a new meta heuristic algorithm for continuous engineering optimization harmony search theory and practice most engineering optimization algorithms are based on numerical linear and nonlinear programming methods that require substantial gradient information and usually seek to improve the solution in the neighborhood of a starting point these algorithms however reveal a limited approach to complicated real world optimization problems if there is more than one local optimum in the problem the result may depend on the selection of an initial point and the obtained optimal solution may not necessarily be the global optimum this paper describes a new harmony search hs meta heuristic algorithm based approach for engineering optimization problems with continuous design variables this recently developed hs algorithm is conceptualized using the musical process of searching for a perfect state of harmony it uses a stochastic random search instead of a gradient search so that derivative information is unnecessary various engineering optimization problems including mathematical function minimization and structural engineering optimization problems are presented to demonstrate the effectiveness and robustness of the hs algorithm the results indicate that the proposed approach is a powerful search and optimization technique that may yield better solutions to engineering problems than those obtained using current algorithms 2004 elsevier b v all rights reserved

logic based fuzzy networks a study in system modeling with triangular norms and uninorms the ultimate challenges of system modeling concern designing accurate yet highly transparent and user centric models we have witnessed a plethora of neurofuzzy architectures which are aimed at addressing these two highly conflicting requirements this study is concerned with the design and the development of transparent logic networks realized with the aid of fuzzy neurons and fuzzy unineurons the construction of networks of this form requires a formation of efficient interfaces that constitute a conceptually appealing bridge between the model and the real world experimental environment in which the model is to be used in general the interfaces are constructed by invoking some form of granulation of information and binary boolean discretization in particular we introduce a new discretization environment that is realized by means of particle swarm optimization pso and data clustering implemented by the k means algorithm the underlying structure of the network is optimized by invoking a combination of the pso and the mechanisms of conventional gradient based learning we discuss various optimization strategies by considering boolean as well as fuzzy data coming as the result of discretization of original experimental data and then involving several learning strategies we elaborate on the interpretation aspects of the network and show how those could be strengthened through efficient pruning we also show how the interpreted network leads to a simpler and more accurate logic description of the experimental data a number of experimental studies are included

algorithm and architecture for a galois field multiplicative arithmetic processor we present a new algorithm for generic multiplicative computations of the form ab c in gf p sup m including multiplication inversion squaring and division the algorithm is based on solving a sequence of congruences that are derived from the theory of grobner bases in modules over the polynomial ring gf p x its corresponding hardware and software architectures can be successfully used in applications such as error control coding and cryptography we describe a versatile circuit associated with the algorithm for the most important case p 2 the same hardware can be used for a range of field sizes thus permitting applications in which different levels of error control or of security are required by different classes of user the operations listed are all performed by the hardware in the same number of clock cycles which prevents certain side channel attacks the loss in performance by having 2m iterations for multiplication is compensated by the full parameterization of the galois field and the ability to perform division and multiplication in parallel

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

at sup 2 optimal galois field multiplier for vlsi vlsi designs for galois field multipliers which are central in many encoding and decoding procedures for error detecting and error correcting codes are presented an at sup 2 optimal galois field multiplier based on at sup 2 optimal integer multipliers for a synchronous vlsi model is exhibited galois field multiplication is done in two steps first two polynomials of degree n 1 over z sub p are multiplied and then the resulting polynomial is reduced modulo a fixed irreducible polynomial of degree n multiplication of polynomials is done by discrete fourier transform dft for p 2 the procedure is more involved for z sub p x than for z x an extension to the case of variable p is included and some open problems are stated

systolic gaussian elimination over gf p with partial pivoting a systolic architecture is proposed for the triangularization by means of the gaussian elimination algorithm of large dense n n matrices over gf p where p is a prime number the solution of large dense linear systems over gf p is the major computational step in various algorithms issuing from arithmetic number theory and computer algebra the proposed architecture implements the elimination with partial pivoting although the operation of the array remains purely systolic extension of the array to the complete solution of a linear system ax b over gf p is also considered

a search of minimal key functions for normal basis multipliers the circuit complexity of a massey omura normal basis multiplier for a finite field gf 2 sup m depends on the key function for multiplication key functions with minimum complexity called minimal key functions are desirable this paper investigates the complexity of a key function and reports search results of minimal key functions a table of minimal key functions for m up to 31 is included

k means type algorithms a generalized convergence theorem and characterization of local optimality the k means algorithm is a commonly used technique in cluster analysis in this paper several questions about the algorithm are addressed the clustering problem is first cast as a nonconvex mathematical program then a rigorous proof of the finite convergence of the k means type algorithm is given for any metric it is shown that under certain conditions the algorithm may fail to converge to a local minimum and that it converges under differentiability conditions to a kuhn tucker point finally a method for obtaining a local minimum solution is given

an efficient constraint handling method for genetic algorithms many real world search and optimization problems involve inequality and or equality constraints and are thus posed as constrained optimization problems in trying to solve constrained optimization problems using genetic algorithms gas or classical optimization methods penalty function methods have been the most popular approach because of their simplicity and ease of implementation however since the penalty function approach is generic and applicable to any type of constraint linear or nonlinear their performance is not always satisfactory thus researchers have developed sophisticated penalty functions specific to the problem at hand and the search algorithm used for optimization however the most difficult aspect of the penalty function approach is to find appropriate penalty parameters needed to guide the search towards the constrained optimum in this paper ga s population based approach and ability to make pair wise comparison in tournament selection operator are exploited to devise a penalty function approach that does not require any penalty parameter careful comparisons among feasible and infeasible solutions are made so as to provide a search direction towards the feasible region once sufficient feasible solutions are found a niching method along with a controlled mutation operator is used to maintain diversity among feasible solutions this allows a real parameter ga s crossover operator to continuously find better feasible solutions gradually leading the search near the true optimum solution gas with this constraint handling approach have been tested on nine problems commonly used in the literature including an engineering design problem in all cases the proposed approach has been able to repeatedly find solutions closer to the true optimum solution than that reported earlier

an improved adaptive binary harmony search algorithm harmony search hs inspired by the music improvisation process is a new meta heuristic optimization method and has been successfully used to tackle the optimization problems in discrete or continuous space although the standard hs algorithm is able to solve binary coded optimization problems the pitch adjustment operator of hs is degenerated in the binary space which spoils the performance of the algorithm based on the analysis of the drawback of the standard hs an improved adaptive binary harmony search abhs algorithm is proposed in this paper to solve the binary coded problems more effectively various adaptive mechanisms are examined and investigated and a scalable adaptive strategy is developed for abhs to enhance its search ability and robustness the experimental results on the benchmark functions and 0 1 knapsack problems demonstrate that the proposed abhs is efficient and effective which outperforms the binary harmony search the novel global harmony search algorithm and the discrete binary particle swarm optimization in terms of the search accuracy and convergence speed

a new representation of elements of finite fields gf 2 sup m yielding small complexity arithmetic circuits let f sub 2 denote the binary field and f sub 2m an algebraic extension of degree m 1 over f sub 2 traditionally elements of f sub 2m are either represented as powers of a primitive element of f sub 2m together with 0 or by an expansion in a basis of the vector space f sub 2m over f sub 2 we propose a new representation based on an isomorphism from f sub 2m into the residue polynomial ring module x sup n 1 the new representation simultaneously satisfies the properties of various traditional representations which leads in some cases to architectures of parallel in parallel out arithmetic circuits adder multiplier exponentiator inverter squarer divider with average to small complexity we show that the implementation of all the arithmetic circuits designed for the new representation on an integrated circuit sometimes has smaller complexity than the implementation of all the arithmetic circuits designed for other representations in addition we derive a serial multiplier for the field f sub 2m which comprises the least number of gates of all the serial multipliers known to the author when m 1 is a prime such that 2 is primitive in the field z sub m 1

automatic clustering using genetic algorithms abstract in face of the clustering problem many clustering methods usually require the designer to provide the number of clusters as input unfortunately the designer has no idea in general about this information beforehand in this article we develop a genetic algorithm based clustering method called automatic genetic clustering for unknown k agcuk in the agcuk algorithm noising selection and division absorption mutation are designed to keep a balance between selection pressure and population diversity in addition the davies bouldin index is employed to measure the validity of clusters experimental results on artificial and real life data sets are given to illustrate the effectiveness of the agcuk algorithm in automatically evolving the number of clusters and providing the clustering partition

supervised hierarchical clustering in fuzzy model identification this paper presents a new supervised hierarchical clustering algorithm suhiclust for fuzzy model identification the presented algorithm solves the problem of global model accuracy together with the interpretability of local models as valid linearizations of the modeled nonlinear system the algorithm combines the advantages of supervised hierarchical algorithms which are based on heuristic tree construction algorithms together with the advantages of fuzzy product space clustering the high flexibility of the validity functions that is obtained by fuzzy clustering combined with supervised learning results in an efficient partitioning algorithm which is independent of initialization and results in a parsimonious fuzzy model furthermore the usability of suhiclust is very undemanding because it delivers in contrast with many other methods reproducible results in order to get reasonable results the user only has to set either a threshold for the maximum number of local models or a value for the maximum allowed global model error as a termination criterion for fine tuning the interpolation smoothness controls the degree of regularization the performance is illustrated on both analytical examples and benchmark problems from the literature

a systolic power sum circuit for gf 2 sup m a systolic power sum circuit designed to perform ab sup 2 c computations in the finite field gf 2 sup m is presented where a b and c are arbitrary elements of gf 2 sup m this new circuit is constructed by m sup 2 identical cells each of which consists of three 2 input and logical gates one 2 input xor gate one 3 input xor gate and ten latches the ab sup 2 c computation is required in decoding many error correcting codes the paper shows that a decoder implemented using the new power sum circuit will have less complex circuitry and shorter decoding delay than one implemented using conventional product sum multipliers

efficient multiplication beyond optimal normal bases in cryptographic applications the use of normal bases to represent elements of the finite field gf 2 sup m is quite advantageous especially for hardware implementation in this article we consider an important field operation namely multiplication which is used in many cryptographic functions we present a class of algorithms for normal basis multiplication in gf 2 sup m our proposed multiplication algorithm for composite finite fields requires a significantly lower number of bit level operations and hence can reduce the space complexity of cryptographic systems

hierarchical ts fuzzy system and its universal approximation an efficient tool to deal with the rule explosion problem is the hierarchical system by which a fuzzy system can be decomposed into a number of hierarchically connected low dimensional systems in this paper a generalized hierarehical tagaki sugeno ts system is built it is shown that the input output i o relationship of this generalized hierarehical system can be represented as one of a standard ts fuzzy system and the system approximation capability is analyzed by taking piecewise linear functions as a bridge by constructive method it is proven that the hierarchical fuzzy systems hfs s can be universal approximators for the given approximation accuracy an estimation formula about the number of the rules needed in the hfs is established finally some simulation examples confirm that the hfs s with smaller size rule base can approximate the given functions with high accuracy the results obtained here provide us with the theoretical basis for various applications of hfs s

hierarchical fuzzy cmac for nonlinear systems modeling since the fuzzy cerebellar model articulation controller fcmac uses linguistic variables it is highly intuitive and easily comprehended despite the fcmac s good local generalization capability for approximating nonlinear functions and fast learning a normal fcmac requires huge memory and its dimension increases exponentially with the number of inputs in order to overcome the memory explosion problem this paper proposes two types of hierarchical fcmac hfcmac another contribution of the paper is that we give stable learning algorithms for these two hfcmacs backpropagation like approach is applied to train each block with a time varying learning rate which is obtained by the input to state stability technique

high speed architectures for multiplication using reordered normal basis normal basis has been widely used for the representation of binary field elements mainly due to its low cost squaring operation optimal normal basis type ii is a special class of normal basis exhibiting very low multiplication complexity and is considered as a safe choice for hardware implementation of cryptographic applications in this paper high speed architectures for binary field multiplication using reordered normal basis are proposed where reordered normal basis is referred to as a certain permutation of optimal normal basis type ii complexity comparison shows that the proposed architectures are faster compared to previously presented architectures in the open literature using either an optimal normal basis type ii or a reordered normal basis one advantage of the new word level architectures is that the critical path delay is a constant not a function of word size this enables the multipliers to operate at very high clock rates regardless of the field size or the number of words hardware implementation of some practical size multipliers for elliptic curve cryptography is also included

double basis multiplicative inversion over gf 2 sup m inversion over galois fields is much more difficult than the corresponding multiplication efficient computation of inverses in gf 2 sup m is considered by solving a set of linear equations over the ground field gf 2 the proposed algorithm uses two separate bases for the representation of its input and output elements and has low computational complexity the algorithm is also suitable for hardware implementation using vlsi technologies

a comparison of vlsi architecture of finite field multipliers using dual normal or standard bases three different finite field multipliers are presented 1 a dual basis multiplier due to e r berlekamp 1982 the massey omura normal basis multiplier and 3 the scott tavares peppard standard basis multiplier these algorithms are chosen because each has its own distinct features that apply most suitably in particular areas they are implemented on silicon chips with nmos technology so that the multiplier most desirable for vlsi implementation can readily be ascertained

systolic multipliers for finite fields gf 2 m two systolic architectures are developed for performing the product sum computation ab c in the finite field gf 2m of 2melements where a b and c are arbitrary elements of gf 2m the first multiplier is a serial in serial out one dimensional systolic array while the second multiplier is a parallel in parallel out two dimensional systolic array the first multiplier requires a smaller number of basic cells than the second multiplier the second multiplier heeds less average time per computation than the first multiplier if a number of computations are performed consecutively to perform single computations both multipliers require the same computational time in both cases the architectures are simple and regular and possess the properties of concurrency and modularity as a consequence they are well suited for use in vlsi systems

analysis and design of hierarchical fuzzy systems in this letter the hierarchical fuzzy systems are analyzed and designed in the analysis part we prove that the hierarchical fuzzy systems are universal approximators and analyze the sensitivity of the fuzzy system output with respect to small perturbations in its inputs in the design part we derive a gradient descent algorithm for tuning the parameters of the hierarchical fuzzy system to match the input output pairs the algorithm is simulated for two examples and the results show that the algorithm is effective and the hierarchical structure gives good approximation accuracy

an improved harmony search algorithm for solving optimization problems this paper develops an improved harmony search ihs algorithm for solving optimization problems ihs employs a novel method for generating new solution vectors that enhances accuracy and convergence rate of harmony search hs algorithm in this paper the impacts of constant parameters on harmony search algorithm are discussed and a strategy for tuning these parameters is presented the ihs algorithm has been successfully applied to various benchmarking and standard engineering optimization problems numerical results reveal that the proposed algorithm can find better solutions when compared to hs and other heuristic or deterministic methods and is a powerful search algorithm for various engineering optimization problems

bit serial systolic divider and multiplier for finite fields gf 2 sup m a systolic structure for bit serial division over the field gf 2 sup m is developed consideration is given to avoid global data communications and dependency of the time step duration on m this is important for applications where the value of m is large the divider requires only three basic processors and one simple control signal and its circuit and time complexities are proportional to m sup 2 and m respectively it does not depend on the irreducible polynomial and can be expanded easily moreover with m additional simple processors a bit serial systolic multiplier is developed which uses part of the divider structure this is advantageous from the implementation point of view as both the divider and multiplier can be fabricated on a single chip resulting in a reduction of area

a new algorithm for multiplication in finite fields a new algorithm is presented for computing the product of two elements in a finite field by means of sums and products in a fixed subfield the algorithm is based on a normal basis representation of fields and assumes that the dimension m of the finite field over the subfield is a highly composite number a very fast parallel implementation and a considerable reduction in the number of computations are allowed in comparison to some methods discussed in the literature

a word level finite field multiplier using normal basis hardware implementations of finite field arithmetic using normal basis are advantageous due to the fact that the squaring operation can be done at almost no cost in this paper a new word level finite field multiplier using normal basis is proposed the proposed architecture takes d clock cycles to compute the product bits where the value for d 1 d m can be arbitrarily selected by the designer to set the tradeoff between area and speed when there exists an optimal normal basis it is shown that the proposed design has a smaller critical path delay than other word level normal basis multipliers found in the literature while its circuit complexities are moderate and comparable to the others different word size multipliers were implemented in hardware and implementation results are also presented

novel meta heuristic algorithms for clustering web documents clustering the web documents is one of the most important approaches for mining and extracting knowledge from the web recently one of the most attractive trends in clustering the high dimensional web pages has been tilt toward the learning and optimization approaches in this paper we propose novel hybrid harmony search hs based algorithms for clustering the web documents that finds a globally optimal partition of them into a specified number of clusters by modeling clustering as an optimization problem first we propose a pure harmony search based clustering algorithm that finds near global optimal clusters within a reasonable time then we hybridize k means and harmony clustering in two ways to achieve better clustering experimental results reveal that the proposed algorithms can find better clusters when compared to similar methods and also illustrate the robustness of the hybrid clustering algorithms

approximation capabilities of hierarchical fuzzy systems derived from practical application in location analysis and pricing and based on the approach of hierarchical structure analysis of continuous functions this paper investigates the approximation capabilities of hierarchical fuzzy systems by first introducing the concept of the natural hierarchical structure it is proved that continuous functions with natural hierarchical structure can be naturally and effectively approximated by hierarchical fuzzy systems to overcome the curse of dimensionality in both the number of rules and parameters then based on kolmogorov s theorem it is shown that any continuous function can be represented as a superposition of functions with the natural hierarchical structure and can then be approximated by hierarchical fuzzy systems to achieve the universal approximation property further the conditions under which the hierarchical fuzzy approximation is superior to the standard fuzzy approximation in overcoming the curse of dimensionality are analyzed

intermediate variable normalization for gradient descent learning for hierarchical fuzzy system when applying gradient descent learning methods to hierarchical fuzzy systems there is great difficulty in handling the intermediate variables introduced by the hierarchical structures as the intermediate variables may go outside their definition domain that makes gradient descent learning invalid to overcome this difficulty this paper proposes a learning scheme that integrates a normalization process for intermediate variables into gradient descent learning this ensures that gradient descent methods are applicable to and correctly used for learning general hierarchical fuzzy systems benchmark datasets are used to demonstrate the validity and advantages of the proposed learning scheme over other existing methods in terms of better accuracy better transparency and fewer fuzzy rules and parameters

systolic array implementation of euclid s algorithm for inversion and division in gf 2 sup m this paper presents two new systolic arrays to realize euclid s algorithm for computing inverses and divisions in finite fields gf 2 sup m with the standard basis representation one of these two schemes is parallel in parallel out and the other is serial in serial out the former employs o m sup 2 area complexity to provide the maximum throughput in the sense of producing one result every clock cycle while the latter achieves a throughput of one result per m clock cycles using o m log m area complexity both of the proposed architectures are highly regular and thus well suited to vlsi implementation as compared to existing related systolic architectures with the same throughput performance the proposed parallel in parallel out scheme reduces the hardware complexity and thus the area time product by a factor of o m and the proposed serial in serial out scheme by a factor of o m log sub 2 m

multiobjective programming for type 2 hierarchical fuzzy inference trees this paper proposes a design of hierarchical fuzzy inference tree hfit an hfit produces an optimum tree like structure i e a natural hierarchical structure that accommodates simplicity by combining several low dimensional fuzzy inference systems fiss such a natural hierarchical structure provides a high degree of approximation accuracy the construction of the hfit takes place in two phases first a nondominated sorting based multiobjective genetic programming mogp is applied to obtain a simple tree structure a low complexity model with a high accuracy second the differential evolution algorithm is applied to optimize the obtained tree s parameters in the derived tree each node acquires a different input s combination where the evolutionary process governs the input s combination hence hfit nodes are heterogeneous in nature which leads to a high diversity among the rules generated by the hfit additionally the hfit provides an automatic feature selection because it uses mogp for the tree s structural optimization that accepts inputs only relevant to the knowledge contained in data the hfit was studied in the context of both type 1 and type 2 fiss and its performance was evaluated through six application problems moreover the proposed multiobjective hfit was compared both theoretically and empirically with recently proposed fiss methods from the literature such as mcit2fis tscit2fnn sit2fnn rit2fns wb et2fis mrit2nfs it2fnn svr etc from the obtained results it was found that the hfit provided less complex and highly accurate models compared to the models produced by the most of other methods hence the proposed hfit is an efficient and competitive alternative to the other fiss for function approximation and feature selection

montgomery multiplier and squarer for a class of finite fields montgomery multiplication in gf 2 sup m is defined by a x b x r sup 1 x mod f x where the field is generated by a root of the irreducible polynomial f x a x and b x are two field elements in gf 2 sup m and r x is a fixed field element in gf 2 sup m in this paper first a slightly generalized montgomery multiplication algorithm in gf 2 sup m is presented then by choosing r x according to f x we show that efficient architectures of bit parallel montgomery multiplier and squarer can be obtained for the fields generated with an irreducible trinomial complexities of the montgomery multiplier and squarer in terms of gate counts and time delay of the circuits are investigated and found to be as good as or better than that of previous proposals for the same class of fields

efficient algorithms and architectures for field multiplication using gaussian normal bases recently implementations of normal basis multiplication over the extended binary field gf 2 sup m have received considerable attention a class of low complexity normal bases called gaussian normal bases has been included in a number of standards such as ieee and nist for an elliptic curve digital signature algorithm the multiplication algorithms presented there are slow in software since they rely on bit wise inner product operations in this paper we present two vector level software algorithms which essentially eliminate such bit wise operations for gaussian normal bases our analysis and timing results show that the software implementation of the proposed algorithm is faster than previously reported normal basis multiplication algorithms the proposed algorithm is also more memory efficient compared with its look up table based counterpart moreover two new digit level multiplier architectures are proposed and it is shown that they outperform the existing normal basis multiplier structures as compared with similar digit level normal basis multipliers the proposed multiplier with serial output requires the fewest number of xor gates and the one with parallel output is the fastest multiplier

new systolic architectures for inversion and division in gf 2 sup m we present two systolic architectures for inversion and division in gf 2 sup m based on a modified extended euclidean algorithm our architectures are similar to those proposed by others in that they consist of two dimensional arrays of computing cells and control cells with only local intercell connections and have o m sup 2 area time product however in comparison to similar architectures both our architectures have critical path delays that are smaller gate counts that range from being considerably smaller to only slightly larger and latencies that are identical for inversion but somewhat larger for division one architecture uses an adder or an m l bit ring counter inside each control cell while the other architecture distributes the ring counters into the computing cells thereby reducing each control cell to just two gates

concurrent error detection and correction in gaussian normal basis multiplier over gf 2 m fault based cryptanalysis has been developed to effectively break both private key and public key cryptosystems making robust finite field multiplication a very important research topic in recent years however no robust normal basis multiplier has been proposed in the literature therefore this investigation presents a semisystolic gaussian normal basis multiplier based on the proposed gaussian normal basis multiplier both concurrent error detection and correction capabilities can be easily achieved using time redundancy technology with no hardware modification

uncertainty of data fuzzy membership functions and multilayer perceptrons probability that a crisp logical rule applied to imprecise input data is true may be computed using fuzzy membership function mf all reasonable assumptions about input uncertainty distributions lead to mfs of sigmoidal shape convolution of several inputs with uniform uncertainty leads to bell shaped gaussian like uncertainty functions relations between input uncertainties and fuzzy rules are systematically explored and several new types of mfs discovered multilayered perceptron mlp networks are shown to be a particular implementation of hierarchical sets of fuzzy threshold logic rules based on sigmoidal mfs they are equivalent to crisp logical networks applied to input data with uncertainty leaving fuzziness on the input side makes the networks or the rule systems easier to understand practical applications of these ideas are presented for analysis of questionnaire data and gene expression data

projection based approximation and a duality with kernel methods projection pursuit regression and kernel regression are methods for estimating a smooth function of several variables from noisy data obtained at scattered sites methods based on local averaging can perform poorly in high dimensions curse of dimensionality intuition and examples have suggested that projection based approaches can provide better fits for what sorts of regression functions is this true when and by how much do projection methods reduce the curse of dimensionality we make a start by focusing on the two dimensional problem and study the l 2 approximation error bias of the two procedures with respect to gaussian measure let ra stand for a certain ppr type approximation and ka for a particular kernel type approximation building on a simple but striking duality for polynomials we show that ra behaves significantly better than the minimax rate of approximation for radial functions while ka performs significantly better than the minimax rate for harmonic functions in fact the rate improvements carry over to large classes ra behaving very well for functions with enough angular smoothness oscillating slowly with angle while ka behaves very well for functions with enough laplacian smoothness oscillations averaging out locally the rate improvements matter they are equivalent to lowering the dimensionality of the problem for example for functions with nice tail behavior ra behaves as if the dimensionality of the problem were 1 5 rather than its nominal value 2 also ra and ka are complementary for a given function if one method offers a dimensionality reduction the other does not

identification and simplification of t s fuzzy neural networks based on incremental structure learning and similarity analysis abstract this paper proposes a novel identification method and a simplification scheme for t s fuzzy neural networks which consists of two steps the first step refers to the structure design based on an incremental clustering approach whose basic ideas are that the structure identification of fuzzy neural networks is guided by the attenuation of output approximation error in each cluster and processed by a recursive refined clustering iteration with the input space clustering and sub clustering as the main steps once the structure of a t s fuzzy neural network is identified by the incremental clustering approach its parameters are further learned and refined by the levenberg marquardt optimization algorithm the second step refers to the structure simplification including removing redundant fuzzy rules and merging highly similar fuzzy rules furthermore the performances of several similarity calculating methods are analyzed and discussed which provides a basis for the selection of the appropriate similarity analysis and effective calculating method for the merging of fuzzy rules and system simplification that is the given performance analysis provides the methodology basis and design guide for structure simplification based on similarity analysis and merger of fuzzy rules several experiments are implemented to illustrate the feasibility and effectiveness of the proposed approach

gf 2 sup m multiplication and division over the dual basis in this paper an algorithm for gf 2 sup m multiplication division is presented and a new more generalized definition of duality is proposed from these the bit serial berlekamp multiplier is derived and shown to be a specific case of a more general class of multipliers furthermore it is shown that hardware efficient bit parallel dual basis multipliers can also be designed these multipliers have a regular structure are easily extended to different gf 2 sup m and hence suitable for vlsi implementations as in the bit serial case these bit parallel multipliers can also be hardwired to carry out constant multiplication these constant multipliers have reduced hardware requirements and are also simple to design in addition the multiplication division algorithm also allows a bit serial systolic finite field divider to be designed this divider is modular independent of the defining irreducible polynomial for the field easily expanded to different gf 2 sup m and its longest delay path is independent of m

analysis and design of functionally weighted single input rule modules connected fuzzy inference systems the single input rule modules sirms connected fuzzy inference method can efficiently solve the fuzzy rule explosion phenomenon which usually occurs in the multivariable modeling and or control applications however the performance of the sirms connected fuzzy inference system sirm fis is limited due to its simple input output mapping in this paper to further enhance the performance of sirm fis a functionally weighted sirm fis fwsirm fis which adopts multivariable functional weights to measure the important degrees of the sirms is presented then in order to show the fundamental differences of the sirms methods properties of the traditional sirm fis the type 2 sirm fis t2sirm fis the functional sirm fis fsirm fis the sirms model with single variable functional weights sirm fw and fwsirm fis are explored these properties demonstrate that the proposed fwsirm fis has more general and complex input output mapping than the existing sirms methods such properties theoretically guarantee that better performance can be achieved by fwsirm fis furthermore based on the least squares method a novel data driven optimization method is presented for the parameter learning of fwsirm fis it can also be used to optimize the parameters of sirm fis t2sirm fis fsirm fis and sirm fw due to the properties of the least squares method the proposed parameter learning algorithm can overcome the drawbacks of the gradients based parameter learning methods and obtain both smallest training errors and smallest parameters finally to show the effectiveness and superiority of fwsirm fis and the proposed optimization method six examples and detailed comparisons are given simulation results show that fwsirm fis can obtain better performance than the other sirms methods and compared with some well known methods fwsirm fis can achieve similar or better performance but has much less parameters and faster training speed

hierarchical fuzzy systems for function approximation on discrete input spaces with application this paper investigates the capabilities of hierarchical fuzzy systems to approximate functions on discrete input spaces first it is shown that any function on a discrete space has an arbitrary separable hierarchical structure and can be naturally approximated by hierarchical fuzzy systems as a by product of this result a discrete version of kolmogorov s theorem is obtained second it is proven that any function on a discrete space can be approximated to any degree of accuracy by hierarchical fuzzy systems with any desired separable hierarchical structure that is functions on discrete spaces can be approximated more simply and flexibly than those on continuous spaces third a hierarchical fuzzy system identification method is proposed in which human knowledge and numerical data are combined for system construction and identification finally the proposed method is applied to the market condition performance modeling problem in site selection decision support and shows the better performance in both accuracy and interpretability than the regression and neural network approaches in additions the reason and mechanism why hierarchical fuzzy systems outperform regression and neural networks in this type of application are analyzed

universal approximation by hierarchical fuzzy system with constraints on the fuzzy rule this paper presents a special hierarchical fuzzy system where the outputs of the previous layer are not used in the if parts but used only in the then parts of the fuzzy rules of the current layer the proposed scheme can be shown to be a universal approximator to any continuous function on a compact set if complete fuzzy sets are used in the if parts of the fuzzy rules with singleton fuzzifier and center average defuzzifier from the simulation of ball and beam control system it is demonstrated that the proposed scheme approximates with good accuracy the model nonlinear controller with fewer fuzzy rules than the centralized fuzzy system and its control performance is comparable to that of the nonlinear controller

sufficient conditions on general fuzzy systems as function approximators abstract in a constructive way we have found sufficient conditions under which general fuzzy systems can uniformly approximate any real continuous function on a compact domain to any degree of accuracy more importantly we have revealed the underlying mechanism of such function approximation the fuzzy systems can uniformly approximate any polynomials which according to the weierstrass approximation theorem can uniformly approximate any continuous function on a compact domain these findings lead to the following practical results 1 explicit fuzzy rules of fuzzy systems can now be easily obtained according to functions to be approximated and 2 formulas are derived which can calculate the number of input fuzzy sets output fuzzy sets and fuzzy rules needed in order to satisfy any given approximation accuracy the number increases as required approximation error decreases and as approximation error approaches zero the number approaches these formulas also reveal that the number is minimal when the maximum number of intersection between adjacent input fuzzy sets is one practical implications of these results will be discussed especially in the context of fuzzy control and fuzzy modeling two examples are provided to demonstrate how to design fuzzy systems to approximate given functions with required approximation accuracy

efficient bit serial multiplication and the discrete time wiener hopf equation over finite fields discrete time wiener hopf equations dtwhes over finite fields are considered it is shown that solving the dtwhe is equivalent to performing division over finite fields the proof provides a new interpretation of the relationship between bit serial multiplication and dtwhes the interpretation enables bit serial multiplication over gf 2 sup m to be understood more easily as an example bit serial multiplication methods for multiplying any two elements that can be done without performing any transformation or with only a simple transformation of the bases are presented

structure of parallel multipliers for a class of fields gf 2 m abstract this paper presents a configuration of parallel multipliers for gf 2 m based on canonical bases the possible parallel multipliers by the proposed configuration are limited to a class of fields gf 2 m however they can be constructed by o m 2 and gates and o m 2 eor gates with the structural modularity this is a desirable feature for the hardware implementation and their operation time is about log m t where m is the dimension of gf 2 m and t is the delay time of an eor gate in order to construct such parallel multipliers we define two types of polynomials of special form over gf 2 one is called all one polynomial denoted by aop and the other is called equally spaced polynomial denoted by esp furthermore we show a necessary and sufficient condition for esps to be irreducible over gf 2 and the uniqueness of the irreducible esps over gf 2 finally we propose the configuration of parallel multipliers for a class of fields gf 2 m based on irreducible aops and esps over gf 2

support vector learning for fuzzy rule based classification systems to design a fuzzy rule based classification system fuzzy classifier with good generalization ability in a high dimensional feature space has been an active research topic for a long time as a powerful machine learning approach for pattern recognition problems the support vector machine svm is known to have good generalization ability more importantly an svm can work very well on a high or even infinite dimensional feature space this paper investigates the connection between fuzzy classifiers and kernel machines establishes a link between fuzzy rules and kernels and proposes a learning algorithm for fuzzy classifiers we first show that a fuzzy classifier implicitly defines a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function fuzzy inference on the if part of a fuzzy rule can be viewed as evaluating the kernel function the kernel function is then proven to be a mercer kernel if the reference functions meet a certain spectral requirement the corresponding fuzzy classifier is named positive definite fuzzy classifier pdfc a pdfc can be built from the given training samples based on a support vector learning approach with the if part fuzzy rules given by the support vectors since the learning process minimizes an upper bound on the expected risk expected prediction error instead of the empirical risk training error the resulting pdfc usually has good generalization moreover because of the sparsity properties of the svms the number of fuzzy rules is irrelevant to the dimension of input space in this sense we avoid the curse of dimensionality finally pdfcs with different reference functions are constructed using the support vector learning approach the performance of the pdfcs is illustrated by extensive experimental results comparisons with other methods are also provided

global best harmony search harmony search hs is a new meta heuristic optimization method imitating the music improvisation process where musicians improvise their instruments pitches searching for a perfect state of harmony a new variant of hs called global best harmony search ghs is proposed in this paper where concepts from swarm intelligence are borrowed to enhance the performance of hs the performance of the ghs is investigated and compared with hs and a recently developed variation of hs the experiments conducted show that the ghs generally outperformed the other approaches when applied to ten benchmark problems the effect of noise on the performance of the three hs variants is investigated and a scalability study is conducted the effect of the ghs parameters is analyzed finally the three hs variants are compared on several integer programming test problems the results show that the three approaches seem to be an efficient alternative for solving integer programming problems 2007 elsevier inc all rights reserved

chaotic harmony search algorithms harmony search hs is one of the newest and the easiest to code music inspired heuristics for optimization problems like the use of chaos in adjusting note parameters such as pitch dynamic rhythm duration tempo instrument selection attack time etc in real music and in sound synthesis and timbre construction this paper proposes new hs algorithms that use chaotic maps for parameter adaptation in order to improve the convergence characteristics and to prevent the hs to get stuck on local solutions this has been done by using of chaotic number generators each time a random number is needed by the classical hs algorithm seven new chaotic hs algorithms have been proposed and different chaotic maps have been analyzed in the benchmark functions it has been detected that coupling emergent results in different areas like those of hs and complex dynamics can improve the quality of results in some optimization problems it has been also shown that some of the proposed methods have somewhat increased the solution quality that is in some cases they improved the global searching capability by escaping the local solutions

architectures for exponentiation over gd 2 sup n adopted for smartcard application two exponentiation circuits are proposed using the fact that squaring is a linear operation over gf 2 sup n a time space tradeoff in smartcard based circuitry is presented it is shown how multiplication is performed by a single shift based on replacing the public key alpha sup a in gf 2 sup n by its minimal polynomial other considerations related to structure regularity and the possible use of dynamic shift registers are also treated

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

fault tolerant decoders for cyclic error correcting codes high speed cyclic code decoders which are central to modern communication systems when implemented in dense very large scale integration vlsi are susceptible to pernicious momentary internal soft fails presenting a demanding error control challenge however special structures inherent in such decoders offer new methods for incorporating distributed error control throughout their designs the underlying design principles and motivations are emphasized providing a variety of options to meet various requirements bose chaudhuri hocquenghem bch codes are used to exemplify the new techniques as applied to the usual three standard subsystems present in a decoder the first and last parts syndrome calculations and transform inversion both involve finite field transforms suggesting the effective application of fast transform algorithms error control features are based upon the chord properties of the transform coefficients including even fast algorithms the third subsystem the berlekamp massey algorithm can be protected through a chord recursion property affiliated with the error location connection polynomial the propagation and spread of internal errors are studied and a special sink register compares several quantities available in close proximity to their easily recomputed counterparts most of these results are applicable to generalized to decoders for codes over higher ordered alphabets such as reed solomon codes

comb architectures for finite field multiplication in f 2 m two high speed bit serial word parallel or comb style finite field multipliers are proposed in this paper the first proposal utilizes a redundant representation for any binary field and the other uses a reordered normal basis for the binary field where a type ii optimal normal basis exists the proposed redundant representation architecture has a smaller critical path delay compared to the previous methods while the complexities remain about the same the proposed reordered normal basis multiplier has a significantly smaller critical path delay compared to the previous methods using the same basis or normal basis field programmable gate array fpga implementation results of the proposed multipliers are compared to those of the previous methods using the same basis which confirms that the proposed multipliers allow a much higher clock rate

a vlsi design for computing exponentiations in gf 2 sup m and its application to generate pseudorandom number sequences a vlsi design for computing exponentiation in finite fields is developed an algorithm to generate a relatively long pseudorandom number sequence is presented it is shown that the period of this sequence is significantly increased compared to that of the sequence generated by the most commonly used maximal length shift register scheme

on computing multiplicative inverses in gf 2 sup m the design of a modular standard basis inversion for galois fields gf 2 sup m based on euclid s algorithm for computing the greatest common divisor of two polynomials is presented the asymptotic complexity is linear with m both in computation time and area requirement thus resulting in an at complexity of o m sup 2 this is a significant improvement over the best previous proposal which achieves at complexity of only o m sup 3

efficient multiplier architectures for galois fields gf 2 sup 4n this contribution introduces a new class of multipliers for finite fields gf 2 sup n sup 4 the architecture is based on a modified version of the karatsuba ofman algorithm koa by determining optimized field polynomials of degree four the last stage of the koa and the module reduction can be combined this saves computation and area in vlsi implementations the new algorithm leads to architectures which show a considerably improved gate complexity compared to traditional approaches and reduced delay if compared with koa based architectures with separate module reduction the new multipliers lead to highly modular architectures and are thus well suited for vlsi implementations three types of field polynomials are introduced and conditions for their existence are established for the small fields where n 2 3 8 which are of primary technical interest optimized field polynomials were determined by an exhaustive search for each field order exact space and time complexities are provided

modeling of hierarchical fuzzy systems a new kind of mapping rule base scheme is proposed to get the fuzzy rules of hierarchical fuzzy systems the algorithm of this scheme is developed such that one can easily design the involved fuzzy rules in the middle layers of the hierarchical structure in contrast with the conventional single layer fuzzy controller the present method has approximate performance using the same scaling factors next examples are given at last simulated results demonstrate that the algorithm is effective and feasible

a nested tensor product model transformation the tensor product model transformation tpmt is an emerging numerical framework of the takagi sugeno t s fuzzy or polytopic system modeling for a linear matrix inequality based system control design a nested tpmt ntpmt is proposed in this paper which merges the dimensions of the tensors and performs the tpmt iteratively the resultant fuzzy model is in a multilevel nested tensor product tp structure the vertex tensor obtained by ntpmt has fewer dimension results than the original tpmt so the number of vertices or fuzzy rules which have been the main bottleneck for further application of the tpmt in higher dimensional systems is expected to decrease significantly it is also proven that the ntpmt contains the hierarchical fuzzy logic meaning that the ntpmt is capable of conducting hierarchical fuzzy modeling and reduction furthermore because the inclusion of multiple tpmts is prone to augment the conservativeness of the resultant fuzzy model a suboptimal convex hull rectification algorithm for the tpmt is developed based on a newly defined tightness measure and then extended to render the ntpmt as less conservative as possible finally numerical simulations on two real physical systems two and four parameter dimension are verified to demonstrate the performance of the methods

universal approximation by hierarchical fuzzy systems abstract a serious problem limiting the applicability of standard fuzzy controllers is the rule explosion problem that is the number of rules increases exponentially with the number of input variables to the fuzzy controller a way to deal with this curse of dimensionality is to use the hierarchical fuzzy systems a hierarchical fuzzy system consists of a number of hierarchically connected low dimensional fuzzy systems it can be shown that the number of rules in the hierarchical fuzzy system increases linearly with the number of input variables in this paper we prove that the hierarchical fuzzy systems are universal approximators that is they can approximate any nonlinear function on a compact set to arbitrary accuracy our proof is constructive that is we first construct a hierarchical fuzzy system in a step by step manner then prove that the constructed fuzzy system satisfies an error bound and finally show that the error bound can be made arbitrarily small

new systolic arrays for c ab sup 2 inversion and division in gf 2 sup m in this paper we present a new parallel in parallel out systolic array with unidirectional data flow for performing the power sum operation c ab sup 2 in finite fields gf 2 sup m the architecture employs the standard basis representation and can provide the maximum throughput in the sense of producing new results at a rate of one per clock cycle it is highly regular modular and thus well suited to vlsi implementation as compared to a previous systolic power sum circuit with bidirectional data flow and the same throughput performance the proposed one has smaller latency consumes less chip area and can more easily incorporate fault tolerant design based on the new power sum circuit we also propose a parallel in parallel out systolic array with the maximum throughput for computing inverses divisions in gf 2 sup m the proposed systolic divider gains advantages over an existing system with the same throughput performance in terms of chip area latency and fault tolerance

necessary conditions for some typical fuzzy systems as universal approximators abstract we investigate necessary conditions for general single input single output fuzzy systems and a class of typical multiple input single output fuzzy systems as universal approximators for continuous functions defined on compact domains with arbitrarily small uniform approximation error bounds considering the case where the only available data about the continuous function to be approximated are a finite set of its extrema we have established some necessary conditions for the fuzzy systems to be universal approximators of the function the conditions can be used practically to determine input fuzzy sets output fuzzy sets and fuzzy rules of the fuzzy systems furthermore these necessary conditions provide a basis for insightful analysis of the strengths as well as the limitations of the fuzzy systems the main strength is that only a small number of fuzzy rules may be needed to uniformly approximate continuous functions that have a complicated formulation but a relatively small number of extrema the limitation is that in order to approximate highly oscillatory continuous functions the number of fuzzy rules must be large 1997 elsevier science ltd

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

approximation theory of fuzzy systems spl minus siso case in this paper the approximation properties of mimo fuzzy systems generated by the product inference are discussed we first give an analysis of fuzzy basic functions fbf s and present several properties of fbf s based on these properties of fbf s we obtain several basic approximation properties of fuzzy systems 1 basic approximation property which reveals the basic approximation mechanism of fuzzy systems 2 uniform approximation bounds which give the uniform approximation bounds between the desired control or decision functions and fuzzy systems 3 uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions and 4 universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect the similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds

two mode indirect adaptive control approach for the synchronization of uncertain chaotic systems by the use of a hierarchical interval type 2 fuzzy neural network two mode adaptive controllers have two phases of operation a learning phase and an operation phase this paper presents a two mode indirect adaptive control approach for the synchronization of chaotic systems using a hierarchical interval type 2 fuzzy neural network ht2fnn its contribution to the existing literature is the adaptation laws derived for the parameters of the membership functions based on lyapunov stability analysis since in hierarchical case each t2fnn has only two inputs the computing of the derivatives is much simpler than the case in classical interval type 2 fnn moreover a novel approach is presented for the compensation of the approximation error the tuning of the parameters of the membership functions mf and the use of an interval type 2 fnn ensures that the estimation error is very small so that it can be negligible furthermore the number of mf required is seen to be less than that needed with type 1 fuzzy sets the simulation results confirm the efficacy of the proposed scheme in the synchronization of a uncertain nonidentical chaotic systems

automatic design of hierarchical takagi sugeno type fuzzy systems using evolutionary algorithms this paper presents an automatic way of evolving hierarchical takagi sugeno fuzzy systems ts fs the hierarchical structure is evolved using probabilistic incremental program evolution pipe with specific instructions the fine tuning of the if then rule s parameters encoded in the structure is accomplished using evolutionary programming ep the proposed method interleaves both pipe and ep optimizations starting with random structures and rules parameters it first tries to improve the hierarchical structure and then as soon as an improved structure is found it further fine tunes the rules parameters it then goes back to improve the structure and the rules parameters this loop continues until a satisfactory solution hierarchical ts fs model is found or a time limit is reached the proposed hierarchical ts fs is evaluated using some well known benchmark applications namely identification of nonlinear systems prediction of the mackey glass chaotic time series and some classification problems when compared to other neural networks and fuzzy systems the developed hierarchical ts fs exhibits competing results with high accuracy and smaller size of hierarchical architecture

provably sublinear point multiplication on koblitz curves and its hardware implementation we describe algorithms for point multiplication on koblitz curves using multiple base expansions of the form k sigmaplusmntaua tau 1 b and k sigmaplusmntaua tau mu b tau2 mutau 1 c we prove that the number of terms in the second type is sublinear in the bit length of k which leads to the first provably sublinear point multiplication algorithm on koblitz curves for the first type we conjecture that the number of terms is sublinear and provide numerical evidence demonstrating that the number of terms is significantly less than that of tau adic nonadjacent form expansions we present details of an innovative fpga implementation of our algorithm and performance data demonstrating the efficiency of our method we also show that implementations with very low computation latency are possible with the proposed method because parallel processing can be exploited efficiently

a class of hierarchical fuzzy systems with constraints on the fuzzy rules this paper presents a class of hierarchical fuzzy systems where previous layer outputs are used not in if parts but only in then parts of the fuzzy rules of the current layer the proposed scheme is shown to be a universal approximator to any real continuous function on a compact set if complete fuzzy sets are used in the if parts of the fuzzy rules with singleton fuzzifier and center average defuzzifier from the example of ball and beam control system simulation it is demonstrated that the proposed scheme approximates with high accuracy a model nonlinear controller with fewer fuzzy rules than a centralized fuzzy system and its control performance is comparable to that of a nonlinear controller

hybridising harmony search with a markov blanket for gene selection problems gene selection which is a well known np hard problem is a challenging task that has been the subject of a large amount of research especially in relation to classification tasks this problem addresses the identification of the smallest possible set of genes that could achieve good predictive performance many gene selection algorithms have been proposed however because the search space increases exponentially with the number of genes finding the best possible approach for a solution that would limit the search space is crucial metaheuristic approaches have the ability to discover a promising area without exploring the whole solution space hence we propose a new method that hybridises the harmony search algorithm hsa and the markov blanket mb called hsa mb for gene selection in classification problems in this proposed approach the hsa as a wrapper approach improvises a new harmony that is passed to the mb treated as a filter approach for further improvement the addition and deletion of operators based on gene ranking information is used in the mb algorithm to further improve the harmony and to fine tune the search space the hsa mb algorithm method works especially well on selected genes with higher correlation coefficients based on symmetrical uncertainty ten microarray datasets were experimented on and the results demonstrate that the hsa mb has a performance that is comparable to state of the art approaches hsa mb yields very small sets of genes while preserving the classification accuracy the results suggest that hsa mb has a high potential for being an alternative method of gene selection when applied to microarray data and can be of benefit in clinical practice

division and accumulation over gf 2 sup m the galois field division is a complex arithmetic operation the corresponding division and accumulation daa is not only complex but also a time consuming operation in this article the daa over gf 2 sup m is considered and a simple scheme for its sequential operation is presented a multiple stream daa structure is developed which supports pipeline operations and yields an increased throughput with only a modest increase in the hardware as an application the use of the sequential daa algorithm is shown for the high speed encoding of reed solomon codes

an improved variant of the conventional harmony search algorithm the harmony search algorithm hs has been used for optimization in different fields and despite the relative short time it has been around it already has many variants this article presents a new modification of hs based on variable parameters which is able to yield better results than previously reported data and with the additional benefit of not requiring prior knowledge of the maximum number of iterations in this research a comparison is made with the original hs algorithm and with its improved version i e ihs finding that the proposed variants not only reduce convergence time of the algorithm but they also increase its precision some commonly used benchmark functions were used as a testing scenario and the performance of the novel approach is evaluated for an objective function in up to 1000d where it was found to converge appropriately these findings are important since they indicate that the proposed version could be used for different kinds of optimization problems thus allowing a broader use of the hs algorithm