry allows the ADC to operate with different input range from 0.6 V to 2.1 V. The ADC consumed less than 7.5 mW power with a 3 V supply.

Keywords: Successive Approximation Register Analog-to- Digital Converter, SAR ADC, Resistive DAC, Programmable Reference.

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Authors: Yongxin Yuan

Abstract:

The problem of updating damped gyroscopic systems using measured modal data can be mathematically formulated as following two problems. Problem I: Given Ma ∈ Rn×n, Λ = diag{λ1, ··· , λp} ∈ Cp×p, X = [x1, ··· , xp] ∈ Cn×p, where p<n and both Λ and X are closed under complex conjugation in the sense that λ2j = λ¯2j−1 ∈ C, x2j = ¯x2j−1 ∈ Cn for j = 1, ··· , l, and λk ∈ R, xk ∈ Rn for k = 2l + 1, ··· , p, find real-valued symmetric matrices D,K and a real-valued skew-symmetric matrix G (that is, GT = −G) such that MaXΛ2 + (D + G)XΛ + KX = 0. Problem II: Given real-valued symmetric matrices Da, Ka ∈ Rn×n and a real-valued skew-symmetric matrix Ga, find (D, ˆ G, ˆ Kˆ ) ∈ SE such that Dˆ −Da2+Gˆ−Ga2+Kˆ −Ka2 = min(D,G,K)∈SE (D− Da2 + G − Ga2 + K − Ka2), where SE is the solution set of Problem I and · is the Frobenius norm. This paper presents an iterative algorithm to solve Problem I and Problem II. By using the proposed iterative method, a solution of Problem I can be obtained within finite iteration steps in the absence of roundoff errors, and the minimum Frobenius norm solution of Problem I can be obtained by choosing a special kind of initial matrices. Moreover, the optimal approximation solution (D, ˆ G, ˆ Kˆ ) of Problem II can be obtained by finding the minimum Frobenius norm solution of a changed Problem I. A numerical example shows that the introduced iterative algorithm is quite efficient.

Keywords: Model updating, iterative algorithm, gyroscopic system, partially prescribed spectral data, optimal approximation.

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Authors: Clark Van Dam, Gagan Mirchandani

Abstract:

The Continuously Adaptive Mean-Shift (CamShift) algorithm, incorporating scene depth information is combined with the l1-minimization sparse representation based method to form a hybrid kernel and state space-based tracking algorithm. We take advantage of the increased efficiency of the former with the robustness to occlusion property of the latter. A simple interchange scheme transfers control between algorithms based upon drift and occlusion likelihood. It is quantified by the projection of target candidates onto a depth map of the 2D scene obtained with a low cost stereo vision webcam. Results are improved tracking in terms of drift over each algorithm individually, in a challenging practical outdoor multiple occlusion test case.

Keywords: CamShift, l1-minimization, particle filter, stereo vision, video tracking.

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Authors: Jeff D. Colwell, Benjamin W. Knox

Abstract:

While fire and explosion incidents are often observed by eyewitnesses, the weight that fire investigators should place on those observations in their investigations is a complex issue. There is no doubt that eyewitness statements can be an important component to an investigation, particularly when other evidence is sparse, as is often the case when damage to the scene is severe. However, it is well known that eyewitness statements can be incorrect for a variety of reasons, including deception. In this paper, we reviewed factors that can have an effect on the complex processes associated with the perception, retention, and retrieval of an event. We then review the accuracy of eyewitness statements from unique criminal and civil incidents, including fire and explosion incidents, in which the accuracy of the statements could be independently evaluated. Finally, the motives for deceptive eyewitness statements are described, along with techniques that fire and explosion investigators can employ, to increase the accuracy of the eyewitness statements that they solicit.

Keywords: Explosion, eyewitness, fire, reliability.

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Authors: Daniel I. Morariu, Lucian N. Vintan, Volker Tresp

Abstract:

Text categorization is the problem of classifying text documents into a set of predefined classes. After a preprocessing step, the documents are typically represented as large sparse vectors. When training classifiers on large collections of documents, both the time and memory restrictions can be quite prohibitive. This justifies the application of feature selection methods to reduce the dimensionality of the document-representation vector. In this paper, we present three feature selection methods: Information Gain, Support Vector Machine feature selection called (SVM_FS) and Genetic Algorithm with SVM (called GA_SVM). We show that the best results were obtained with GA_SVM method for a relatively small dimension of the feature vector.

Keywords: Feature Selection, Learning with Kernels, Support Vector Machine, Genetic Algorithm, and Classification.

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Authors: Daniel Morariu, Lucian N. Vintan, Volker Tresp

Abstract:

Text categorization is the problem of classifying text documents into a set of predefined classes. After a preprocessing step, the documents are typically represented as large sparse vectors. When training classifiers on large collections of documents, both the time and memory restrictions can be quite prohibitive. This justifies the application of feature selection methods to reduce the dimensionality of the document-representation vector. In this paper, three feature selection methods are evaluated: Random Selection, Information Gain (IG) and Support Vector Machine feature selection (called SVM_FS). We show that the best results were obtained with SVM_FS method for a relatively small dimension of the feature vector. Also we present a novel method to better correlate SVM kernel-s parameters (Polynomial or Gaussian kernel).

Keywords: Feature Selection, Learning with Kernels, SupportVector Machine, and Classification.

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Authors: Nur Zakiah Mohd Saat, Abdul Aziz Jemain

Abstract:

Saddlepoint approximations is one of the tools to obtain an expressions for densities and distribution functions. We approximate the densities of the observed gaps between the hypopnea events using the Huzurbazar saddlepoint approximation. We demonstrate the density of a maximum likelihood estimator in exponential families.

Keywords: Exponential, maximum likehood estimators, observed gap, Saddlepoint approximations.

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Authors: Weizhi Xu, Zhiyong Liu, Dongrui Fan, Shuai Jiao, Xiaochun Ye, Fenglong Song, Chenggang Yan

Abstract:

Many-core GPUs provide high computing ability and substantial bandwidth; however, optimizing irregular applications like SpMV on GPUs becomes a difficult but meaningful task. In this paper, we propose a novel method to improve the performance of SpMV on GPUs. A new storage format called HYB-R is proposed to exploit GPU architecture more efficiently. The COO portion of the matrix is partitioned recursively into a ELL portion and a COO portion in the process of creating HYB-R format to ensure that there are as many non-zeros as possible in ELL format. The method of partitioning the matrix is an important problem for HYB-R kernel, so we also try to tune the parameters to partition the matrix for higher performance. Experimental results show that our method can get better performance than the fastest kernel (HYB) in NVIDIA-s SpMV library with as high as 17% speedup.

Keywords: GPU, HYB-R, Many-core, Performance Tuning, SpMV

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Authors: Xian Ming Gu, Ting Zhu Huang, Hou Biao Li

Abstract:

In this paper, getting an high-efficiency parallel algorithm to solve sparse block pentadiagonal linear systems suitable for vectors and parallel processors, stair matrices are used to construct some parallel polynomial approximate inverse preconditioners. These preconditioners are appropriate when the desired target is to maximize parallelism. Moreover, some theoretical results about these preconditioners are presented and how to construct preconditioners effectively for any nonsingular block pentadiagonal H-matrices is also described. In addition, the availability of these preconditioners is illustrated with some numerical experiments arising from two dimensional biharmonic equation.

Keywords: Parallel algorithm, Pentadiagonal matrix, Polynomial approximate inverse, Preconditioners, Stair matrix.

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Authors: Mohamed K. El Daou

Abstract:

We construct an exponentially weighted Legendre- Gauss Tau method for solving differential equations with oscillatory solutions. The proposed method is applied to Sturm-Liouville problems. Numerical examples illustrating the efficiency and the high accuracy of our results are presented.

Keywords: Oscillatory functions, Sturm-Liouville problems, legendre polynomial, gauss points.

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Authors: Minghui Wang, Luping Xu, Juntao Zhang

Abstract:

Based on the conjugate gradient (CG) algorithm, the constrained matrix equation AXB=C and the associate optimal approximation problem are considered for the symmetric arrowhead matrix solutions in the premise of consistency. The convergence results of the method are presented. At last, a numerical example is given to illustrate the efficiency of this method.

Keywords: Iterative method, symmetric arrowhead matrix, conjugate gradient algorithm.

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Authors: Edward J. Kansa, Pavel Holoborodko

Abstract:

Continuously differentiable radial basis functions (RBFs) are meshfree, converge faster as the dimensionality increases, and is theoretically spectrally convergent. When implemented on current single and double precision computers, such RBFs can suffer from ill-conditioning because the systems of equations needed to be solved to find the expansion coefficients are full. However, the Advanpix extended precision software package allows computer mathematics to resemble asymptotically ideal Platonic mathematics. Additionally, full systems with extended precision execute faster graphical processors units and field-programmable gate arrays because no branching is needed. Sparse equation systems are fast for iterative solvers in a very limited number of cases.

Keywords: Meshless spectrally convergent, partial differential equations, extended arithmetic precision, no branching.

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