convergence analysis of an online approach to parameter estimation problems based on binary observations in this paper we present an online identification method to the problem of parameter estimation from binary observations a recursive identification algorithm with low storage requirements and computational complexity is derived we prove the convergence of this method provided that the input signal satisfies a strong mixing property some simulation results are then given in order to illustrate the properties of this method under various scenarios this method is appealing in the context of micro electronic devices since it only requires a 1 bit analog to digital converter with low power consumption and minimal silicon area

a note on some recent methods for solving nonlinear equations abstract in this note we present some comments on the recent results concerning iterative methods for solving nonlinear equations in the first part we compare a new method n ujevic a method for solving nonlinear equations appl math comput 174 2006 1416 1426 with the newton and ostrowski method including an extensive analysis of numerical results together with the choice of initial approximations and the computational efficiency of the considered methods in the second part we give a list of recently derived root finding methods and we study the question of priority namely we find out that all reviewed methods were already derived at the beginning of the 60s of the last century one of them dates from the 18th century some remarks on the comparison analysis and the application of iteration methods are included

on the global convergence of schroder s iteration formula for real zeros of entire functions abstract schroder s formula of the second kind of order m of convergence s2 m formula is a generalization of newton s m 2 and halley s m 3 iterative formulae for finding zeros of functions the authors showed that the s2 m formula of every odd order m 3 converges globally and monotonically to real zeros of polynomials on the real line for halley s formula such the convergence property for real zeros of entire functions had been shown by davies and dawson in this paper we extend both results by showing that the s2 m formula of every odd order m 5 has the same convergence property for real zeros of entire functions by numerical examples we illustrate the monotonic convergence of the formula of odd order m 3 5 and 7 and the non monotonic convergence of even order m 2 4 and 6 further we compare several formulae of both first and second kinds in performance

the solution to the conjecture on properness of weakly relaxed delayed controls in 1986 warga proposed a weak relaxation procedure applicable to fully nonlinear problems with delays in the control variables and showed that the resulting relaxed problem has a solution however in the event of commensurate delays several examples were found for which weakly relaxed controls cannot be approximated with original controls so that this extension fails to be proper although the case of commensurate delays was solved by the introduction of a strong model the question of how to properly relax noncommensurately delayed controls has remained open and a natural candidate has been precisely that of weakly relaxed controls in this paper a general counterexample is constructed which rules out the weak relaxation as a proper relaxation when there are two or more delays it is hoped that this result will provide some insight into the problem of finding a general representation of properly relaxed controls

on a family of symmetric polynomials abstract in this paper we provide two simple approaches to the explicit expression of a family of symmetric polynomials introduced and studied in milovanovic and cvetkovic j math anal appl 311 2005 191 thereby improving on their observations

a note on determinantal representation of a schroder konig like simultaneous method for finding polynomial zeros abstract using pade approximation sakurai torii and sugiura derived in the paper sakurai et al 1991 the generalized iterative method of order n 2 for finding all zeros of a polynomial where n is the highest order of a polynomial derivative involved in the presented iterative formula in this note we give the determinantal representation of this method and analyze procedures for its implementation and some computational aspects

accelerated convergence in newton s method newton s method is based on a linear approximation of the function whose roots are to be determined taken at the current point and the resulting algorithm is known to converge quadratically in a procedure to increase the rate of convergence the author modifies the target function in such a way that newton s method applied to the modified function will yield a faster rate of convergence

on the rediscovery of halley s iterative method for computing the zero of an analytic function abstract we show that halley s basic sequence resulting from accelerating the order of convergence of newton s method is the most efficient way of doing so in terms of usage of certain derivatives this fact could explain why this process of accelerating the convergence of newton s method is so frequently rediscovered then we present an algorithmic way of recognizing halley s family and we apply this algorithm to examples of rediscoveries

a representation of properly relaxed controls with two delays abstract the search of a proper relaxation procedure in the sense that relaxed minimizers can be approximated with ordinary controls for optimal control problems involving delays in the control variables has been the main topic of several recent papers three models of relaxation which we call the weak strong and d procedures have been proposed and in all cases the existence of minimizers has been established it has been proved that the d model is a proper relaxation procedure but determining the set of d relaxed controls for specific problems is very difficult and perhaps even a hopeless task thus there is a need to find more concrete characterizations of the closure of the space of ordinary delayed controls in the event of commensurate delays this is solved through the strong model the space of strongly relaxed controls coincides with the space of d relaxed controls for the noncommensurate case the problem of how to characterize d relaxed controls has remained unsolved and although a natural candidate had been the space of weakly relaxed controls now we know that for either commensurate or noncommensurate delays it may fail to be proper in this paper we are finally able to characterize d relaxed controls for two noncommensurate delays by the introduction of a new model of relaxation the p model which turns out to be a space strictly contained in that of weakly relaxed controls

n th root extraction double iteration process and newton s method a double iteration process already used to find the nth root of a positive real number is analysed and showed to be equivalent to the newton s method these methods are of order two and three higher order methods for finding the nth root are also mentioned

identification of hammerstein systems with quantized observations this work is concerned with identification of hammerstein systems whose outputs are measured by quantized sensors the system consists of a memoryless nonlinearity that is polynomial and possibly noninvertible followed by a linear subsystem the parameters of linear and nonlinear parts are unknown but have known orders input design identification algorithms and their essential properties are presented under the assumptions that the distribution function of the noise is known and the quantization thresholds are known the concept of strongly scaled full rank signals is introduced to capture the essential conditions under which the hammerstein system can be identified with set valued observations under strongly scaled full rank conditions a strongly convergent algorithm is constructed asymptotic consistency and efficiency of the algorithm are investigated

variants of newton s method using fifth order quadrature formulas abstract some variants of newton s method are developed in this work in order to solve nonlinear equations depending on one or several variables based in rules of quadrature of fifth order we prove the third or fifth order of convergence of these methods for dimension one and the second or third order in several variables depending on the behaviour of the second derivative moreover different numeric tests confirm or improve theoretic results and allow us to compare these variants with newton s classical method

on rediscovered methods for solving equations abstract in the recent papers of gerlach siam rev 36 1994 272 276 and ford and pennline siam rev 38 1996 658 659 a class of iterative methods for solving a single equation f x 0 with arbitrary rate of convergence has been presented in this paper we show that this class is equivalent to five other classes of iterative methods derived earlier in various ways and expressed in different forms the proofs of equivalence of all considered iteration formulas are given

a non iterative method for solving non linear equations abstract in this paper using a hyperbolic tangent function tanh x 0 we develop a non iterative method to estimate a root of an equation f x 0 the problem of finding root is transformed to evaluating an integral and thus we need not take account of choosing initial guess the larger the value of the better the approximation to the root alternatively we employ the signum function sgn x instead of the hyperbolic tangent function which results in an exact formula for the root availability of the present method is shown by some numerical examples for which the traditional newton s method is not appropriate

on the characterization of properly relaxed delayed controls abstract in previous papers warga and zhu have proven the conjecture of rosenblueth and vinter that a particular abstract model characterizes the set r of relaxed controls that are limits of ordinary controls with possibly time dependent delays rosenblueth and vinter have also provided a simple and concrete characterization of r for the case of constant delays that are consecutive multiples of a common divisorh in the present paper we provide a concrete characterization of r for the case of delays that are arbitrary and not necessarily consecutive multiples of someh we also exhibit a simple compactification of the set of solutions of controlled differential equations with such delays in control

newton s method s basins of attraction revisited in this paper we revisit the chaotic number of iterations needed by newton s method to converge to a root here we consider a simple modified newton method depending on a parameter it is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions i e regions where the method converges nicely to a root can be complicatedly a function of the parameter

on an application of symbolic computation and computer graphics to root finders the contemporary powerful mathematical software enables a new approach to handling and manipulating complex mathematical expressions and other mathematical objects particularly the use of symbolic computation leads to new contribution to constructing and analyzing numerical algorithms for solving very difficult problems in applied mathematics and other scientific disciplines in this paper we are concerned with the problem of determining multiple zeros when the multiplicity is not known in advance a task that is seldom considered in literature by the use of computer algebra system mathematica we employ symbolic computation through several programs to construct and investigate algorithms which both determine a sought zero and its multiplicity applying a recurrent formula for generating iterative methods of higher order for solving nonlinear equations we construct iterative methods that serve i for approximating a multiple zero of a given function f when the order of multiplicity is unknown and simultaneously ii for finding exact order of multiplicity in particular we state useful cubically convergent iterative sequences that find the exact multiplicity in a few iteration steps such approach combined with a rapidly convergent method for multiple zeros provides the construction of efficient composite algorithms for finding multiple zeros of very high accuracy the properties of the proposed algorithms are illustrated by several numerical examples and basins of attraction

new simultaneous root finding methods with accelerated convergence for analytic functions a new iterative method of the fourth order for the simultaneous determination of zeros of a class of analytic functions is proposed further improvements of the basic method are attained by using newton s and halley s corrections giving the orders of convergence five and six respectively the improved convergence is achieved with negligible number of additional calculations which significantly increases the computational efficiency of the accelerated methods numerical examples demonstrate a good convergence properties fitting very well theoretical results

relationships between order and efficiency of a class of methods for multiple zeros of polynomials the behavior of a class of high order methods for solving polynomial equations is examined it is shown that the number of iterations for local convergence to a multiple zero to the limits of attainable accuracy is bounded independent of the multiplicity of the zero and decreases as the order of the method increases for the higher order methods the number of iterations decreases as the multiplicity increases computational efficiency as a function of degree order and multiplicity is investigated and an effective choice of order is recommended numerical examples are provided

on the global convergence of schroder s iterative formulae for real roots of algebraic equations abstract schroder s formulae of the first s1 and second s2 kind of order m of convergence are generalizations of newton s m 2 and halley s s2 m 3 iterative formulae for finding zeros of functions davies and dawson show that for entire functions with only real zeros halley s formula converges globally and monotonically to their zeros independently of the initial value on the real line we show that the s2 formulae of odd order 5 enjoy the same convergence feature for polynomials with only real zeros numerical examples illustrate this we illustrate no monotonic convergence of the s1 formulae and of the s2 formulae of even order

fast switch and spline scheme for accurate inversion of nonlinear functions the new first choice solution to kepler s equation abstract numerically obtaining the inverse of a function is a common task for many scientific problems often solved using a newton iteration method here we describe an alternative scheme based on switching variables followed by spline interpolation which can be applied to monotonic functions under very general conditions to optimize the algorithm we designed a specific ultra fast spline routine we also derive analytically the theoretical errors of the method and test it on examples that are of interest in physics in particular we compute the real branch of lambert s w y function which is defined as the inverse of xexp x and we solve kepler s equation in all cases our predictions for the theoretical errors are in excellent agreement with our numerical results and are smaller than what could be expected from the general error analysis of spline interpolation by many orders of magnitude namely by an astonishing 3 10 22 factor for the computation of w in the range w y 0 10 and by a factor 2 10 4 for kepler s problem in our tests this scheme is much faster than newton raphson method by a factor in the range 10 4 10 3 for the execution time in the examples when the values of the inverse function over an entire interval or for a large number of points are requested for kepler s equation and tolerance 10 6 rad the algorithm outperforms newton s method for all values of the number of points n 2

problems with relaxed shifted controls and the pointwise maximum principle abstract in previous work j warga a proper relaxation of controls with variable shifts j math anal appl 196 1995 783 793 it has been shown that controls with k variable and unrelated time shifts form a set whose closure in the weak star topology of l 1 t c k 1 is a convex and compact space this provides a basis for establishing the existence of an optimal solution of the shifted control problem by essentially copying the assumptions and arguments used for control problems without shifted controls similarly the necessary optimality conditions for original and relaxed minima that define an extremal previously studied for differential and functional equations apply directly to such problems with shifted controls except that the maximum principle follows directly in integral form only in the present note we first establish the form of dependence of the relaxed shifted controls on the shifts and then use the integral form of the maximum principle to derive a corresponding pointwise maximum principle for optimal ordinary or relaxed shifted controls acting through ordinary differential and certain functional differential equations

input design in worst case system identification using binary sensors this technical note addresses system identification using binary valued sensors in a worst case set membership setting the main contribution is the solution of the optimal input design problem for identification of scalar gains which is instrumental to the construction of suboptimal input signals for identification of fir models of arbitrary order two different cost functions are considered for input design the maximum parametric identification error and the relative uncertainty reduction with respect to the minimum achievable error it is shown that in the latter case the solution enjoys the property of being independent of the length of the identification experiment and as such it can be implemented as an optimal recursive procedure over a time interval of arbitrary length

kepler equation and accelerated newton method we study the efficiency of the accelerated newton method garlach siam rev 36 1994 272 276 for several orders of convergence versus danby s method for the resolution of kepler s equation we find that the cited method of order three is competitive with danby s method and the classical newton s method we also generalize the accelerated newton method for the resolution of system of algebraic equations obtaining a formula of order three and a proof of its convergence its application to several examples shows that its efficiency is greater than newton s method

the instability of optimal control problems to time delay this paper considers a general optimal control problem with a small time delay delta in both the state and control the objective is to find a correct way to neglect the delay or in other words to construct a well posed instantaneous approximation of the delayed problem the natural way to do so is by merely ignoring the delay in the model it is shown that this way is almost always incorrect it gives rise to an error that does not vanish as delta to 0 and so results in an ill posed model a proper approximation of the delayed problem is offered

asymptotically efficient identification of fir systems with quantized observations and general quantized inputs this paper introduces identification algorithms for finite impulse response systems under quantized output observations and general quantized inputs while asymptotically efficient algorithms for quantized identification under periodic inputs are available their counterpart under general inputs has encountered technical difficulties and evaded satisfactory resolutions under quantized inputs this paper resolves this issue with constructive solutions a two step algorithm is developed which demonstrates desired convergence properties including strong convergence mean square convergence convergence rates asymptotic normality and asymptotical efficiency in terms of the cramer rao lower bound some essential conditions on input excitation are derived that ensure identifiability and convergence it is shown that by a suitable selection of the algorithm s weighting matrix the estimates become asymptotically efficient the strong and mean square convergence rates are obtained optimal input design is given also the joint identification of noise distribution functions and system parameters is investigated numerical examples are included to illustrate the main results of this paper

accelerating generators of iterative methods for finding multiple roots of nonlinear equations two accelerating generators that produce iterative root finding methods of arbitrary order of convergence are presented primary attention is paid to algorithms for finding multiple roots of nonlinear functions and in particular of algebraic polynomials first two classes of algorithms for solving nonlinear equations are studied those with a known order of multiplicity and others with no information on multiplicity we also demonstrate the acceleration of iterative methods for the simultaneous approximations of multiple roots of algebraic polynomials a discussion about the computational efficiency of the root solvers considered and three numerical examples are given

means based modifications of newton s method for solving nonlinear equations in this paper we consider a family of six sets of means based modifications of newton s method for solving nonlinear equations each set is a parametric class of methods some well known methods belong to our family for example the arithmetic mean newton s method s weerakoon t g i fernando a variant of newton s method with accelerated third order convergence appl math lett 13 2000 87 93 the harmonic mean newton s method a y ozban some new variants of newton s method appl math lett 17 2004 677 682 the geometric mean newton s method t lukic n m ralevic geometric mean newton s method for simple and multiple roots appl math lett 21 2008 30 36 the power mean newton s method x zhou a class of newton s methods with third order convergence appl math lett 20 2007 1026 1030 the convergence analysis shows third order of our family comparison of the family members shows that there are no big differences between them twelve numerical examples were tested and two characteristic ones are presented

a class of iterative methods with third order convergence to solve nonlinear equations algebraic and differential equations generally co build mathematical models either lack or intractability of their analytical solution often forces workers to resort to an iterative method and face the likely challenges of slow convergence non convergence or even divergence this manuscript presents a novel class of third order iterative techniques in the form of x k 1 g u x k x k f x k u x k to solve a nonlinear equation f with the aid of a weight function u the class currently contains an invert and average g k i a an average and invert g k a i and an invert and exponentiate g k e branch each branch has several members some of which embed second order newton s g n third order chebychev s g c or halley s g h solvers class members surpassed stand alone applications of these three well known methods other methods are also permitted as auxiliaries provided they are at least of second order asymptotic convergence constants are calculated assignment of class parameters to non members carries them to a common basis for comparison this research also generated a one step solver that is usable for post priori analysis trouble shooting and comparison

a first overview on the real dynamics of chebyshev s method in this paper we explore some properties of the well known root finding chebyshev s method applied to polynomials defined on the real field in particular we are interested in showing the existence of extraneous fixed points that is fixed points of the iteration map that are not root of the considered polynomial the existence of such extraneous fixed points is a specific property in the dynamical study of chebyshev s method that does not happen in other known iterative methods as newton s or halley s methods in addition in this work we consider other dynamical aspects of the method as for instance the feigenbaum bifurcation diagrams or the parameter plane

identification of linear models from quantized data a midpoint projection approach in this article we consider the identification of linear models from quantized output data we develop a variational approximation of the likelihood function which allows us to find variationally optimal approximations of the maximum likelihood and maximum a posteriori estimates we show that these estimates are obtained by projecting the midpoint in the quantization interval of each output measurement onto the column space of the input regression matrix interpreting the quantized output as a random variable we derive its moments for generic noise distributions for the case of gaussian noise and gaussian independent identically distributed input we give an analytical characterization of the bias which we use to build a bias compensation scheme that leads to consistent estimates

on the estimation of transfer functions regularizations and gaussian processes revisited intrigued by some recent results on impulse response estimation by kernel and nonparametric techniques we revisit the old problem of transfer function estimation from input output measurements we formulate a classical regularization approach focused on finite impulse response fir models and find that regularization is necessary to cope with the high variance problem this basic regularized least squares approach is then a focal point for interpreting other techniques like bayesian inference and gaussian process regression the main issue is how to determine a suitable regularization matrix bayesian prior or kernel several regularization matrices are provided and numerically evaluated on a data bank of test systems and data sets our findings based on the data bank are as follows the classical regularization approach with carefully chosen regularization matrices shows slightly better accuracy and clearly better robustness in estimating the impulse response than the standard approach the prediction error method maximum likelihood pem ml approach if the goal is to estimate a model of given order as well as possible a low order model is often better estimated by the pem ml approach and a higher order model is often better estimated by model reduction on a high order regularized fir model estimated with careful regularization moreover an optimal regularization matrix that minimizes the mean square error matrix is derived and studied the importance of this result lies in that it gives the theoretical upper bound on the accuracy that can be achieved for this classical regularization approach

pseudocomposition a technique to design predictor corrector methods for systems of nonlinear equations abstract a new technique for designing iterative methods for solving nonlinear systems is presented this procedure called pseudocomposition uses a known method as a predictor and the gaussian quadrature as a corrector the order of convergence of the resulting scheme depends among other factors on the order of the last two steps of the predictor we also introduce a new iterative algorithm of order six and apply the mentioned technique to generate a new method of order ten finally some numerical test are shown

a basic family of iteration functions for polynomial root finding and its characterizations abstract let p x be a polynomial of degree n 2 with coefficients in a subfield k of the complex numbers for each natural number m 2 let lm x be the m m lower triangular matrix whose diagonal entries are p x and for each j 1 m 1 its jth subdiagonal entries are p j x j for i 1 2 let lmi x be the matrix obtained from lm x by deleting its first i rows and its last i columns l1 1 x 1 then the function bm x x p x det l m 1 1 x det l m 1 x defines a fixed point iteration function having mth order convergence rate for simple roots of p x for m 2 and 3 bm x coincides with newton s and halley s respectively the function bm x is a member of s m m n 2 where for any m m s m m is the set of all rational iteration functions g x k x such that for all roots of p x then g x i mm i x x i with i x k x and well defined at any simple root given g s m m and a simple root of p x gi 0 i 1 m 1 and the asymptotic constant of convergence of the corresponding fixed point iteration is m 1 g m m for bm x we obtain m 1 m det l m 1 2 det l m 1 if all roots of p x are simple bm x is the unique member of s m m n 2 by making use of the identity 0 i 0 n p i x i x i we arrive at two recursive formulas for constructing iteration functions within the s m m family in particular the family of bm x can be generated using one of these formulas moreover the other formula gives a simple scheme for constructing a family of iteration functions credited to euler as well as schroder whose mth order member belongs to s m mn m 2 the iteration functions within s m m can be extended to any arbitrary smooth function f with the uniform replacement of p j with f j in g as well as in m

identification of fir systems based on quantized output measurements a quadratic programming based method this technical note proposes a quadratic programming qp based method for identification of finite impulse response fir dynamic systems from quantized or binary data the main idea of the proposed method is to reformulate this identification problem usually viewed as a nonlinear estimation problem with discontinuous nonlinearities in the form of a standard qp problem which is a convex optimization problem and can be solved efficiently the so called complete input conditions to ensure the unique solution of the qp problem are developed and the consistency of the estimated parameters is established under the complete input conditions numerical examples demonstrate the effectiveness of the proposed method

a one parameter family of locally quartically convergent zero finding methods a one parameter family of iteration functions for finding simple and multiple zeros of analytic functions is derived the family includes as a special case traub s quartic square root method and as limiting cases the kiss method of order 4 the halley and the newton methods all the methods of the family are locally quartically convergent for a simple or multiple zero with known multiplicity the asymptotic error constants for the methods of the family are given the decreasing ratio at infinity of iteration functions is discussed the optimum parameter of the family for polynomials is given

extraneous fixed points basin boundaries and chaotic dynamics for schro der and ko nig rational iteration functions the schroder and konig iteration schemes to find the zeros of a polynomial functiong z represent generalizations of newton s method in both schemes iteration functionsf m z are constructed so that sequencesz n 1 f m z n converge locally to a rootz ofg z aso z n z m it is well known that attractive cycles other than the zerosz may exist for newton s method m 2 asm increases the iteration functions add extraneous fixed points and cycles whether attractive or repulsive they affect the julia set basin boundaries the konig functionsk m z appear to minimize such perturbations in the case of two roots e g g z z 2 1 cayley s classical result for the basins of attraction of newton s method is extended for allk m z the existence of chaotic z n sequences is also demonstrated for these iteration methods

identification of sparse fir systems using a general quantisation scheme this paper presents an identification scheme for sparse fir systems with quantised data we consider a general quantisation scheme which includes the commonly deployed static quantiser as a special case to tackle the sparsity issue we utilise a bayesian approach where an l1 a priori distribution for the parameters is used as a mechanism to promote sparsity the general framework used to solve the problem is maximum likelihood ml the ml problem is solved by using a generalised expectation maximisation algorithm

fixed point and newton s methods for solving a nonlinear equation from linear to high order convergence in this paper we revisit the necessary and sufficient conditions for linear and high order convergence of fixed point and newton s methods based on these conditions we extend schroder s process of the first kind to increase the order of convergence of the fixed point method we also obtain two processes to increase the order of convergence of newton s method one of which is schroder s process of the second kind for which several forms are also presented a link between schroder s two processes is given examples and numerical experiments are included

third order methods from quadrature formulae for solving systems of nonlinear equations we extend to p dimensional problems a modification of the newton method based on quadrature formulas of order at least one which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given these new methods may be more efficient then other third order methods as they do not require the use of the second order frechet derivative

accurate computation of weights in classical gauss christoffel quadrature rules for many classical gauss christoffel quadrature rules there does not exist a method which guarantees a uniform level of accuracy for the gaussian quadrature weights at all quadrature nodes unless the nodes are known exactly more disturbing some algebraic expressions for these weights exhibit an excessive sensitivity to even the smallest perturbations in the node location this sensitivity rapidly increases with high order quadrature rules current uses of very high order quadratures are common with the advent of more powerful computers and a loss of accuracy in the weights has become a problem and must be addressed a simple but efficient and general method for improving the accuracy of the computation of the quadrature weights is proposed it ensures a high level of accuracy for these weights even though the nodes may carry a significant large error in addition a highly efficient root finding iterative technique with superlinear converging rates for computing the nodes is developed it uses solely the quadrature polynomials and their first derivatives a comparison of this method with the eigenvalue method of golub and welsh implemented in most standard software libraries is made the proposed method outperforms the latter from the point of view of both accuracy and efficiency the legendre lobatto radau hermite and laguerre quadrature rules are examined

the linear quadratic optimal control problem with delays in state and control variables a state space approach this paper generalizes the results of vinter and kwong to a control operator with a more general delay structure than the one in a ichikawa and to delay systems with both finite and infinite memories the linear quadratic optimal control problem over both finite and infinite time horizons is treated several forms of associated abstract riccati differential and algebraic equations are presented the properties of the kernel of the solution are studied and a coupled set of matrix ordinary and partial differential equations is obtained

symmetries of the julia sets of konig s methods for polynomials abstract let r be the group of symmetries of the julia set of a rational map r and k f n be the konig s method for polynomial f of order n 2 for any given integer n 2 we prove that if f is in normal form then f is a subgroup of k f n we also obtain a necessary and sufficient condition for the julia set of k f n to be a line

identification of wiener systems with clipped observations in this paper we consider the parametric version of wiener systems where both the linear and nonlinear parts are identified with clipped observations in the presence of internal and external noises also the static functions are allowed noninvertible we propose a classification based support vector machine svm and formulate the identification problem as a convex optimization the solution to the optimization problem converges to the true parameters of the linear system if it is an finite impulse response fir system even though clipping reduces a great deal of information about the system characteristics in identifying a wiener system with a stable infinite impulse response iir system an fir system is used to approximate it and the problem is converted to identifying the fir system together with solving a set of nonlinear equations this leads to biased estimates of parameters in the iir system while the bias can be controlled by choosing the order of the approximated fir system

newton s method and generation of a determinantal family of iteration functions abstract it is well known that halley s method can be obtained by applying newton s method to the function f f gerlach siam rev 36 1994 272 276 gives a generalization of this approach and for each m 2 recursively defines an iteration function gm x having order m kalantari et al j comput appl math 80 1997 209 226 and kalantari technical report dcs tr 328 department of computer science rutgers university new brunswick nj 1997 derive and characterize a determinantal family of iteration functions called the basic family bm x m 2 in this paper we prove gm x bm x on the one hand this implies that gm x enjoys the previously derived properties of bm x i e the closed formula its efficient computation an expansion formula which gives its precise asymptotic constant as well as its multipoint versions on the other this gives a new insight on the basic family and newton s method

input design in worst case system identification with quantized measurements this paper addresses the problem of set membership system identification with quantized measurements following the work developed for binary measurements the problem of optimal input design with multiple sensor thresholds is tackled for a fir model of order n the problem is decomposed into n static gain problems the one step optimal input problem is solved both for equispaced and generic sensor threshold distribution moreover the n step optimal input problem for the case of equispaced thresholds is addressed and a solution is provided under a suitable assumption on the sensor range and resolution the obtained results allow us to construct an upper bound on the time complexity of the fir identification problem for the case of equispaced thresholds numerical application examples are reported to show the effectiveness of the proposed algorithms

statistical results for system identification based on quantized observations system identification based on quantized observations requires either approximations of the quantization noise leading to suboptimal algorithms or dedicated algorithms tailored to the quantization noise properties this contribution studies fundamental issues in estimation that relate directly to the core methods in system identification as a first contribution results from statistical quantization theory are surveyed and applied to both moment calculations mean variance etc and the likelihood function of the measured signal in particular the role of adding dithering noise at the sensor is studied the overall message is that tailored dithering noise can considerably simplify the derivation of optimal estimators the price for this is a decreased signal to noise ratio and a second contribution is a detailed study of these effects in terms of the cramer rao lower bound the common additive uniform noise approximation of quantization is discussed compared and interpreted in light of the suggested approaches

recursive projection algorithm on fir system identification with binary valued observations this paper studies the identification of finite impulse response fir systems with binary valued observations combining with the stochastic gradient algorithm and statistical property of the system noise a recursive projection algorithm is proposed to estimate the unknown system parameters under some mild conditions on the a priori knowledge of the unknown parameters and inputs the algorithm is proved to be convergent in the almost sure and mean square sense furthermore the almost sure and mean square convergence rates of estimation errors are also obtained a numerical example is given to demonstrate the effectiveness of the algorithm and the main results obtained

on schroder s families of root finding methods schroder s methods of the first and second kind for solving a nonlinear equation f x 0 originally derived in 1870 are of great importance in the theory and practice of iteration processes they were rediscovered several times and expressed in different forms during the last 130 years it was proved in the paper of petkovic and herceg 1999 7 that even seven families of iteration methods for solving nonlinear equations are mutually equivalent in this paper we show that these families are also equivalent to another four families of iteration methods and find that all of them have the origin in schroder s generalized method of the second kind presented in 1870 in the continuation we consider smale s open problem from 1994 about possible link between schroder s methods of the first and second kind and state the link in a simple way

on the roles of minimization and linearization in least squares finite element models of nonlinear boundary value problems in this paper we examine the roles of minimization and linearization in the least squares finite element formulations of nonlinear boundary values problems the least squares principle is based upon the minimization of the least squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations in the present case we consider l 2 norms since the least squares method is independent of the discretization procedure and the solution scheme the least squares principle suggests that minimization should be performed prior to linearization where linearization is employed in the context of either the picard or newton iterative solution procedures however in the least squares finite element analysis of nonlinear boundary value problems it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations the main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged the assessment is performed mathematically through an examination of the variational setting for the least squares formulation of an abstract nonlinear boundary value problem and also computationally through the numerical simulation of the least squares finite element solutions of both a nonlinear form of the poisson equation and also the incompressible navier stokes equations the assessment suggests that although the least squares principle indicates that minimization should be performed prior to linearization such an approach is often impractical and not necessary

linear approximations of nonlinear fir systems for separable input processes nonlinear systems can be approximated by linear time invariant lti models in many ways here lti models that are optimal approximations in the mean square error sense are analyzed a necessary and sufficient condition on the input signal for the optimal lti approximation of an arbitrary nonlinear finite impulse response nfir system to be a linear finite impulse response fir model is presented this condition says that the input should be separable of a certain order i e that certain conditional expectations should be linear for the special case of gaussian input signals this condition is closely related to a generalized version of bussgang s classic theorem about static nonlinearities it is shown that this generalized theorem can be used for structure identification and for the identification of generalized wiener hammerstein systems

a weighted least squares approach to parameter estimation problems based on binary measurements we present a new approach to parameter estimation problems based on binary measurements motivated by the need to add integrated low cost self test features to microfabricated devices this approach is based on the use of original weighted least squares criteria as opposed to other existing methods it requires no dithering signal and it does not rely on an approximation of the quantizer in this technical note we focus on a simple choice for the weights and establish some asymptotical properties of the corresponding criterion to achieve this the assumption that the quantizer s input is gaussian and centered is made in this context we prove that the proposed criterion is locally convex and that it is possible to use a simple gradient descent to find a consistent estimate of the unknown system parameters regardless of the presence of measurement noise at the quantizer s input

letter to the editor comments on some recent methods for the simultaneous determination of polynomial zeros in this note we give some comments on the recent results concerning a simultaneous method of the fourth order for finding complex zeros in circular interval arithmetic the main discussion is directed to a rediscovered iterative formula and its modification presented recently in sun and kosmol j comput appl math 130 2001 293 the presented comments include some critical parts of the papers jpn j indust appl math 15 1998 295 and sun and kosmol 2001 which treat the same subject

joint identification of plant rational models and noise distribution functions using binary valued observations system identification of plants with binary valued output observations is of importance in understanding modeling capability and limitations for systems with limited sensor information establishing relationships between communication resource limitations and identification complexity and studying sensor networks this paper resolves two issues arising in such system identification problems first regression structures for identifying a rational model contain non smooth nonlinearities leading to a difficult nonlinear filtering problem by introducing a two step identification procedure that employs periodic signals empirical measures and identifiability features rational models can be identified without resorting to complicated nonlinear searching algorithms second by formulating a joint identification problem we are able to accommodate scenarios in which noise distribution functions are unknown convergence of parameter estimates is established recursive algorithms for joint identification and their key properties are further developed

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

a new modified king werner method for solving nonlinear equations in this paper a new method for solving nonlinear equations f x 0 is presented analysis of the convergence shows that the asymptotic convergence order of this method is 1 3 some numerical results are given to demonstrate its efficiency

letter to the editor accelerated convergence in newton s method for approximating square roots in this paper we construct a modification of newton s method to accelerate the convergence of this method to the approximation of the positive square root of a positive real number from this modification we can define a new iterative process with prefixed order q n q 2

a proper relaxation of shifted and delayed controls abstract we consider optimal control problems defined by functional integral equations with variable shifts in the controls and with the shifted controls nonadditively coupled as a special case we obtain differential equations with delays and or advances in both the state and the control functions we prove that if the original controls are measurable functions u with values in a compact metric space then the k 1 tuples u u u h 1 u h k of shifted controls have relaxed limits that form a compact and convex set this enables us to prove an existence theorem to approximate any relaxed solution with original solutions and to derive necessary conditions generalizeing pontryagin s maximum principle in integral form even for nonsmooth problems we also discuss the dependence on the shifts hj of the set of original controls u of the set of their relaxed limits and of the set of the corresponding solutions of the functional integral equation our results include the proof that the d model of rosenblueth and vinter is proper

relaxation procedures for time delay systems abstract despite the attention that optimal control problems involving time delays have received over the last two decades little attention has been paid to the important question of how such problems should be relaxed in order to assure existence of minimizers recently warga has proposed a relaxation procedure for fully nonlinear problems with delays in the state and control variables and showed that the resulting relaxed problem has a solution nevertheless the effect of this relaxation may reduce the infimum cost as we show through an example which belongs to a class of problems with commensurate delays i e the quotient of any two delays is rational for problems of this nature we provide a new relaxation procedure for which the extension is proper i e the infimum costs coincide we also present an abstract relaxation procedure applicable to problems with possibly noncommensurate delays

third order modifications of newton s method based on stolarsky and gini means in this paper we consider third order modifications of newton s method for solving nonlinear equations considered methods are based on stolarsky and gini means stolarsky 1975 13 stolarsky 1980 14 chen 2008 12 and depend on two parameters some known methods are special cases of our methods for example the power mean newton s method zhou 2007 10 the arithmetic mean newton s method weerakoon and fernando 2000 5 the harmonic mean newton s method ozban 2004 8 and the geometric mean newton s method lukic and ralevic 2008 9 third order convergence of the considered methods is proved and corresponding asymptotic error constants are expressed in terms of two parameters numerical examples obtained using mathematica with high precision arithmetic support the theoretical results some numerical tests were performed and it was shown that our methods yield better numerical results i e a smaller error x email protected when compared to halley euler hansen patrick ostrowski and inverse interpolation methods