Contraharmonic Mean Research Articles

A hypergraph-based algorithm for image restoration from salt and pepper noise

An algorithm is designed for the hypergraph (HG) representation of an image, subsequent detection of Salt and Pepper (SP) noise in the image and finally the restoration of the image from this noise. The image is first represented as the set union of hyperedges. As for the hyperedges themselves, these are determined by two Image Neighborhood Hypergraph (INHG) parameters, with the concepts of 8-bit neighborhood and INHG of a graph being central. The images taken up for experimental analyses are subjected to the Contra Harmonic Mean (CHM) filter for SP noise removal. The proposed algorithm exhibits superiority over traditional algorithms and recently proposed ones in terms of visual quality, Peak Signal to Noise Ratio (PSNR) and Mean Absolute Error (MAE). This superior performance of the CHM Filter is solely due to the HG representation of the test images.

A Six-order Variant of Newton’s Method for Solving Nonlinear Equations

A new variant of Newton's method based on contra harmonic mean has been developed and its convergence properties have been discussed. The order of convergence of the proposed method is six. Starting with a suitably chosen x0, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. In terms of computational cost, it requires evaluations of only two functions and two first order derivatives per iteration. This implies that efficiency index of our method is 1.5651. The proposed method is comparable with the methods of Parhi, and Gupta (15) and that of Kou and Li (8). It does not require the evaluation of the second order derivative of the given function as required in the family of Chebyshev-Halley type methods. The efficiency of the method is tested on a number of numerical examples. It is observed that our method takes lesser number of iterations than Newton's method and the other third order variants of Newtons method. In comparison with the sixth order methods, it behaves either similarly or better for the examples considered.

A new fourth order embedded RKAHeM(4,4) method with error control on multilayer raster cellular neural network

We introduce a new technique for solving initial value problems (IVPs) with error control by formulating an embedded method involving RK methods based on arithmetic mean (AM) and Heronian mean (HeM). The function of the simulator is that it is capable of performing raster simulation for any kind as well as any size of input image. It is a powerful tool for researchers to examine the potential applications of CNN. By using the newly proposed embedded method, a versatile algorithm for simulating multilayer CNN arrays is implemented. This article proposes an efficient pseudo code for exploiting the latency properties of CNN along with well known RK-fourth order embedded numerical integration algorithms. Simulation results and comparison have also been presented to show the efficiency of the numerical integration algorithms. It is found that the RK-embedded Heronian mean outperforms well in comparison with the RK-embedded centroidal mean, harmonic mean and contra-harmonic mean. A more quantitative analysis has been carried out to clearly visualize the goodness and robustness of the proposed algorithm.

The inverse mean problem of geometric mean and contraharmonic means

In this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices (proposed in [W.N. Anderson, M.E. Mays, T.D. Morley, G.E. Trapp, The contraharmonic mean of HSD matrices, SIAM J. Algebra Disc. Meth. 8 (1987) 674–682]) A = X + Y - 2 ( X - 1 + Y - 1 ) - 1 , B = X # Y , by proving its equivalence to the well-known nonlinear matrix equation X = T − BX −1 B where T = 1 2 ( A + A # ( A + 8 BA - 1 B ) ) is the unique positive definite solution of X = A + 2 BX −1 B. The inverse mean problem is solvable if and only if B ⩽ A.

On the numerical solution of initial value problems for nonlinear trapezoidal formulas with different types

In this paper, the local truncation errors of the trapezoidal formulas such as arithmetic mean (AM), geometric mean (GM), heronian mean (HeM), harmonic mean (HaM), contraharmonic mean (CoM), root mean square (RMS), logarithmic mean (LM) and centroidal mean (CeM) are investigated and the stability analysis of these formulas are found. Finally, it is applied to various initial value problems.

A comparison of extended runge-kutta formulae based on variety of means to solve system of ivps

Extended fourth order Runge-Kutta (RK) formulae based on Arithmetic Mean (AM), Geometric Mean (GM), Harmonic Mean (HaM), Heronian Mean (HeM), Root Mean Square (RM), Centroidal Mean (CeM) and Contraharmonic Mean (CoM) are developed to solve the system of linear differential equations.The accuracy of the new formulae are tested for five major Initial Value Problems (IVPs) from the real world applications.The algorithms governing the methods can easily be computerized.The investigation undertaken in this study reveals that the extended fourth order RK methods based on AM, and CeM suit well for the system of Initial Value Problems (IVPs).

A new fourth order runge-kutta formula based on the contra-harmonic (C0M) mean

In this paper new Runge-Kutta formulae of orders 3 and 4 based on the contra harmonic mean are derived. Numerical evidence confirming the accuracy of the new method and comparison with alternative Runge-Kutta formula based on arithmetic, geometric and harmonic means are also presented.

THE RUNGE-KUTTA CONTRAHARMONIC MEAN METHOD WITH EXTRAPOLATION FOR THE PARALLEL SOLUTION OF O.D.E.'S

In this paper the implementation of the Runge-Kutta Contraharmonic Mean Method using Romberg extrapolation on a parallel computer is presented.

The Contraharmonic Mean of HSD Matrices

For positive scalars a and b the contraharmonic mean of a and b, $C(a,b)$, is defined by \[ C(a,b) = (a^2 + b^2 )/(a + b). \] In this paper we consider a natural matrix generalization of the contraharmonic mean, fit this into the matrix analogue of some of the classical scalar inequalities for means, develop computational procedures which let us generate the matrix analogues of an infinite family of scalar means, and study fixed point problems. Finally, we mention a relationship between least squares problems and the contraharmonic mean.

Four Variational Formulations of the Contraharmonic Mean of Operators

Previous article Next article Four Variational Formulations of the Contraharmonic Mean of OperatorsW. L. Green and T. D. MorleyW. L. Green and T. D. Morleyhttps://doi.org/10.1137/0608055PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract The authors establish four variational expressions, each related to the parallel sum, for the contraharmonic mean of two positive operators or matrices.[1] W. N. Anderson and , R. J. Duffin, Series and parallel addition of matrices, J. Math. Anal. Appl., 26 (1969), 576–594 10.1016/0022-247X(69)90200-5 39:3904 0177.04904 CrossrefISIGoogle Scholar[2] W. N. Anderson, , M. E. Mays and , G. E. Trapp, The contraharmonic mean of HSD matrices and electrical networks, Proc. 27th Midwest Symposium on Circuits and Systems, 1984, 665–668, June Google Scholar[3] W. N. Anderson, , M. E. Mays, , T. D. Morley and , G. E. Trapp, The contraharmonic mean of HSD matrices, SIAM J. Algebraic Discrete Methods, 8 (1987), 674–682 89b:47030 0641.15009 LinkISIGoogle Scholar[4] W. N. Anderson and , G. E. Trapp, Shorted operators. II, SIAM J. Appl. Math., 28 (1975), 60–71 10.1137/0128007 50:9417 0295.47032 LinkISIGoogle Scholar[5] T. Ando, Topics on operator inequalities, Division of Applied Mathematics, Research Institute of Applied Electricity, Hokkaido University, Sapporo, 1978ii+44, Japan 58:2451 0696.47001 Google Scholar[6] P. A. Fillmore and , J. P. Williams, On operator ranges, Advances in Math., 7 (1971), 254–281 45:2518 0224.47009 CrossrefISIGoogle Scholar[7] William L. Green and , T. D. Morley, Operator means, fixed points, and the norm convergence of monotone approximants, Math. Scand., 60 (1987), 202–218 88k:47018 0663.47017 CrossrefISIGoogle Scholar[8] T. D. Morley, Parallel summation, Maxwell's principle and the infimum of projections, J. Math. Anal. Appl., 70 (1979), 33–41 10.1016/0022-247X(79)90073-8 80g:94075 0456.47021 CrossrefISIGoogle Scholar[9] Sujit Kumar Mitra and , Madan Lal Puri, On parallel sum and difference of matrices, J. Math. Anal. Appl., 44 (1973), 92–97 10.1016/0022-247X(73)90027-9 48:3989 0271.15008 CrossrefISIGoogle Scholar[10] George E. Trapp, Hermitian semidefinite matrix means and related matrix inequalities—an introduction, Linear and Multilinear Algebra, 16 (1984), 113–123 86f:15009 0548.15013 CrossrefGoogle ScholarKeywordscontraharmonic meanparallel sumpositive operatorpositive semi-definite matrixvariational formulation Previous article Next article FiguresRelatedReferencesCited byDetails The Contraharmonic Mean of HSD MatricesWilliam N. Anderson, Jr., Michael E. Mays, Thomas D. Morley, and George E. Trapp17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 8, No. 4AbstractPDF (766 KB) Volume 8, Issue 4| 1987SIAM Journal on Algebraic Discrete Methods History Submitted:29 December 1986Accepted:21 April 1987Published online:17 July 2006 InformationCopyright © 1987 Society for Industrial and Applied MathematicsKeywordscontraharmonic meanparallel sumpositive operatorpositive semi-definite matrixvariational formulationMSC codes15A4549A2947D9926D20PDF Download Article & Publication DataArticle DOI:10.1137/0608055Article page range:pp. 670-673ISSN (print):0196-5212ISSN (online):2168-345XPublisher:Society for Industrial and Applied Mathematics