a remez type algorithm for spline functions a remez type algorithm for computing best spline approximations of degreen withk fixed knots is developed it is shown that the sequence constructed in the algorithm converges to a best approximation ifk n 1 and at least to a nearly best approximation ifk n 1 in contrast to the standard interpolation methods no restriction to the position of the knots is required this fact may be important to treat approximation problems for splines with free knots

new eighth order iterative methods for solving nonlinear equations in this paper three new families of eighth order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative this implies that the efficiency index of the developed methods is 1 682 which is optimal according to kung and traub s conjecture 7 for four function evaluations per iteration notice that bi et al s method in 2 and 3 are special cases of the developed families of methods in this study several new examples of eighth order methods with efficiency index 1 682 are provided after the development of each family of methods numerical comparisons are made with several other existing methods to show the performance of the presented methods

a new family of optimal eighth order methods with dynamics for nonlinear equations we propose a simple yet efficient family of three point iterative methods for solving nonlinear equations each method of the family requires four evaluations namely three functions and one derivative per full iteration and possesses eighth order of convergence thus the family is optimal in the sense of kung traub conjecture and has the efficiency index 1 682 which is better than that of newton s and many other higher order methods various numerical examples are considered to check the performance and to verify the theoretical results computational results including the elapsed cpu time confirm the efficient and robust character of proposed technique moreover the presented basins of attraction also confirm better performance of the methods as compared to other established methods in literature

on computational efficiency for multi precision zero finding methods an adapted definition related to multi precision arithmetic zero finding methods is considered and several improvements to iterative methods to compute nonlinear equation solutions which increase the local order of convergence are revisited furthermore a higher efficiency when we use a definition related to adaptive multi precision arithmetic is also given numerical results computed with a floating point system representing 200 and 1000 decimal digits support this theory

a family of three point methods of optimal order for solving nonlinear equations a family of three point iterative methods for solving nonlinear equations is constructed using a suitable parametric function and two arbitrary real parameters it is proved that these methods have the convergence order eight requiring only four function evaluations per iteration in this way it is demonstrated that the proposed class of methods supports the kung traub hypothesis 1974 3 on the upper bound 2 n of the order of multipoint methods based on n 1 function evaluations consequently this class of root solvers possesses very high computational efficiency numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations

predictor corrector halley method for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative methods for solving nonlinear equations of the type f x 0 by using the technique of updating the solution this method can be viewed as a predictor corrector iterative halley s method we also consider the convergence analysis of the proposed method to illustrate the efficiency of this new method we give several examples and a comparison with the method of grau and diaz barrero m grau j l diaz barrero an improvement to ostrowski root finding method appl math comput 173 2006 450 456 is given this method can be considered as a refinement and improvement of the halley method

ostrowski type methods for solving systems of nonlinear equations abstract four generalized algorithms builded up from ostrowski s method for solving systems of nonlinear equations are written and analyzed a development of an inverse first order divided difference operator for functions of several variables is presented as well as a direct computation of the local order of convergence for these variants of ostrowski s method furthermore a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced

motion compensating prediction with fractional pel accuracy the effect of fractional pel accuracy on the efficiency of motion compensating predictors is studied using various spatial prediction interpolation filters in model calculations the power spectral density of the prediction error is related to the probability density function of the displacement error prediction can be improved both by higher accuracy of motion compensation and by spatial wiener filtering in the predictor beyond a critical accuracy the possibility of further improving prediction by more accurate motion compensation is small experiments with videophone signals and with broadcast tv signals confirm these model calculations sinc interpolation bilinear interpolation and wiener filtering are compared at integer pel 1 2 pel 1 4 pel and 1 8 pel accuracies a three state technique for reliable displacement estimation with fractional pel accuracy is introduced it is based on phase correlation for motion compensation with block size of 16 pels 16 pels 1 4 pel accuracy appears to be sufficient for broadcast tv signals whereas for videophone signals 1 2 pel accuracy is desirable

improvements of the efficiency of some three step iterative like newton methods an improvement of the local order of convergence is presented to increase the efficiency of the iterative method with an appropriate number of evaluations of the function and its derivative the third and fourth order of known two step like newton methods have been improved and the efficiency has also been increased

modified householder iterative method free from second derivatives for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative method free from second derivatives for solving nonlinear equations of the type f x 0 this new method includes the two step newton method as a special case we prove that the new iterative method is of fourth order several examples are given to illustrate the efficiency of this new method and its comparison with other iterative methods this method can be considered as a significant improvement of the newton method and its variant forms

a variant of jarratt method with sixth order convergence in this paper we present a variant of jarratt method for solving non linear equations per iteration the new method adds the evaluation of the function at another point in the procedure iterated by jarratt method as a consequence the local order of convergence is improved from four for jarratt method to six for the new method a new multistep iteration scheme based on the new method is developed and numerical tests verifying the theory are also given

some sixth order variants of ostrowski root finding methods abstract in this paper we present some sixth order class of modified ostrowski s methods for solving nonlinear equations per iteration each class member requires three function and one first derivative evaluations and is shown to be at least sixth order convergent several numerical examples are given to illustrate the performance of some of the presented methods

a note on halley s method we introduce the degree of logarithmic convexity which provides a measure of the convexity of a function at each point making use of this concept we obtain a new theorem of global convergence for halley s method

modifications of higher order convergence for solving nonlinear equations abstract in liang fang guoping he some modifications of newton s method with higher order convergence for solving nonlinear equations j comput appl math 228 2009 296 303 the authors pointed out that the iteration constructed in y m ham c b chun and s g lee some higher order modifications of newton s method for solving nonlinear equations j comput appl math 222 2008 477 486 failed when p 2 they gave some counterexamples and obtained a modified result however they did not show the essential reason which leads to the incorrect result in this paper we shall show that reason and present more general results than the above mentioned papers

a one parameter square root family of two step root finders a one parameter family of iterative methods for solving nonlinear equations is constructed all the methods of the proposed family are cubically convergent for a simple root except one particular method which attains the fourth order without the increase of computational cost these methods belong to the class of two step methods and require three function evaluations per iteration the square root structure of the family provides finding a complex zero of real functions in some cases a comparison analysis shows that the presented family generates the methods which are comparable or even superior than the existing two step iterative methods of the third order

on a general class of multipoint root finding methods of high computational efficiency a general class of n point iterative methods for solving nonlinear equations is constructed by combining methods of newton s type and an arbitrary two point method of the fourth order of convergence it is proved that these methods have the convergence order 2 n requiring only n 1 function evaluations per iteration in this way it is demonstrated that the proposed class of methods supports the kung traub hypothesis 1974 on the upper bound 2 n of the order of multipoint methods based on n 1 function evaluations consequently this class possess as high as possible computational efficiency in the sense of the kung traub hypothesis numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations

a convergence improvement factor and higher order methods for solving nonlinear equations abstract a convergence improvement factor is introduced for solving nonlinear equations so as to increase the efficiency of iterative methods many well known iterative methods are covered

new iterative methods for solving nonlinear equation by using homotopy perturbation method in this paper we suggest and analyze a new class of iterative methods for solving nonlinear equations by using the homotopy perturbation method convergence of their method is also considered here we also discuss the efficiency index and computational order of convergence of new methods several numerical examples are given to illustrate the efficiency and performance of these new methods these new iterative methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations

a variant of super halley method with accelerated fourth order convergence in this paper we present a new variant of super halley method for solving non linear equations analysis of convergence shows that the method has the order of convergence four per iteration the new method requires the same evaluations of the function the first derivative and second derivative as its classical predecessor super halley method numerical results show that the method has definite practical utility

approximation by generalized splines the theory of best approximation by polynomial splines has been the subject of a substantial number of papers over the past fifteen years see e g l 2 5 9 19 22 321 these papers discuss all of the usual approximation questions including existence characterization uniqueness strong uniqueness the existence of selections and computational methods much of the theory has been extended to the class of tchebychehian splines the purpose of this paper is to develop a similar theory for more general classes of spline functions in this paper we deal only with the case of fixed knots and with the uniform and norms it turns out that in order to get results analogous to those for polynomial splines the right class of generalized spines to look at consists of those spline spaces which have a certain interlacing property connected with the solvability of interpolation problems a complete characterization of such spline spaces was recently given in 13

a multi parameter family of three step eighth order iterative methods locating a simple root a multi parameter family of three step eighth order iterative methods free from second derivatives are proposed in this paper to find a simple root of nonlinear algebraic equations convergence analysis as well as numerical experiments confirms the eighth order convergence and asymptotic error constants

mmse design of interpolation and downsampling fir filters in the context of periodic nonuniform sampling the generalized sampling theorem states that any analog signal whose spectrum is limited to 1 t can be exactly recovered from n sequences of samples taken at a rate 2 nt and all having a different sampling phase when n 2 the exact interpolation formula can be derived quite easily the ideal interpolation filters have infinite impulse responses iirs this paper addresses first the question of recovering from the two initial sequences any other sequence taken at the same rate 1 t and with a different sampling phase the design problem is dealt with for finite length filters and the criterion is the minimization of the mean squared interpolation error next the problem of computing from the two initial sequences a third one at a lower rate is addressed fir decimation filters are also designed for an mmse criterion these problems are illustrated for two typical covariance functions

a family of higher order iterations free from second derivative for nonlinear equations in r abstract the aim of this article is to develop a family of higher order iterations free from second derivative for solving nonlinear equations in r their theoretical computational and dynamical aspects are fully investigated and theorems are established to provide their order of convergence and asymptotic error constant it is observed that the family includes sixth order methods and for a particular case its eighth order can be achieved in this family methods use three functions and one first derivative evaluations the family of methods can be shown to be optimal by using kung traub conjecture a number of numerical examples are worked out to demonstrate the applicability of these methods the improved results are obtained in comparison to some of the existing robust methods on the considered test examples local convergence analysis and dynamical study of the proposed family of methods are also carried out

uniqueness and strong uniqueness of best approximations by spline subspaces and other subspaces abstract a complete characterization is given of those functions in c a b which have a unique best approximation from the subspace of spline functions of degree n with k fixed knots also the relationship between unique and strongly unique best approximations from arbitrary finitedimensional subspaces of c0 t is investigated

steffensen type methods for solving nonlinear equations in the present paper by approximating the derivatives in the well known fourth order ostrowski s method and in a sixth order improved ostrowski s method by central difference quotients we obtain new modifications of these methods free from derivatives we prove the important fact that the methods obtained preserve their convergence orders 4 and 6 respectively without calculating any derivatives finally numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical newton s method

an improvement of the jarratt method in this paper we present a variant of jarratt method for solving non linear equations per iteration the new method adds the evaluation of the function at another point in the procedure iterated by jarratt method as a consequence the local order of convergence is improved from four for jarratt method to six for the new method the new multistep iteration scheme based on the new method is developed and numerical tests verifying the theory are also given

some sixth order zero finding variants of chebyshev halley methods a zero finding technique in which the order of convergence is improved and nonlinear equations are solved more efficiently than they are solved by traditional iterative methods is derived composing a modified chebyshev halley method with a variant of this method that just introduces one evaluation of the function the iterative methods presented are obtained by carrying out this procedure the output numerical results show that the new methods compete in both order and efficiency with the modified chebyshev halley methods

fifth order iterative methods for solving nonlinear equations abstract in this paper we suggest and analyze a new two step iterative method for solving nonlinear equation f x 0 by rewriting the given nonlinear equation as a coupled system of equations and using the taylor series it is shown that this new iterative method is of fifth order several examples are given to illustrate its performance and efficiency comparison with other methods is also given this method can be viewed as an improvement of the halley s method

a hummel seebeck family of iterative methods with improved convergence and efficiency several improvements to a family of iterative methods deduced from a hummel seebeck theorem are presented a symbolic computation that allows to find the best coefficients respect to the local order of convergence is also given the theoretical and computational order of convergence for all methods is increased furthermore the efficiency of these methods applied to the functions tested is improved adapting the strategy presented here a new iteration function with a new evaluation of the function is obtained and using adaptive multi precision arithmetic a smaller cost is got the numerical results computed carrying out this procedure with a floating point system representing 1000 decimal digits support this theory

new family of seventh order methods for nonlinear equations abstract a family of seventh order iterative methods for the solution of nonlinear equations is presented the new methods are based on king s fourth order methods and without using the second derivatives per iteration the new methods require three evaluations of the function and one evaluation of its first derivative therefore this family of methods has efficiency index equal to 1 627 numerical comparisons are made with several other existing methods to show the performance of the presented methods

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

a new family of eighth order iterative methods for solving nonlinear equations a family of eighth order iterative methods with four evaluations for the solution of nonlinear equations is presented kung and traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2 n 1 the new family of eighth order methods agrees with the conjecture of kung traub for the case n 4 therefore this family of methods has efficiency index equal to 1 682 numerical comparisons are made with several other existing methods to show the performance of the presented methods

on the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods a development of an inverse first order divided difference operator for functions of several variables is presented two generalized derivative free algorithms built up from ostrowski s method for solving systems of nonlinear equations are written and analyzed a direct computation of the local order of convergence for these variants of ostrowski s method is given in order to preserve the local order of convergence any divided difference operator is not valid two counterexamples of computation of a classical divided difference operator without preserving the order are presented a rigorous study to know a priori if the new method will preserve the order of the original modified method is presented the conclusion is that this fact does not depend on the method but on the systems of equations and if the associated divided difference verifies a particular condition a new divided difference operator solving this problem is proposed furthermore a computation that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced this study can be applied directly to other newton s type methods where derivatives are approximated by divided differences

an improvement to ostrowski root finding method an improvement to the iterative method based on the ostrowski one to compute nonlinear equation solutions which increases the local order of convergence is suggested the adaptation of a strategy presented here gives a new iteration function with an additional evaluation of the function it also shows a smaller cost if we use adaptive multi precision arithmetic the numerical results computed using this system with a floating point system representing 200 decimal digits support this theory

a note on some quadrature based three step iterative methods for non linear equations in a recent paper n a mir t zaman some quadrature based three step iterative methods for non linear equations appl math comput 193 2007 366 373 some new three step iterative methods for non linear equations have been proposed in this note we show that the algorithm 2 2 and algorithm 2 3 given by the authors have twelfth order and ninth order convergence respectively not seventh order one as claimed in their work

a fifth order iterative method for solving nonlinear equations in this paper we present a new fifth order method for solving nonlinear equations per iteration the new method requires two function and two first derivative evaluations it is shown that the new method is fifth order convergent several numerical examples are given to illustrate the performance of the presented method

statistical properties of coded interlaced and progressive image sequences we investigate the statistical properties of interlaced and progressive image sequences in order to determine what is the most efficient format from a coding point of view intra and inter coding modes are both considered the main result is that for both intra and inter images the coding of progressive images outperforms the coding of interlaced ones for low bit rates and large motion vectors particular attention is paid to inter coding with subpel accuracy where the effects of nonideal interpolation filters and finite precision of the motion vector are taken into account

modified householder iterative method for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative method for solving nonlinear equations of the type f x 0 this new method includes the two step newton method as a special case we show that this new two step method is a sixth order convergent method several examples are given to illustrate the efficiency of this new method and its comparison with other sixth order methods this method can be considered as a significant improvement of the newton method and its variant forms

some improvements of jarratt s method with sixth order convergence abstract in this paper we present a one parameter family of variants of jarratt s fourth order method for solving nonlinear equations it is shown that the order of convergence of each family member is improved from four to six even though it adds one evaluation of the function at the point iterated by jarratt s method per iteration several numerical examples are given to illustrate the performance of the presented methods

alternation for best spline approximations forf c a b the existence and uniqueness of best spline approximationss for which the errorf s has certain alternation properties are investigated such best approximations play an important role to develop a remez type algorithm for computing best spline approximations

some modification of newton s method by the method of undetermined coefficients in this paper we construct some modifications of newton s method for solving nonlinear equations which is based on the method of undetermined coefficients it is shown by way of illustration that the method of undetermined coefficients is a promising tool for developing new methods and reveals its wide applicability by obtaining some existing methods as special cases two new sixth order methods are developed numerical examples are given to support that the methods thus obtained can compete with other iterative methods

improved interpolation motion estimation and compensation for interlaced pictures this paper introduces a new method for the interpolation that has to be performed when motion estimation and compensation are applied to interlaced sequences with subpel accuracy it is based on the assumption that a uniform motion exists between two successive frames the exact formulas for the estimation are derived they show that in order to obtain a correct interpolation of each field of one frame it is necessary to use the information of both fields of this frame because the ideal filters have infinite impulse responses the filter design is discussed and the efficiency is measured for typical sequences

some higher order modifications of newton s method for solving nonlinear equations in this paper we consider constructing some higher order modifications of newton s method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units this construction can be applied to any iteration formula and per iteration the resulting methods add only one additional function evaluation to increase the order some illustrative examples are provided and several numerical results are given to show the performance of the presented methods

a family of modified ostrowski methods with accelerated sixth order convergence based on ostrowski fourth order multipoint method we derive a one parameter family of sixth order methods for solving equations each member of the family requires three evaluations of the given function and one evaluation of its derivative per iteration numerical examples are presented and performance is compared with ostrowski method

sixteenth order method for nonlinear equations modification of newton s method with higher order convergence is presented the modification of newton s method is based on king s fourth order method the new method requires three step per iteration analysis of convergence demonstrates that the order of convergence is 16 some numerical examples illustrate that the algorithm is more efficient and performs better than classical newton s method and other methods

three step iterative methods with eighth order convergence for solving nonlinear equations a family of eighth order iterative methods for the solution of nonlinear equations is presented the new family of eighth order methods is based on king s fourth order methods and the family of sixth order iteration methods developed by chun et al per iteration the new methods require three evaluations of the function and one evaluation of its first derivative therefore this family of methods has the efficiency index which equals 1 682 kung and traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2 n 1 thus we provide a new example which agrees with the conjecture of kung traub for n 4 numerical comparisons are made to show the performance of the presented methods

an acceleration of newton s method super halley method from a study of the convexity we give an acceleration for newton s method and obtain a new third order method then we use this method for solving non linear equations in banach spaces establishing conditions on convergence existence and uniqueness of solution as well as error estimates

eighth order methods with high efficiency index for solving nonlinear equations in this paper we derive a new family of eighth order methods for solving simple roots of nonlinear equations by using weight function methods per iteration these methods require three evaluations of the function and one evaluation of its first derivative which implies that the efficiency indexes are 1 682 numerical comparisons are made to show the performance of the derived methods as shown in the illustration examples

some iterative methods free from second derivatives for nonlinear equations in a recent paper noor m aslam noor new classes of iterative methods for nonlinear equations appl math comput 2007 doi 10 1016 j amc 2007 suggested and analyzed a generalized one parameter halley method for solving nonlinear equations using in this paper we modified this method which has fourth order convergence as special cases we obtain a family of third order iterative methods for appropriate and suitable choice of the parameter we have compared this modified noor method with some other iterative methods which shows that this new iterative method is robust and efficient one several examples are given to illustrate the efficiency and the performance of this new method

some quadrature based three step iterative methods for non linear equations in this paper we present three step quadrature based iterative methods for solving non linear equations the convergence analysis of the methods is discussed it is established that the new methods have convergence order six seven and eight respectively numerical tests show that the new methods are comparable with the well known existing methods and in many cases give better results our results can be considered as an improvement and refinement of the previously known results in the literature

a family of modified super halley methods with fourth order convergence in this paper we present a family of modified super halley methods for solving non linear equations analysis of convergence shows that the methods have fourth order convergence the superiority of the new methods is that they require no additional evaluations of the function the first derivative or second derivative as compared with the classical third order methods although their order is improved numerical results show that the new methods can be efficient

three step iterative methods with optimal eighth order convergence in this paper based on ostrowski s method a new family of eighth order methods for solving nonlinear equations is derived in terms of computational cost each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative so that their efficiency indices are 1 682 which is optimal according to kung and traub s conjecture numerical comparisons are made to show the performance of the new family

some variants of ostrowski s method with seventh order convergence in this paper we present a class of new variants of ostrowski s method with order of convergence seven per iteration the new methods require three evaluations of the function and one evaluation of its first derivative and therefore this class of methods has the efficiency index equal to 1 627 numerical tests verifying the theory are given and multistep iterations based on the present methods are developed

an infinite family of one step iterators for solving nonlinear equations to increase the order of convergence and a new algorithm of global convergence in this paper we present an infinite family of one step iterative formulas for solving nonlinear equations present method one from now on pmi that can be expressed as x n 1 f m x n with email protected m 1 we will prove that the corresponding iteration formula of the family x n 1 f m 0 x n has an order of convergence m 0 1 the increment of the velocity of convergence of the sequence of the iterator family x n 1 f m 1 x n with respect to the previous one x n 1 f m x n is attained at the expense of one derivative evaluation more besides we introduce a new algorithm present method two from now on pmii that plays the role of seeker for an initial value to guarantee the local convergence of the pmi both of them can be combined as an algorithm of global convergence included the case of singular roots that does not depend on the chosen initial value and that allows to find all the roots in a feasible interval in a general and complete way these are in my opinion the main results of this work