some further results on the eccentric distance sum abstract let g v g e g be a simple connected graph then the eccentric distance sum of g which is a novel graph invariant by offering a great potential for structure activity property relationships is defined as d g v v e g v d g v where e g v is the eccentricity of the vertex v and d g v is the sum of all distance from the vertex v as a continuation to the parts of 22 li et al j math anal appl 43 1149 1162 2015 and 17 hua et al discrete appl math 160 170 180 2012 this paper answers one of some remaining problems in 22 is how to determine the bipartite graphs of even diameter with the minimum eds and gives a nordhaus gaddum bound for eccentric distance sum which is the generalization of the corresponding results in 17 moreover some sharp lower bounds on the eds of halin graphs and of triangle free and quadrangle free graphs are respectively presented

an extension of the conjugate residual method to nonsymmetric linear systems the conjugate gradient cg method and the conjugate residual cr method are krylov subspace methods for solving symmetric positive definite linear systems to solve nonsymmetric linear systems the bi conjugate gradient bi cg method has been proposed as an extension of cg bi cg has attractive short term recurrences and it is the basis for the successful variants such as bi cgstab in this paper we extend cr to nonsymmetric linear systems with the aim of finding an alternative basic solver numerical experiments show that the resulting algorithm with short term recurrences often gives smoother convergence behavior than bi cg hence it may take the place of bi cg for the successful variants

on the minimum eccentric distance sum of bipartite graphs with some given parameters the eccentric distance sum is a novel graph invariant with vast potential in structure activity property relationships this graph invariant displays high discriminating power with respect to both biological activity and physical properties if g vg eg is a simple connected graph then the eccentric distance sum eds of g is defined as d g v vgeg v dg v where eg v is the eccentricity of the vertex v and dg v u vgdg u v is the sum of all distances from the vertex v much extremal work has been done on trees with some given parameters by yu et al 2011 25 li et al 2012 19 and geng et al 2013 6 it is natural to consider this extremal problem on bipartite graphs with some given parameters in this paper sharp lower bound on the eds in the class of all connected bipartite graphs with a given matching number q is determined the minimum eds is realized only by the graph kq n q the extremal graph with the minimum eds in the class of all the n vertex connected bipartite graphs of odd diameter is characterized all the extremal graphs having the minimum eds in the class of all connected n vertex bipartite graphs with a given vertex connectivity are identified as well

high speed low complexity systolic designs of novel iterative division algorithms in gf 2 sup m we extend the binary algorithm invented by stein and propose novel iterative division algorithms over gf 2 sup m for systolic vlsi realization while algorithm ebg is a basic prototype with guaranteed convergence in at most 2m 1 iterations its variants algorithms ebd and ebdf are designed for reduced complexity and fixed critical path delay respectively we show that algorithms ebd and ebdf can be mapped to parallel in parallel out systolic circuits with low area time complexities of o m sup 2 loglogm and o m sup 2 respectively compared to the systolic designs based on the extended euclid s algorithm our circuits exhibit significant speed and area advantages

on the eccentric distance sum of trees and unicyclic graphs abstract let g be a simple connected graph with the vertex set v g the eccentric distance sum of g is defined as d g v v g e v d g v where e v is the eccentricity of the vertex v and d g v u v g d u v is the sum of all distances from the vertex v in this paper we characterize the extremal unicyclic graphs among n vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum in addition we characterize the extremal trees with given diameter and minimal eccentric distance sum

transpose free multiple lanczos and its application in pade approximation a transpose free two sided nonsymmetric lanczos method is developed for multiple starting vectors on both the left and right the method is mathematically equivalent to the two sided methods without look ahead or deflation steps but avoids the use of the transpose of the system matrix the method is applied to the computation of the matrix pade approximation to a linear dynamical system the result is a method which can be labeled transpose free matrix pade via lanczos tfmpvl under certain circumstances tfmpvl will actually reduce the total number of matrix vector products needed it is illustrated with some numerical examples

a simplified and flexible variant of gcrot for solving nonsymmetric linear systems there is a need for flexible iterative solvers that can solve large scale 10 6 unknowns nonsymmetric sparse linear systems to a small tolerance among flexible solvers flexible gmres fgmres is attractive because it minimizes the residual norm over a particular subspace in practice fgmres is often restarted periodically to keep memory and work requirements reasonable however like restarted gmres restarted fgmres can suffer from stagnation this has led us to develop a flexible variant of the krylov linear solver gcrot generalized conjugate residual with inner orthogonalization and outer truncation unlike the original gcrot algorithm the proposed gcrot variant uses a simplified truncation strategy similar to loose gmres lgmres this modification is motivated by numerical experiments that suggest the specific subspace retained in the outer iteration of gcrot is less important than its size the flexible gcrot variant appears to be well suited for advection dominated problems in particular when applied to an adjoint problem from computational aerodynamics the proposed gcrot variant is robust and efficient compared with several popular truncated krylov subspace methods finally a flexible version of lgmres is easily constructed by recognizing algorithmic similarities to gcrot

interpreting idr as a petrov galerkin method the induced dimension reduction idr method of sonneveld and van gijzen siam j sci comput 31 2008 pp 1035 1062 is shown to be a petrov galerkin projection method with a particular choice of left krylov subspaces these left subspaces are rational krylov spaces consequently other methods such as bicgstab and ml s bicgstab which are mathematically equivalent to some versions of idr can also be interpreted as petrov galerkin methods the connection with rational krylov spaces inspired a new version of idr called ritz idr where the poles of the rational function are chosen as certain ritz values experiments are presented illustrating the effectiveness of this new version

three dimensional visco acoustic modeling using a renormalized integral equation iterative solver we present a frequency domain renormalized integral equation formulation for solving a three dimensional visco acoustic medium using an iterative solver upon applying this special renormalization the resulting integral equation operator can be proven to have a contraction property hence solving the linear system of equations using a krylov optimization method will result in a good convergence rate furthermore since the matrix vector multiplication can be done using a fast fourier transform fft technique its operation is of the order of o n log n where n is the size of the discretization grid this technique also allows us to use matrix free implementation hence the memory usage is about o n numerical tests show that the computational time and memory usage of this renormalized integral equation approach can be quite competitive with the frequency domain finite difference iterative solver further the numerical examples demonstrate that it is possible to solve a problem with over 100 million unknowns using an integral equation approach

time domain modeling of electromagnetic diffusion with a frequency domain code we modeled time domain em measurements of induction currents for marine and land applications with a frequency domain code an analysis of the computational complexity of a number of numerical methods shows that frequency domain modeling followed by a fourier transform is an attractive choice if a sufficiently powerful solver is available a recently developed robust multigrid solver meets this requirement an interpolation criterion determined the automatic selection of frequencies the skin depth controlled the construction of the computational grid at each frequency tests of the method against exact solutions for some simple problems and a realistic marine example demonstrate that a limited number of frequencies suffice to provide time domain solutions after piecewise cubic hermite interpolation and a fast fourier transform

an efficient montgomery exponentiation algorithm by using signed digit recoding and folding techniques abstract the motivation for designing fast modular exponentiation algorithms comes from their applications in computer science in this paper a new csd ef montgomery binary exponentiation algorithm is proposed it is based on the montgomery algorithm using the canonical signed digit csd technique and the exponent folding ef binary exponentiation technique by using the exponent folding technique of computing the common parts in the folded substrings the same common part in the folding substrings can be simply computed once we can thus improve the efficiency of the binary exponentiation algorithm by decreasing the number of modular multiplications moreover the signed digit representation has less occurrence probability of the nonzero digit than binary number representation taking this advantage we can further effectively decrease the amount of modular multiplications and we can therefore decrease the computational complexity of modular exponentiation as compared with the ha moon s algorithm 1 261718m multiplications and the lou chang s algorithm 1 375m multiplications the proposed csd ef montgomery algorithm on average only takes 0 5m multiplications to evaluate modular exponentiation where m is the bit length of the exponent

a new taxonomy of conjugate gradient methods abstract in this paper a framework is proposed for conjugate gradient and conjugate direction algorithms based on an indefinite quadratic form by the choice of the hessian g of this form and of another arbitrary matrix k no fewer than nineteen distinct cg algorithms may be succinctly defined four of these are believed to be new the choice of certain other vectors leads either to a two term or a three term recurrence formula and it is shown that to every two term formula there corresponds a three term formula although the converse is not generally true the two matrices g and k also determine the stability characteristics of the two term methods

fdem how we make the fdm more flexible than the fem the finite difference element method fdem is a black box solver for the solution of nonlinear systems of elliptic and parabolic pdes an algorithm has been developed to generate on an unstructured fem grid difference formulas of arbitrary consistency order q from the difference of difference formulas of different consistency order an estimate of the discretization error is obtained an error equation permits the explicit following of all errors and gives the prescriptions for the selfadaptation of the method coupled domains with different pdes and different nonmatching grids that slide relative to each other can be treated and a global error estimate is computed thus we get an fdm that is in all aspects more flexible than the fem the whole code is efficiently parallelized on distributed memory parallel computers

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

vector sequence transformations methodology and applications to linear systems in this paper a methodology for the construction of various vector sequence transformations is formulated leading to a unified presentation of the subject and to new results the connections to the general interpolation problem and to projections are discussed various particular cases are examined in more details applications to the solution of systems of linear equations will end the paper and in particular their relation with lanczos method will be studied some numerical examples will be given

on the extremal values of the eccentric distance sum of trees abstract let g v g e g be a simple connected graph the eccentric distance sum of g is defined as d g v v g e g v d g v where e g v is the eccentricity of the vertex v and d g v u v g d u v is the sum of all distances from the vertex v in this paper we first characterize the extremal trees among n vertex conjugated trees trees with a perfect matching having the minimal and second minimal eccentric distance sums then we identify the trees with the minimal and second minimal eccentric distance sums among the n vertex trees with matching number m finally we characterize the extremal tree with the second minimal eccentric distance sum among the n vertex trees of a given diameter consequently we determine the trees with the third and fourth minimal eccentric distance sums among the n vertex trees which is a continuance study as the results in g h yu l h feng a ilic on the eccentric distance sum of trees and unicyclic graphs j math anal appl 375 2011 99 107

qmr smoothing for lanczos type product methods based on three term rrecurrences for the solution of large sparse non hermitian linear systems lanczos type product methods that are based on the lanczos three term recurrence are derived in a systematic way these methods either square the lanczos process or combine it with a local minimization of the residual for them a quasi minimal residual qmr smoothing is proposed that can also be implemented by short term recurrences the practical performance of these methods and the qmr smoothing is demonstrated in a number of numerical examples

ml n bicgstabt a ml n bicgstab variant with a transpose the 1980 idr method wesseling and sonneveld 1980 12 plays an important role in the history of krylov subspace methods it started the research of transpose free krylov subspace methods the ml n bicgstab method yeung 2012 is one of such methods in this paper we present a new ml n bicgstab variant that involves a transpose in its implementation comparison of this new algorithm with the existing ml n bicgstab algorithms and some other krylov subspace algorithms will be presented

on augmentation block triangular preconditioners for regularized saddle point problems two augmentation block triangular preconditioners were introduced by shen et al 2012 for the regularized saddle point problem however the spectral analysis of the preconditioner based on the augmentation of the 2 2 block was not throughly derived there in this short paper we give a detailed spectral analysis on this preconditioner a numerical example is employed to validate the efficiency of the presented theoretical results

a hybridized iterative algorithm of the bicorstab and gpbicor methods for solving non hermitian linear systems in this study we derive a new iterative algorithm including its preconditioned version which is a hybridized variant of the biconjugate a orthogonal residual stabilized bicorstab method and the generalized product type solvers based on bicor gpbicor method the proposed method which is named gpbicor m similarly to the gpbicg m method proposed by fujino 2002 can be regarded as an extension of the bicorstab2 method introduced by zhao and huang 2013 inspired by fujino s idea for improving the bicgstab2 method in the established gpbicor m method the parameters computed by the bicorstab method are chosen at successive m iteration steps and afterwards the parameters of the gpbicor method are utilized in the subsequent iteration steps therefore the proposed method can inherit the low computational cost of bicorstab and the attractive convergence of gpbicor extensive numerical convergence results on selected real and complex matrices are shown to assess the performances of the proposed gpbicor m method also against other popular non hermitian krylov subspace methods

a variant algorithm of the orthomin m method for solving linear systems abstract we propose a variant of o rthomin m for solving linear systems a x b it is mathematically equivalent to the original o rthomin m method but uses recurrence formulas that are different from those of o rthomin m they contain alternative expressions for the auxiliary vectors and the recurrence coefficients our implementation has the same computational costs as o rthomin m as a result of numerical experiments on nonsingular linear systems we have confirmed the equivalence of our proposed variant of o rthomin m with the original o rthomin m using finite precision arithmetic numerical experiments on singular linear systems show that our proposed algorithm is more accurate and less affected by rounding errors than the original o rthomin m

ml k bicgstab a bicgstab variant based on multiple lanczos starting vectors we present a variant of the popular bicgstab method for solving nonsymmetric linear systems the method which we denote by ml k bicgstab is derived from a variant of the bicg method based on a lanczos process using multiple k 1 starting left lanczos vectors compared with the original bicgstab method our new method produces a residual polynomial which is of lower degree after the same number of steps but which also requires fewer matrix vector products to generate on average requiring only 1 1 k matvecs per step empirically it also seems to be more stable and more quickly convergent the new method can be implemented as a k term recurrence and can be viewed as a bridge connecting the arnoldi based fom gmres methods and the lanczos based bicgstab methods

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

an error indicator monitor function for an r adaptive finite element method an r adaptive finite element method based on moving mesh partial differential equations pdes and an error indicator is presented the error indicator is obtained by applying a technique developed by bank and weiser to elliptic equations which result in this case from temporal discretization of the underlying physical pdes on moving meshes the construction of the monitor function based on the error indicator is discussed numerical results obtained with the current method and the commonly used method based on solution gradients are presented and analyzed for several examples c 2001 academic press

on the maximal connective eccentricity index of bipartite graphs with some given parameters abstract the connective eccentricity index is a novel graph invariant with vast potential in structure activity property relationships this graph invariant displays high discriminating power with respect to both biological activity and physical properties given a simple connected graph g the connective eccentricity index cei of g is defined as e e g u v e g 1 e g u 1 e g v where e g denotes the eccentricity of the corresponding vertex in this paper we first determine the sharp upper bound on the cei of graphs in the class of all n vertex connected bipartite graphs with matching number q the maximum cei is realized only by the graph k q n q second we characterize the graph with the maximum cei in the class of all the n vertex connected bipartite graphs of given diameter finally all the extremal graphs having the maximum cei in the class of all the connected n vertex bipartite graphs with a given connectivity s are identified as well

exploiting bicgstab ell strategies to induce dimension reduction idr s p sonneveld and m b van gijzen siam j sci comput 31 2008 pp 1035 1062 and bicgstab ell g l g sleijpen and d r fokkema electron trans numer anal 1 1993 pp 11 32 are two of the most efficient short recurrence iterative methods for solving large nonsymmetric linear systems of equations which of the two is best depends on the specific problem class in this paper we describe idrstab a new method that combines the strengths of idr s and bicgstab ell to derive idrstab we extend the results that we reported on in g l g sleijpen p sonneveld and m b van gijzen appl numer math 2009 doi 10 1016 j apnum 2009 07 001 where we considered bi cgstab as an induced dimension reduction idr method we will analyze the relation between hybrid bi cg methods and idr and introduce the new concept of the sonneveld subspace as a common framework through numerical experiments we will show that idrstab can outperform both idr s and bicgstab ell

on the eccentric distance sum of graphs the eccentric distance sum is a novel topological index that offers a vast potential for structure activity property relationships for a graph g it is defined as d g v ve v d v where e v is the eccentricity of the vertex v and d v u v g d u v is the sum of all distances from the vertex v motivated by g yu l feng a ilic on the eccentric distance sum of trees and unicyclic graphs j math anal appl 375 2011 934 944 in this paper we characterize the extremal trees and graphs with maximal eccentric distance sum various lower and upper bounds for the eccentric distance sum in terms of other graph invariants including the wiener index the degree distance eccentric connectivity index independence number connectivity matching number chromatic number and clique number are established in addition we present explicit formulae for the values of eccentric distance sum for the cartesian product applied to some graphs of chemical interest like nanotubes and nanotori

connective steiner 3 eccentricity index and network similarity measure abstract for a set s v g in a network g the steiner distance dg s of s is the minimum size among all connected subnetworks whose vertex sets contain s the steiner k eccentricity k v of a vertex v of g is the maximum steiner distance among all k vertex set s which contains the vertex v i e e k v max d s s v g s k v s based on steiner k eccentricity the connective steiner k eccentricity index is introduced as a newly structural invariant some properties of the connective steiner 3 eccentricity index are investigated firstly we present an o n2 polynomial time algorithm to calculate the connective steiner 3 eccentricity index of trees secondly some optimal problems among some network classes are discussed as its application finally we consider the network similarity measure based on the connective steiner 3 eccentricity index by two different methods we study its advantages numerical results show that the measure based on the connective steiner 3 eccentricity index has more advantages than the ones based on other topological indices graph energy randic index the largest adjacent eigenvalue the largest laplacian eigenvalue

application of the multiplicative regularized gauss newton algorithm for three dimensional microwave imaging we apply the so called multiplicative regularized gauss newton inversion algorithm for solving three dimensional electromagnetic microwave inverse problems this inversion algorithm automatically adjusts the regularization parameter and when combined with the total variation type regularization function it can provide inversion results with excellent edge preserving characteristics in addition in order to deal with an extensive memory requirement for the gauss newton method we employ an implicit jacobian calculation scheme by using this scheme we do not have to explicitly store the jacobian matrix hence we are able to significantly reduce the memory requirement of the gauss newton method albeit at an additional computational overhead furthermore in order to be able to handle a large scale computational problem both the forward and the inversion algorithms are parallelized using the mpi library where we obtain a nearly linear speedup factor we demonstrate efficiency and robustness of this algorithm by inverting synthetic data fresnel experimental data and biomedical experimental data

metric tensors for anisotropic mesh generation it has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features in practice an anisotropic mesh is commonly generated as a quasi uniform mesh in the metric determined by a tensor specifying the shape size orientation of elements thus it is crucial to choose an appropriate metric tensor for anisotropic mesh generation and adaptation in this paper we develop a general formula for the metric tensor for use in any spatial dimension the formulation is based on error estimates for polynomial preserving interpolation on simiplicial elements numerical results in two dimensions are presented to demonstrate the ability of the metric tensor to produce anisotropic meshes with correct mesh concentration and good overall quality the procedure developed in this paper for defining the metric tensor can also be applied to other types of error estimates

low complexity bit parallel multiplier for gf 2 sup m defined by all one polynomials using redundant representation this paper presents a new bit parallel multiplier for the finite field gf 2 sup m defined by an irreducible all one polynomial in order to reduce the complexity of the multiplier we introduce a redundant representation and use the well known multiplication method proposed by karatsuba the main idea is to combine the redundant representation and the karatsuba method to design an efficient bit parallel multiplier as a result the proposed multiplier requires about 25 percent fewer and xor gates than the previously proposed multipliers using an all one polynomial while it has almost the same time delay as the previously proposed ones

eccentric distance sum a novel graph invariant for predicting biological and physical properties abstract eccentric distance sum a novel graph invariant with vast potential in structure activity property relationships has been conceptualized in the present study this graph invariant displayed high discriminating power with respect to both biological activity and physical properties the structure activity relationship of eccentric distance sum was investigated with regard to anti hiv activity of dihydroseselins the values of eccentric distance sum of each analogue in the data set were computed and active range identified subsequently biological activity was assigned to each analogue in the data set which was then compared with the reported anti hiv activity of dihydroseselin analogues surprisingly the accuracy of prediction was found to be more than 88 with regard to anti hiv activity on the other hand investigations pertaining to quantitative structure property relationship of the novel graph invariant with regard to various physical properties of diverse nature for data sets consisting of primary amines secondary amines and alcohols revealed correlation percentages ranging from 93 to 99 the over all results with regard to structure activity and quantitative structure property studies using eccentric distance sum were better than the corresponding values obtained using wiener s index

vlsi architectures for computing multiplications and inverses in gf 2 m finite field arithmetic logic is central in the implementation of reed solomon coders and in some cryptographic algorithms there is a need for good multiplication and inversion algorithms that can be easily realized on vlsi chips massey and omura 1 recently developed a new multiplication algorithm for galois fields based on a normal basis representation in this paper a pipeline structure is developed to realize the massey omura multiplier in the finite field gf 2m with the simple squaring property of the normal basis representation used together with this multiplier a pipeline architecture is also developed for computing inverse elements in gf 2m the designs developed for the massey omura multiplier and the computation of inverse elements are regular simple expandable and therefore naturally suitable for vlsi implementation

general transmission lemma and wiener complexity of triangular grids abstract the transmission lemma from rajasingh et al 2016 is extended to the general transmission lemma it gives a formula for the transmission of a vertex u as a function of a collection of edge cuts and a u routing that uniformly intersects the edge cuts the applicability of the general transmission lemma is demonstrated by computing the wiener complexity of triangular grid networks

on iterative approaches for electromagnetic rough surface scattering problems iterative techniques developed for solving general systems of linear equations have been applied to systems resulting from electromagnetic rough surface scattering problems previously used iterative procedures that model the multiple scattering of the electromagnetic energy are shown to be mathematically equivalent to the application of stationary iterative procedures to the system of equations resulting from the standard moment method convergence difficulties that are sometimes observed with these approaches are due to the inherent limitations of the stationary techniques the performances of the stationary approaches are compared with that of several conjugate direction based nonstationary iterative procedures through the application to a series of scattering surfaces that yield rapidly changing conditioning of the moment method interaction matrix the stationary algorithms give the quickest convergence when applied to the systems with the best conditioning but the nonstationary techniques prove much more robust in other more ill conditioned situations

fast inversion in schmi gf 2 m with normal basis using hybrid double multipliers fast inversion in finite fields is crucial for high performance cryptography and codes we present techniques to exploit the recently proposed hybrid double multipliers for fast inversions in binary fields gf 2m with normal bases a hybrid double multiplier computes a double multiplication the product of three elements in gf 2m with a latency comparable to the latency of single multiplication of two elements traditional approaches such as itoh tsujii cannot utilize hybrid double multipliers we devise a new inversion algorithm based on ternary representations that exploits their potential the algorithm reduces the latency of inversion significantly for the fields recommended by nist if hybrid double multipliers are employed for example the algorithm computes an inversion in gf 2163 with only five double multiplications whereas the itoh tsujii algorithm requires nine single or double multiplications we propose a new inverter architecture using this new algorithm and a hybrid double multiplier we show that it is faster than the existing techniques by providing asic synthesis results using 65 nm cmos technology for example our inverter for gf 2163 achieves about 34 percent shorter computation time than an inverter using the itoh tsujii algorithm and a single multiplier

vpastab j l an iterative method with look ahead for the solution of large sparse linear systems vpastab j l denotes an iterative method designed to give an accurate numerical solution of a large sparse system of linear equations the iteration comprises l steps and possibly jump steps an l step involves stabilisation minimisation over an l dimensional subspace following l levels of vector pade approximation a jump step look ahead stage avoids near breakdowns by jumping past them by j levels simultaneously a numerical implementation of vpastab 2 l is considered in the context of some well known examples

some extremal results on the connective eccentricity index of graphs abstract the connective eccentricity index cei of a graph g is defined as ce g v i v g d v i e v i where e v i and d v i are the eccentricity and the degree of vertex v i respectively in g in this paper we obtain some lower and upper bounds on the connective eccentricity index for all trees of order n and with matching number and characterize the corresponding extremal trees and the maximal graphs of order n and with matching number and n edges have been determined which maximize the connective eccentricity index also the extremal graphs with maximal connective eccentricity index are completely characterized among all connected graphs of order n and with matching number moreover we establish some relations between connective eccentricity index and eccentric connectivity index as another eccentricity based invariant of graphs

towards monte carlo preconditioning approach and hybrid monte carlo algorithms for matrix computations an enhanced version of a stochastic sparse approximate inverse spai preconditioner for general matrices is presented in this paper this method is used in contrast to the standard deterministic preconditioners computed by the modified sparse approximate inverse preconditioner mspai thus we present a monte carlo preconditioner that relies on the use of markov chain monte carlo mcmc methods to compute a rough approximate matrix inverse first which can further be optimized by an iterative filter process and a parallel refinement to enhance the accuracy of the inverse and the preconditioner respectively the advantage of the proposed approach is that finding the sparse monte carlo matrix inversion has a computational complexity linear of the size of the matrix it is inherently parallel and thus can be obtained very efficiently for large matrices and can be used also as an efficient preconditioner while solving systems of linear algebraic equations the behaviour of the proposed algorithm is studied and its performance measured evaluated and compared with mspai

iterative solution of 3d acoustic wave equation with perfectly matched layer boundary condition and multigrid preconditioner abstractwe tested a biconjugate gradient stabilized bicgstab solver using a multigrid based preconditioner for solving the acoustic wave helmholtz equation in the frequency domain the perfectly matched layer pml method was used as the radiation boundary condition rbc the equation was discretized using either a second or fourth order finite difference fd scheme the convergence of an iterative solver depended strongly on the rbc used because the spectrum of the discretized equation also depends on it we used a geometric multigrid approach to construct a preconditioner for our fd frequency domain fdfd forward solver equipped with the pml boundary condition for efficiency this preconditioner was only constructed using a second order fd scheme with negligible attenuation inside the pml domain the preconditioner was used for accelerating the convergence rate of the fdfd forward solver for cases when the discretization grids were oversampled i e when the number of discretization points per

application of graph theory relationship of eccentric connectivity index and wiener s index with anti inflammatory activity abstract graph theory was successfully applied in developing a relationship between chemical structure and biological activity the relationship of two graph invariants the eccentric connectivity index and the wiener s index was investigated with regard to anti inflammatory activity for a data set consisting of 76 pyrazole carboxylic acid hydrazide analogues the values of the eccentric connectivity index and the wiener s index of each analogue in the data set were computed and active ranges were identified subsequently each analogue was assigned a biological activity that was compared with the anti inflammatory activity reported as percent reduction in paw swelling prediction with an accuracy of 90 was obtained using the eccentric connectivity index as compared to 84 in the case of wiener s index

an efficient common multiplicand multiplication method to the montgomery algorithm for speeding up exponentiation the modular exponentiation is a common operation for scrambling secret data and is used by several public key cryptosystems such as the rsa scheme and dss digital signature scheme however the calculations involved in modular exponentiation are time consuming especially when performed in software in this paper we propose an efficient cmm msd montgomery algorithm by utilizing the montgomery modular reduction method common multiplicand multiplication cmm method and minimal signed digit msd recoding technique for fast modular exponentiation by using the technique of recording the common signed digit representations in the grouped substrings of exponent our algorithm can improve the efficiency of both the original cmm exponentiation algorithm and the montgomery multiplication algorithm the fast modular exponentiation algorithm developed in this paper can be easily implemented in general signed digit computing machine and is therefore well suited for parallel implementation to fast evaluating modular exponentiation moreover by using the proposed cmm msd montgomery algorithm on average the total number of single precision multiplications can be reduced by about 38 9 and 26 68 as compared with dusse kaliski s montgomery algorithm and ha moon s montgomery algorithm respectively

a quasi minimal residual variant of the bi cgstab algorithm for nonsymmetric systems motivated by a recent method of freund siam j sci comput 14 1993 pp 470 482 who introduced a quasi minimal residual qmr version of the conjugate gradients squared cgs algorithm a qmr variant of the biconjugate gradient stabilized bi cgstab algorithm of van der vorst that is called qmrcgstab is proposed for solving nonsymmetric linear systems the motivation for both qmr variants is to obtain smoother convergence behavior of the underlying method the authors illustrate this by numerical experiments that also show that for problems on which bi cgstab performs better than cgs the same advantage carries over to qmrcgstab

a short and unified proof of yu et al ʼs two results on the eccentric distance sum abstract the eccentric distance sum eds is a novel topological index that offers a vast potential for structure activity property relationships for a connected graph g the eccentric distance sum is defined as d g v v g ec g v d g v where ec g v is the eccentricity of a vertex v in g and d g v is the sum of distances of all vertices in g from v more recently yu et al g yu l feng a ilic on the eccentric distance sum of trees and unicyclic graphs j math anal appl 375 2011 99 107 proved that for an n vertex tree t d t 4 n 2 9 n 5 with equality holding if and only if t is the n vertex star s n and for an n vertex unicyclic graph g d g 4 n 2 9 n 1 with equality holding if and only if g is the graph obtained by adding an edge between two pendent vertices of n vertex star in this note we give a short and unified proof of the above two results

a fast algorithm for computing multiplicative inverses in gf 2 m using normal bases abstract this paper proposes a fast algorithm for computing multiplicative inverses in gf 2 m using normal bases normal bases have the following useful property in the case that an element x in gf 2 m is represented by normal bases 2 k power operation of an element x in gf 2 m can be carried out by k times cyclic shift of its vector representation c c wang et al proposed an algorithm for computing multiplicative inverses using normal bases which requires m 2 multiplications in gf 2 m and m 1 cyclic shifts the fast algorithm proposed in this paper also uses normal bases and computes multiplicative inverses iterating multiplications in gf 2 m it requires at most 2 log 2 m 1 multiplications in gf 2 m and m 1 cyclic shifts which are much less than those required in the wang s method the same idea of the proposed fast algorithm is applicable to the general power operation in gf 2 m and the computation of multiplicative inverses in gf q m q 2 n

on the connective eccentricity index of trees and unicyclic graphs with given diameter abstract the connective eccentricity index cei of a graph g is defined as ce g v v g d v e v where e v d v denote the eccentricity and the degree of v respectively in this paper we investigate the connective eccentricity index of trees and unicyclic graphs on n vertices with diameter d we also characterize the extremal graphs

a generalization of addition chains and fast inversions in binary fields in this paper we study a generalization of addition chains where k previous values are summed together on each step instead of only two values as in traditional addition chains such chains are called k chains and we show that they have applications in finding efficient parallelizations in problems that are known to be difficult to parallelize in particular 3 chains improve computations of inversions in finite fields using hybrid double multipliers recently it was shown that this operation can be efficiently computed using a ternary algorithm but we show that 3 chains provide a significantly more efficient solution

on the convergence behavior of idr s and related methods an explanation is given of the convergence behavior of idr s methods the convergence mechanism of these algorithms has two components the first consists of damping properties of certain factors in the residual polynomials which becomes less important for large values of s the second component depends on the behavior of lanczos polynomials that occur in the theoretical description this part of the residual polynomials is related to lanczos methods with s left starting vectors as described in a paper by yeung and chan on their ml k bicgstab method in siam j sci comput 21 1999 pp 1263 1290 in this paper the behavior of the second component is compared with the full gmres method siam j sci statist comput 7 1986 pp 856 869 and an expected rate of convergence is given based on a random choice of the s shadow vectors

lanczos type variants of the cocr method for complex nonsymmetric linear systems motivated by the celebrated extending applications of the well established complex biconjugate gradient cbicg method to deal with large three dimensional electromagnetic scattering problems by pocock and walker m d pocock s p walker the complex bi conjugate gradient solver applied to large electromagnetic scattering problems computational costs and cost scalings ieee trans antennas propagat 45 1997 140 146 three lanczos type variants of the recent conjugate a orthogonal conjugate residual cocr method of sogabe and zhang t sogabe s l zhang a cocr method for solving complex symmetric linear systems j comput appl math 199 2007 297 303 are explored for the solution of complex nonsymmetric linear systems the first two can be respectively considered as mathematically equivalent but numerically improved popularizing versions of the bicr and crs methods for complex systems presented in sogabe s ph d dissertation and the last one is somewhat new and is a stabilized and more smoothly converging variant of the first two in some circumstances the presented algorithms are with the hope of obtaining smoother and hopefully faster convergence behavior in comparison with the cbicg method as well as its two corresponding variants this motivation is demonstrated by numerical experiments performed on some selective matrices borrowed from the university of florida sparse matrix collection by davis

on the extremal total reciprocal edge eccentricity of trees abstract the total reciprocal edge eccentricity is a novel graph invariant with vast potential in structure activity property relationships this graph invariant displays high discriminating power with respect to both biological activity and physical properties if g v g e g is a simple connected graph then the total reciprocal edge eccentricity ree of g is defined as e e g u v e g 1 e g u 1 e g v where e g v is the eccentricity of the vertex v in this paper we first introduced four edge grafting transformations to study the mathematical properties of the reciprocal edge eccentricity of g using these elegant mathematical properties we characterize the extremal graphs among n vertex trees with given graphic parameters such as pendants matching number domination number diameter vertex bipartition et al some sharp bounds on the reciprocal edge eccentricity of trees are determined

a fast algorithm for multiplicative inversion in gf 2 sup m using normal basis a fast algorithm for multiplicative inversion in gf 2 sup m using normal basis is proposed it is an improvement on those proposed by itoh and tsujii and by chang et al which are based on fermat s theorem and require o logm multiplications the number of multiplications is reduced by decomposing m 1 into several factors and a small remainder

the eccentric connectivity index of nanotubes and nanotori let g be a molecular graph the eccentric connectivity index x c g is defined as x c g email protected u v g deg g u e g u where deg g u denotes the degree of vertex u and e g u is the largest distance between u and any other vertex v of g in this paper exact formulas for the eccentric connectivity index of tuc 4c 8 s nanotube and tc 4c 8 s nanotorus are given

stationary nonstationary and hybrid iterative method of moments solution schemes the purpose of this paper is to report on the convergence rates of two iterative matrix solution methods individually and then to combine the two methods into a hybrid scheme to achieve additional convergence rate benefits one iterative matrix solver investigated is the sparse iterative method sim which is a stationary jacobi like solver but uses a sparse and not a banded matrix with matrix elements corresponding to strong interactions rather than position in the matrix in this paper the sim is modified to include an adaptive relaxation scheme to improve its convergence speed and numerical stability another iterative scheme investigated is the nonstationary biconjugate gradient stabilized bicgstab method it is shown that the bicgstab is considerably improved when the method is preconditioned by the sparse matrix used in the sim method finally a hybrid scheme is proposed which combines both sim ar and bicgstab precon and it is shown that the hybrid gives best results on the problems considered examples giving convergence time versus accuracy are presented for two problems a wire grid plate and a wire grid partial helicopter

weighted average geodesic distance of vicsek network abstract this paper concerns weighted average distances of vicsek fractal and vicsek networks by using the integral of geodesic distance in terms of self similar measure with respect to the weight vector