Convergence theory of exact interpolation scheme for computing several eigenvectors

The Eigen-FLS approach using an eigenspace-based scheme for fast fuzzy logic system (FLS) establishments has been attempted successfully in speech pattern recognition. However, speech pattern recognition by Eigen-FLS will still encounter a dissatisfactory recognition performance when the collected data for eigen value calculations of the FLS eigenspace is scarce. To tackle this issue, this paper proposes two improved-versioned Eigen-FLS methods, incremental MLED Eigen-FLS and EigenMLLR-like Eigen-FLS, both of which use a linear interpolation scheme for properly adjusting the target speaker’s Eigen-FLS model derived from an FLS eigenspace. Developed incremental MLED Eigen-FLS and EigenMLLR-like Eigen-FLS are superior to conventional Eigen-FLS especially in the situation of insufficient data from the target speaker.

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Optimal Parameters for Modified Butterfly Interpolation Scheme Inspired Configurations as Hole-filling Method in 3D Volume Reconstruction

2020 IEEE Graphics and Multimedia (GAME) ◽

10.1109/game50158.2020.9315085 ◽

2020 ◽

Author(s):

Chan Vei Siang ◽

Farhan Mohamed ◽

Mohd Yazid Idris ◽

Mohd Sharizal Bin Sunar ◽

Ali Bin Selamat ◽

...

Keyword(s):

Optimal Parameters ◽

Interpolation Scheme ◽

Hole Filling ◽

Volume Reconstruction ◽

Filling Method ◽

3D Volume

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Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections

Mathematics ◽

10.3390/math9131551 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1551

Author(s):

Bothina El-Sobky ◽

Yousria Abo-Elnaga ◽

Abd Allah A. Mousa ◽

Mohamed A. El-Shorbagy

Keyword(s):

Nonlinear Programming ◽

Global Convergence ◽

Programming Problem ◽

Trust Region ◽

Convergence Theory ◽

Benchmark Problems ◽

Nonlinear Programming Problem ◽

Barrier Method ◽

Starting Point ◽

Constrained Nonlinear Programming

In this paper, a penalty method is used together with a barrier method to transform a constrained nonlinear programming problem into an unconstrained nonlinear programming problem. In the proposed approach, Newton’s method is applied to the barrier Karush–Kuhn–Tucker conditions. To ensure global convergence from any starting point, a trust-region globalization strategy is used. A global convergence theory of the penalty–barrier trust-region (PBTR) algorithm is studied under four standard assumptions. The PBTR has new features; it is simpler, has rapid convergerce, and is easy to implement. Numerical simulation was performed on some benchmark problems. The proposed algorithm was implemented to find the optimal design of a canal section for minimum water loss for a triangle cross-section application. The results are promising when compared with well-known algorithms.

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Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

Axioms ◽

10.3390/axioms10030169 ◽

2021 ◽

Vol 10 (3) ◽

pp. 169

Author(s):

Avram Sidi

Keyword(s):

Nonlinear Equations ◽

Local Convergence ◽

Real Root ◽

Numerical Procedure ◽

Secant Method ◽

Convergence Theory ◽

Simple Complex ◽

Real Roots ◽

Complex Roots ◽

Appl Math

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

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