scholarly journals Advances in the Approximation of the Matrix Hyperbolic Tangent

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1219
Author(s):  
Javier Ibáñez ◽  
José M. Alonso ◽  
Jorge Sastre ◽  
Emilio Defez ◽  
Pedro Alonso-Jordá
Keyword(s):  
Taylor Series ◽  
State Of The Art ◽  
Matrix Exponential ◽  
The Other ◽  
Hyperbolic Tangent ◽  
Matrix Polynomials ◽  
Matrix Products ◽  
The Matrix ◽  
Computationally Expensive ◽  
Exponential Matrix

In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second approximation, we analyse two different alternatives to evaluate the corresponding matrix polynomials. This resulted in three stable and accurate codes, which we implemented in MATLAB and numerically and computationally compared by means of a battery of tests composed of distinct state-of-the-art matrices. Our results show that the Taylor series-based methods were more accurate, although somewhat more computationally expensive, compared with the approach based on the exponential matrix. To avoid this drawback, we propose the use of a set of formulas that allows us to evaluate polynomials in a more efficient way compared with that of the traditional Paterson–Stockmeyer method, thus, substantially reducing the number of matrix products (practically equal in number to the approach based on the matrix exponential), without penalising the accuracy of the result.

scholarly journals Sampling for Approximate Maximum Search in Factorized Tensor

2017 ◽  
Author(s):  
Zhi Lu ◽  
Yang Hu ◽  
Bing Zeng
Keyword(s):  
Theoretical Analysis ◽  
Collaborative Filtering ◽  
Real World ◽  
State Of The Art ◽  
The Other ◽  
Data Sets ◽  
Real World Data ◽  
Parafac Model ◽  
The Matrix ◽  
Special Case

Factorization models have been extensively used for recovering the missing entries of a matrix or tensor. However, directly computing all of the entries using the learned factorization models is prohibitive when the size of the matrix/tensor is large. On the other hand, in many applications, such as collaborative filtering, we are only interested in a few entries that are the largest among them. In this work, we propose a sampling-based approach for finding the top entries of a tensor which is decomposed by the CANDECOMP/PARAFAC model. We develop an algorithm to sample the entries with probabilities proportional to their values. We further extend it to make the sampling proportional to the $k$-th power of the values, amplifying the focus on the top ones. We provide theoretical analysis of the sampling algorithm and evaluate its performance on several real-world data sets. Experimental results indicate that the proposed approach is orders of magnitude faster than exhaustive computing. When applied to the special case of searching in a matrix, it also requires fewer samples than the other state-of-the-art method.

scholarly journals Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1174 ◽  
Author(s):  
Philipp Bader ◽  
Sergio Blanes ◽  
Fernando Casas
Keyword(s):  
Numerical Experiments ◽  
State Of The Art ◽  
Matrix Exponential ◽  
Superior Performance ◽  
Taylor Polynomial ◽  
Polynomial Evaluation ◽  
The Matrix ◽  
Similar Accuracy ◽  
Matrix Norms ◽  
Range Of Values

A new way to compute the Taylor polynomial of a matrix exponential is presented which reduces the number of matrix multiplications in comparison with the de-facto standard Paterson-Stockmeyer method for polynomial evaluation. Combined with the scaling and squaring procedure, this reduction is sufficient to make the Taylor method superior in performance to Padé approximants over a range of values of the matrix norms. An efficient adjustment to make the method robust against overscaling is also introduced. Numerical experiments show the superior performance of our method to have a similar accuracy in comparison with state-of-the-art implementations, and thus, it is especially recommended to be used in conjunction with Lie-group and exponential integrators where preservation of geometric properties is at issue.

scholarly journals Generalizations of two-index two-variable Hermite matrix polynomials

2009 ◽  
Vol 42 (4) ◽  
Author(s):  
M. S. Metwally ◽  
M. T. Mohamed ◽  
A. Shehata
Keyword(s):  
Wide Class ◽  
Recurrence Formula ◽  
Explicit Representation ◽  
Matrix Exponential ◽  
Matrix Functions ◽  
Differential Systems ◽  
Matrix Polynomials ◽  
The Matrix ◽  
Hermite Matrix ◽  
Class Of Matrices

AbstractIn this paper, we introduce a new generalization of the Hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems. An explicit representation and an expansion of the matrix exponential in a series of these matrix polynomials is obtained. Properties of Hermite matrix polynomials such as the recurrence formula permit an efficient computations of matrix functions are established. A new expansions of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.

scholarly journals Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1600
Author(s):  
Jorge Sastre ◽  
Javier Ibáñez
Keyword(s):  
Evaluation Method ◽  
Matrix Polynomial ◽  
Matrix Product ◽  
Product Evaluations ◽  
Polynomial Evaluation ◽  
General Matrix ◽  
Polynomial Degree ◽  
Matrix Polynomials ◽  
Matrix Products ◽  
The Matrix

Recently, two general methods for evaluating matrix polynomials requiring one matrix product less than the Paterson–Stockmeyer method were proposed, where the cost of evaluating a matrix polynomial is given asymptotically by the total number of matrix product evaluations. An analysis of the stability of those methods was given and the methods have been applied to Taylor-based implementations for computing the exponential, the cosine and the hyperbolic tangent matrix functions. Moreover, a particular example for the evaluation of the matrix exponential Taylor approximation of degree 15 requiring four matrix products was given, whereas the maximum polynomial degree available using Paterson–Stockmeyer method with four matrix products is 9. Based on this example, a new family of methods for evaluating matrix polynomials more efficiently than the Paterson–Stockmeyer method was proposed, having the potential to achieve a much higher efficiency, i.e., requiring less matrix products for evaluating a matrix polynomial of certain degree, or increasing the available degree for the same cost. However, the difficulty of these family of methods lies in the calculation of the coefficients involved for the evaluation of general matrix polynomials and approximations. In this paper, we provide a general matrix polynomial evaluation method for evaluating matrix polynomials requiring two matrix products less than the Paterson-Stockmeyer method for degrees higher than 30. Moreover, we provide general methods for evaluating matrix polynomial approximations of degrees 15 and 21 with four and five matrix product evaluations, respectively, whereas the maximum available degrees for the same cost with the Paterson–Stockmeyer method are 9 and 12, respectively. Finally, practical examples for evaluating Taylor approximations of the matrix cosine and the matrix logarithm accurately and efficiently with these new methods are given.

Pembobotan Berdasarkan Tingkat Kesamaan Semantik pada Metode Fuzzy Semi-Supervised Co-Clustering untuk Pengelompokkan Dokumen Teks

2014 ◽  
Vol 6 (2) ◽  
pp. 46-51
Author(s):  
Galang Amanda Dwi P. ◽  
Gregorius Edwadr ◽  
Agus Zainal Arifin
Keyword(s):  
Supervised Learning ◽  
Semantic Similarity ◽  
The Other ◽  
Classification Result ◽  
Document Similarity ◽  
The Matrix ◽  
Index Terms ◽  
Membership Value ◽  
Degree Of Similarity

Nowadays, a large number of information can not be reached by the reader because of the misclassification of text-based documents. The misclassified data can also make the readers obtain the wrong information. The method which is proposed by this paper is aiming to classify the documents into the correct group.  Each document will have a membership value in several different classes. The method will be used to find the degree of similarity between the two documents is the semantic similarity. In fact, there is no document that doesn’t have a relationship with the other but their relationship might be close to 0. This method calculates the similarity between two documents by taking into account the level of similarity of words and their synonyms. After all inter-document similarity values obtained, a matrix will be created. The matrix is then used as a semi-supervised factor. The output of this method is the value of the membership of each document, which must be one of the greatest membership value for each document which indicates where the documents are grouped. Classification result computed by the method shows a good value which is 90 %. Index Terms - Fuzzy co-clustering, Heuristic, Semantica Similiarity, Semi-supervised learning.

scholarly journals Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1483
Author(s):  
Shanqin Chen
Keyword(s):  
Large Time ◽  
Hyperbolic Equations ◽  
Strong Stability ◽  
Matrix Exponential ◽  
Linear Component ◽  
Nonlinear Hyperbolic Equations ◽  
Time Step ◽  
Integrating Factor ◽  
The Matrix ◽  
Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

Computer Vision and robotics in postal automation

1999 ◽  
Vol 18 (3-4) ◽  
pp. 265-273
Author(s):  
Giovanni B. Garibotto
Keyword(s):  
Image Processing ◽  
Computer Vision ◽  
Pattern Recognition ◽  
Material Handling ◽  
State Of The Art ◽  
Short Description ◽  
The Other ◽  
Functional Requirements ◽  
Postal Automation ◽  
And Robotics

The paper is intended to provide an overview of advanced robotic technologies within the context of Postal Automation services. The main functional requirements of the application are briefly referred, as well as the state of the art and new emerging solutions. Image Processing and Pattern Recognition have always played a fundamental role in Address Interpretation and Mail sorting and the new challenging objective is now off-line handwritten cursive recognition, in order to be able to handle all kind of addresses in a uniform way. On the other hand, advanced electromechanical and robotic solutions are extremely important to solve the problems of mail storage, transportation and distribution, as well as for material handling and logistics. Finally a short description of new services of Postal Automation is referred, by considering new emerging services of hybrid mail and paper to electronic conversion.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

Occupant pre-crash kinematics in rotated seat arrangements

2021 ◽  
pp. 095440702110045
Author(s):  
Alexander Diederich ◽  
Christophe Bastien ◽  
Karthikeyan Ekambaram ◽  
Alexis Wilson
Keyword(s):  
State Of The Art ◽  
The Other ◽  
Human Model ◽  
Multi Objective ◽  
Emergency Braking ◽  
Seating Arrangement ◽  
Current State ◽  
Occupant Comfort ◽  
Occupant Kinematics ◽  
Belt System

The introduction of automated L5 driving technologies will revolutionise the design of vehicle interiors and seating configurations, improving occupant comfort and experience. It is foreseen that pre-crash emergency braking and swerving manoeuvres will affect occupant posture, which could lead to an interaction with a deploying airbag. This research addresses the urgent safety need of defining the occupant’s kinematics envelope during that pre-crash phase, considering rotated seat arrangements and different seatbelt configurations. The research used two different sets of volunteer tests experiencing L5 vehicle manoeuvres, based in the first instance on 22 50th percentile fit males wearing a lap-belt (OM4IS), while the other dataset is based on 87 volunteers with a BMI range of 19 to 67 kg/m2 wearing a 3-point belt (UMTRI). Unique biomechanics kinematics corridors were then defined, as a function of belt configuration and vehicle manoeuvre, to calibrate an Active Human Model (AHM) using a multi-objective optimisation coupled with a Correlation and Analysis (CORA) rating. The research improved the AHM omnidirectional kinematics response over current state of the art in a generic lap-belted environment. The AHM was then tested in a rotated seating arrangement under extreme braking, highlighting that maximum lateral and frontal motions are comparable, independent of the belt system, while the asymmetry of the 3-point belt increased the occupant’s motion towards the seatbelt buckle. It was observed that the frontal occupant kinematics decrease by 200 mm compared to a lap-belted configuration. This improved omnidirectional AHM is the first step towards designing safer future L5 vehicle interiors.

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