a new family of extended gauss quadratures with an interior interval constraint starting from two sequences ga c n and gd b n of ordinary gauss quadrature formulae with an orthogonality measure d on the open intervals a c and d b respectively we construct a new sequence ga b e n of extended gaussian quadrature formulae for d on a b which is based on some preassigned points the nodes of ga c n gd b n and the e n zeros contained in c d of a nonclassical orthogonal polynomial on a b with respect to a linear functional the principal result gives explicit formulae relating these polynomials and shows how their recurrence coefficients in the three term recurrence formulae are related thus a new class of gaussian quadratures having some nodes contained in a given interior interval can be computed directly by standard software for ordinary gauss quadrature formulae

a stable gauss kronrod algorithm for cauchy principal value integrals abstract it is shown that the ratio of the precision of the stable kronrod extension to the precision of the stable version of the gauss rule for cauchy principal value integrals is approximately the same as the ratio of the precision of the kronrod extension to that of the gauss rule for ordinary integrals

on the numerical resolution of the generalized airfoil equation with possio kernel in this paper we consider the so called generalized airfoil singular integral equation with a smooth input function and present ad hoc quadrature rules to compute efficiently the elements of collocation and galerkin matrices

on orthogonal polynomials for certain nondefinite linear functionals abstract we consider the non definite linear functionals l n f r w x f n x d x and prove the nonexistence of orthogonal polynomials with all zeros real in several cases the proofs are based on the connection with moment preserving spline approximation

suboptimal kronrod extension formulae for numerical quadrature we consider cases where the stieltjes polynomial associated with a gaussian quadrature formula has complex zeros in such cases a kronrod extension of the gaussian rule does not exist a method is described for modifying the stieltjes polynomial so that the resulting polynomial has no complex zeros the modification is performed in such a way that the kronrod type extension rule resulting from the addition of then 1 zeros of the modified stieltjes polynomial to the original knots of the gaussian rule has only slightly lower degree of precision than normally achieved when the kronrod extension rule exists as examples of the use of the method we present some new formulae extending the classical gauss hermite quadrature rules we comment on the limited success of the method in extending gauss laguerre rules and show that several modified extensions of the gauss gegenbauer formulae exist in cases where the standard kronrod extension does not

kronrod extensions with multiple nodes of quadrature formulas for fourier coefficients abstract we continue with analyzing quadrature formulas of high degree of precision for computing the fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials started recently by bojanov and petrova quadrature formulae for fourier coefficients j comput appl math 231 2009 378 391 and we extend their results construction of new gaussian quadrature formulas for the fourier coefficients of a function based on the values of the function and its derivatives is considered we prove the existence and uniqueness of kronrod extensions with multiple nodes of standard gaussian quadrature formulas with multiple nodes for several weight functions in order to construct some new generalizations of quadrature formulas for the fourier coefficients for the quadrature formulas for the fourier coefficients based on the zeros of the corresponding orthogonal polynomials we construct kronrod extensions with multiple nodes and highest algebraic degree of precision for this very desirable kind of extension there do not exist any results in the theory of standard quadrature formulas

stieltjes polynomials and functions of the second kind abstract in this paper we discuss known and new facts on the asymptotic behaviour of the stieltjes polynomials and the close relation to the asymptotic behaviour of the boundary values of the functions of the second kind on 1 1 most of the results are derived by new methods of proof where mainly the theory of real orthogonal polynomials is used

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

on the remainder term of gauss radau quadratures for analytic functions for analytic functions the remainder term of gauss radau quadrature formulae can be represented as a contour integral with a complex kernel we study the kernel on elliptic contours with foci at the points 1 and a sum of semi axes r 1 for the chebyshev weight function of the second kind starting from explicit expressions of the corresponding kernels the location of their maximum modulus on ellipses is determined the corresponding gautschi s conjecture from on the remainder term for analytic functions of gauss lobatto and gauss radau quadratures rocky mountain j math 21 1991 209 226 is proved

stopping functionals for gaussian quadrature formulas abstract gaussian formulas are among the most often used quadrature formulas in practice in this survey an overview is given on stopping functionals for gaussian formulas which are of the same type as quadrature formulas i e linear combinations of function evaluations in particular methods based on extended formulas like the important gauss kronrod and patterson schemes and methods which are based on gaussian nodes are presented and compared

gauss kronrod integration rules for cauchy principal value integrals kronrod extensions to two classes of gauss and lobatto integration rules for the evaluation of cauchy principal value integrals are derived since in one frequently occurring case the kronrod extension involves evaluating the derivative of the integrand a new extension is introduced using n 2 points which requires only values of the integrand however this new rule does not exist for all n and when it does several significant figures are lost in its use 1 introduction in this paper we shall consider kronrod extensions ke to integration rules based on gauss and lobatto points for the evaluation of cauchy principal value cpv integrals of the form fix vyx 1 since the existence theory of ke s for regular integrals insures such extensions for only certain classes of weight functions w x we shall restrict our attention here to the most important class namely w x 1 x2y x 2 where 0 p 2 in the gauss case and p 1 in the lobatto case 9 the only other relevant weight functions are the jacobi weight functions w x 1 x a l x with a i s or s y a f and w x x2 i rx2 oo r 1 5 for our case w x 1 x2 m 1 2 the corresponding orthogonal polynomials are the gegenbauer polynomials c x usually written c x which have the following normalization 11 p 174 2 w x c x cmil x dx 8 mhnil where 3 hn i ft n 2p r p n u n r p r 2p

δ entropy definition properties and applications in system identification with quantized data recently the minimum error entropy criterion an information theoretic alternative to the traditional mean square error criterion has been successfully used in the contexts of machine learning and signal processing for system identification however the mee criterion will be no longer suitable if the training data are discrete valued since minimizing error s discrete entropy cannot constrain error s dispersion in this paper to make the mee criterion suitable for the discrete valued data cases we give a new entropy definition for the discrete random variables i e the d entropy based on riemann sums for finite size partitions a probability weighted formula is established to calculate the average partition this new entropy retains some important properties of the differential entropy and reduces to discrete entropy under certain conditions unlike discrete entropy the d entropy is sensitive to the dynamic range of the data and can be used as a superior optimality criterion in system identification problems also we present a plug in estimate of d entropy analyze its asymptotic behavior and explore the links to the kernel based and m spacing based estimates for differential entropy finally the d entropy criterion is applied in system identification with coarsely quantized input output data to search for the optimum parameter set monte carlo simulations demonstrate the performance improvement that may be achieved with the d entropy criterion

stieltjes polynomials and related quadrature rules in this paper we present a survey of theoretical results some of them new and numerical evidence of others concerning stieltjes polynomials kronrod schemes and their generalizations applied to integrals with a classical nonnegative weight function the use of these schemes in automatic integration routines is briefly outlined

stieltjes polynomials and lagrange interpolation bounds are proved for the stieltjes polynomial e n 1 and lower bounds are proved for the distances of consecutive zeros of the stieltjes polynomials and the legendre polynomials p n this sharpens a known interlacing result of szego as a byproduct bounds are obtained for the geronimus polynomials g n applying these results convergence theorems are proved for the lagrange interpolation process with respect to the zeros of e n 1 and for the extended lagrange interpolation process with respect to the zeros of p n e n 1 in the uniform and weighted l p norms the corresponding lebesgue constants are of optimal order

on the minimum entropy of a mixture of unimodal and symmetric distributions progressive encoding of a signal generally involves an estimation step designed to reduce the entropy of the residual of an observation over the entropy of the observation itself oftentimes the conditional distributions of an observation given already encoded observations are well fit within a class of symmetric and unimodal distributions e g the two sided geometric distributions in images of natural scenes or symmetric paretian distributions in models of financial data it is common practice to choose an estimator that centers or aligns the modes of the conditional distributions since it is common sense that this will minimize the entropy and hence the coding cost of the residuals but with the exception of a special case there has been no rigorous proof here we prove that the entropy of an arbitrary mixture of symmetric and unimodal distributions is minimized by aligning the modes the result generalizes to unimodal and rotation invariant distributions in rn we illustrate the result through some experiments with natural images

marcinkiewicz inequalities based on stieltjes zeros the authors find necessary and sufficient conditions for gdt weighted marcinkiewicz inequalities based at stieltjes zeros

modified optimal quadrature extensions optimal extensions of quadrature rules are of importance in the construction of automatic integrators but many sequences fail to exist in usable form the paper considers some techniques for overcoming the problem of inextensibility with a minimal effect on integrating efficiency a modification to the extension procedure proposed recently by begumisa and robinson is shown to be just a special case of the standard theory for quadrature extension some illustrative examples are included

on polynomials orthogonal on a circular arc abstract polynomials rk orthogonal on a circular arc with respect to the complex inner product f g f1 g1 w1 d where 0 1 2 and for f z the function f1 is defined by f1 f ir ei r2 1 1 2 r tan have been introduced by de bruin 1990 in this paper the functions of the second kind as well as the corresponding associated polynomials are introduced some recurrence relations and identities of christoffel darboux type are proved also the corresponding stieltjes polynomials which are orthogonal to all lower degree polynomials with respect to a complex measure on r z c z ir ei r2 1 1 2 tan r are investigated a class of polynomials orthogonal on a symmetrical circular arc in the down half plane is also introduced finally in the jacobi case w z 1 z 1 z 1 a linear second order differential equation for rn z is obtained

gauss kronrod quadrature formulae for weight functions of bernstein szego type ii we study the kronrod extensions of gaussian quadrature rules whose weight functions on 1 l consist of any one of the four chebyshev weights divided by an arbitrary quadratic polynomial that remains positive on 1 11 we show that in almost all cases these extended gauss kronrod quadrature rules have all the desirable properties kronrod nodes interlacing with gauss nodes all nodes contained in 1 11 and all weights positive and representable by semiexplicit formulas exceptions to these properties occur only for small values of n the number of gauss nodes namely n 3 and are carefully identified the precise degree of exactness of each of these gauss kronrod formulae is determined and shown to grow like 4n rather than 3n as is normally the case our findings are the result of a detailed analysis of the underlying orthogonal polynomials and stieltjes polynomials the paper concludes with a study of the limit case of a linear divisor polynomial in the weight function

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

some new formulae for the stieltjes polynomials relative to classical weight functions using known limit formulae which connect the gegenbauer and jacobi polynomials with the hermite and laguerre polynomials respectively similar formulae are derived for the corresponding stieltjes polynomials the usefulness of these formulae is demonstrated by employing them to show the nonexistence of gauss kronrod quadrature formulae for the gegenbauer and jacobi weight functions with real nodes and positive weights

on stratified extensions of gauss laguerre and gauss hermite quadrature formulas for the practical estimation of the error of gauss laguerre and gauss hermite quadrature formulas it is well known that real positive kronrod formulas do not exist among the alternatives which are available in the literature the stratified rules introduced by laurie are of particular interest in this note we determine the degree optimal stratified extensions for the gauss laguerre and gauss hermite formulas and we investigate their properties

stieltjes and geronimus type polynomials abstract this work is an extension to the formal case of a previous work by monegato 5 a stieltjes type polynomial is a polynomial of degree n 1 e n 1 orthogonal with respect to the functional s c i c p n x x i i 0 1 2 s where c is a functional defined by the series f t i 0 c i t i e n 1 is of important use in the estimation of the error in pade approximation an iteration of the construction of e n 1 is attempted sections 1 2 3 4 in the last section we study the properties of the polynomial s n associated with e n 1 with respect to another functional c s n is called a geronimus type polynomial it is shown that s n verifies the system c p n s n g k 0 for k 1 n where the g k s are orthogonal with respect to c

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

stieltjes polynomials and related quadrature formulae for a class of weight functions consider a nonnegative measure d with support in the interval a b such that the respective orthogonal polynomials above a specific index l satisfy a three term recurrence relation with constant coefficients we show that the corresponding stieltjes polynomials above the index 2l 1 have a very simple and useful representation in terms of the orthogonal polynomials as a result of this the gauss kronrod quadrature formulae for d have all the desirable properties namely the interlacing of nodes their inclusion in the closed interval a b under an additional assumption on d and the positivity of all weights furthermore the interpolatory quadrature formulae based on the zeros of the stieltjes polynomials have positive weights and both of these quadrature formulae have elevated degrees of exactness

minimum entropy of error estimation for discrete random variables the principle of minimum entropy of error estimation meee is formulated for discrete random variables in the case when the random variable to be estimated is binary we show that the meee is given by a neyman pearson type strictly monotonous test in addition the asymptotic behavior of the error probabilities is proved to be equivalent to that of the bayesian test

uncertainty quantification in discrete fracture network models we consider flows in fractured media described by discrete fracture network dfn models we perform an uncertainty quantification analysis assuming the fractures transmissivity coefficients to be random variables two probability distributions log uniform and log normal are used within different laws that express the coefficients in terms of a family of independent stochastic variables truncated karhunenlove expansions provide instances of such laws the approximate computation of quantities of interest such as mean value and variance for outgoing fluxes is based on a stochastic collocation approach that uses suitable sparse grids in the range of the stochastic variables whose number defines the stochastic dimension of the problem this produces a non intrusive computational method in which the dfn flow solver is applied as a black box a very fast error decay related to the analytical dependence of the observed quantities upon the stochastic variables is obtained in the low dimensional cases using isotropic sparse grids comparisons with monte carlo results show a clear gain in efficiency for the proposed method for increasing dimensions attained via successive truncations of karhunenlove expansions results are still good although the rates of convergence are progressively reduced resorting to suitably tuned anisotropic grids is an effective way to contrast such curse of dimensionality in the explored range of dimensions the resulting convergence histories are nearly independent of the dimension

asymptotic behaviour of stieltjes polynomials for ultraspherical weight functions abstract for the ultraspherical weight functions w x 1 x 2 1 2 an asymptotic representation of the stieltjes polynomials is proved for 1 n n the main result is applied to prove positivity results for kronrod extensions of gauss and lobatto quadrature formulae

modified stieltjes polynomials and gauss kronrod quadrature rules modified stieltjes polynomials are defined and used to construct suboptimal extensions of gaussian rules with one or two degrees less of polynomial exactness than the corresponding kronrod extension we prove that for wide classes of weight functions and a sufficiently large number of nodes the extended quadratures have positive weights and simple nodes on the interval 1 1 the classes of weight functions considered complement those for which the gauss kronrod rule is known to exist also strong asymptotic representations on the whole interval 1 1 are given for the modified stieltjes polynomials which prove that they behave asymptotically as orthogonal polynomials finally we provide some numerical examples

large deviations and maximum entropy principle for interacting random fields on mathbb z d we present a new approach to the principle of large deviations for the empirical field of a gibbsian random field on the integer lattice zd this approach has two main features first we can replace the traditional weak topology by the finer topology of convergence of cylinder probabilities and thus obtain estimates which are finer and more widely applicable second we obtain as an immediate consequence a limit theorem for conditional distributions under conditions on the empirical field the limits being those predicted by the maximum entropy principle this result implies a general version of the equivalence of gibbs ensembles stating that every microcanonical limiting state is a grand canonical equilibrium state we also prove a converse to the last statement and discuss some applications 0 introduction as is well known the study of the asymptotic probabilities of large fluctuations of time averages or space averages away from the mean is based on two fundamental principles the principle of large deviations and the maximum entropy principle the former provides the exact rate of exponential decay of the fluctuation probabilities whereas the latter predicts the limiting conditional distribution under the condition that the fluctuations are large it is obvious that these principles are intimately related in this paper we investigate these principles in the case of interacting random fields on the integer lattice 7zd the setup is the following first we let e 4 be any measurable space we shall assume throughout that e a is standard borel but we shall avoid making any explicit topological assumptions on e so we do not assume that e is polish next we let s zd be the d dimensional integer lattice and q f e gas the associated product space in lattice models of statistical mechanics e is called the state space or single spin space and q f the configuration space we let j s f denote the set of all probability

an algebraic study of gauss kronrod quadrature formulae for jacobi weight functions we study gauss kronrod quadrature formulae for the jacobi weight function w d t 1 t a 1 t and its special case a f a of the gegenbauer weight function we are interested in delineating regions in the a f plane resp intervals in a for which the quadrature rule has a the interlacing property i e the gauss nodes and the kronrod nodes interlace b all nodes contained in 1 1 c all weights positive d only real nodes not necessarily satisfying a and or b we determine the respective regions numerically for n 1 1 20 4 40 in the gegenbauer case and for n 1 1 10 in the jacobi case where n is the number of gauss nodes algebraic criteria in particular the vanishing of appropriate resultants and discriminants are used to determine the boundaries of the regions identifying properties a and d the regions for properties b and c are found more directly a number of conjectures are suggested by the numerical results finally the gauss kronrod formula for the weight w al 2 is obtained from the one for the weight w ca and similarly the gauss kronrod formula with an odd number of gauss nodes for the weight function w t itlv 1 t2 c is derived from the gauss kronrod formula for the weight w a l y 2

minimum entropy of error principle in estimation abstract the principle of minimum error entropy estimation as found in the work of weidemann and stear is reformulated as a problem of finding optimum locations of probability densities in a given mixture such that the resulting differential entropy is minimized new results concerning the entropy lower bound are derived continuity of the entropy and attaining the minimum entropy are proved in the case where the mixture is finite some other examples and situations in particular that of symmetric unimodal densities are studied in more detail

smolyak cubature of given polynomial degree with few nodes for increasing dimension some recent investigations see e g gerstner and griebel 5 novak and ritter 9 and 10 novak ritter and steinbauer 11 wasilkowski and wo niakowski 18 or petras 13 show that the so called smolyak algorithm applied to a cubature problem on the d dimensional cube seems to be particularly useful for smooth integrands the problem is still that the numbers of nodes grow polynomially but fast for increasing dimensions we therefore investigate how to obtain smolyak cubature formulae with a given degree of polynomial exactness and the asymptotically minimal number of nodes for increasing dimension d and obtain their characterization for a subset of smolyak formulae error bounds and numerical examples show their good behaviour for smooth integrands a modification can be applied successfully to problems of mathematical finance as indicated by a further numerical example

hermite and hermite fejer interpolation for stieltjes polynomials let x 1 x 2 1 2 and p n be the ultraspherical polynomials with respect to x then we denote by e n 1 the stieltjes polynomials with respect to x satisfying in this paper we show uniform convergence of the hermite fejer interpolation polynomials h n 1 and h 2n 1 based on the zeros of the stieltjes polynomials e n 1 and the product e n 1 p n for 0 1 and 0 a 1 2 respectively to prove these results we prove that the lebesgue constants of hermite fejer interpolation operators for the stieltjes polynomials e n 1 and the product e n 1 p n are optimal that is the lebesgue constants h n 1 0 a 1 and h 2n 1 0 a 1 2 have optimal order o 1 in the case of the hermite fejer interpolation polynomials h 2n 1 for 1 2 1 we prove weighted uniform convergence moreover we give some convergence theorems of hermite fejer and hermite interpolation polynomials for 0 1 in weighted lp norms

pointwise simultaneous convergence of extended lagrange interpolation with additional knots in numerical analysis it is important to construct interpolating polynomials approximating a given function and its derivatives simultaneously the authors define some new good interpolating matrices with many nodes close to the endpoints of the interval and also give error estimates

stieltjes polynomials and gauss kronrod quadrature for jacobi weight functions let w beta t 1 t 1 t 1 denote the jacobi weight function for 0 5 2 the stieltjes polynomial has only few real zeros and that gauss kronrod quadrature is not possible so far results of the above type have been known only for ultraspherical weight functions w lambda t big 1 t 2 big lambda 1 2 for 0 2 for 3 the impossibility of gauss kronrod quadrature has been proved by the authors recently

computation of gauss kronrod of quadrature rules recently laurie presented a new algorithm for the computation of 2n 1 point gauss kronrod quadrature rules with real nodes and positive weights this algorithm first determines a symmetric tridiagonal matrix of order 2n 1 from certain mixed moments and then computes a partial spectral factorization we describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined and thereby avoids computations that can be sensitive to perturbations our algorithm uses the consolidation phase of a divide and conquer algorithm for the symmetric tridiagonal eigenproblem we also discuss how the algorithm can be applied to compute kronrod extensions of gauss radau and gauss lobatto quadrature rules throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible numerical examples illustrate the performance of the algorithm

stieltjes polynomials and related quadrature formulae for a class of weight functions ii consider a nonnegative measure d sigma with support in the interval a b such that the respective orthogonal polynomials satisfy a three term recurrence relation with coefficients alpha n left begin array ll alpha e n hbox even alpha o n hbox odd end array right beta n beta hbox for n ge ell where alpha e alpha o beta and ell are specific constants we show that the corresponding stieltjes polynomials above the index 2 ell 1 have a very simple and useful representation in terms of the orthogonal polynomials as a result of this the gauss kronrod quadrature formula for d sigma has all the desirable properties namely the interlacing of nodes their inclusion in the closed interval a b under an additional assumption on d sigma and the positivity of all weights while the formula enjoys an elevated degree of exactness furthermore the interpolatory quadrature formula based on the zeros of the stieltjes polynomials has positive weights and also elevated degree of exactness it turns out that this formula is the anti gaussian formula for d sigma while the resulting averaged gaussian formula coincides with the gauss kronrod formula for this measure moreover we show that the only positive and even measure d sigma on a a for which the gauss kronrod formula is almost of chebyshev type i e it has almost all of its weights equal is the measure d sigma t a 2 t 2 1 2 dt

on the computing gauss kronrod quadrature formulae on discute l application de la methode de newton au calcul des formules de quadrature de gauss kronrod a partir de moments modifies et on analyse les applications non lineaires sous jacentes on determine le polynome dont les zeros sont les n uds de kronrod on traite des exemples de formules pour les integrales a singularite logarithmique ou algebrique

uniform convergence of derivatives of extended lagrange interpolation the authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm they prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation

a note on peano constants of gauss kronrod quadrature schemes abstract for the error r 2 n 1 gk of the gauss kronrod quadrature formula q 2 n 1 gk we prove the precise asymptotic value of the low order peano constants c s r 2 n 1 gk where s is fixed for increasing n the result is applied for comparisons of the gauss kronrod formula q 2 n 1 gk with the gauss formula q n g and with the gauss formula q 2 n 1 g with respect to this often used quality measure

gauss quadratures and jacobi matrices for weight functions not of one sign construction of gauss quadratures with prescribed knots via jacobi matrices is extended to the case where not all orthogonal polynomials exist due to the weight function changing sign an algorithm is described and is demonstrated by calculating the knots of kronrod schemes and other gauss quadratures with prescribed knots

quadrature formulas for fourier coefficients we consider quadrature formulas of high degree of precision for the computation of the fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials in particular we show the uniqueness of a multiple node formula for the fourier tchebycheff coefficients given by micchelli and sharma and construct new gaussian formulas for the fourier coefficients of a function based on the values of the function and its derivatives

nested quadrature rules for cauchy principal value integrals abstract we consider quadrature formulas for the numerical evaluation with error estimate of integrals of the form a b f x x d x and we discuss them pointing out the computational aspects then we present an automatic integrator and we compare our numerical results with the ones of another published integrator for cauchy principal value integrals

calculation of gauss kronrod quadrature rules the jacobi matrix of the 2n 1 point gauss kronrod quadrature rule for a given measure is calculated efficiently by a five term recurrence relation the algorithm uses only rational operations and is therefore also useful for obtaining the jacobi kronrod matrix analytically the nodes and weights can then be computed directly by standard software for gaussian quadrature formulas

stieltjes type polynomials on the unit circle stieltjes type polynomials corresponding to measures supported on the unit circle t are introduced and their asymptotic properties away from t are studied for general classes of measures as an application we prove the convergence of an associated sequence of interpolating rational functions to the corresponding caratheodory function in turn this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the gauss kronrod rule provided that the integrand is analytic in a neighborhood of t

the rank and kernel of extended 1 perfect spl zopf sub 4 linear and additive non spl zopf sub 4 linear codes a binary extended 1 perfect code of length n 1 2 sup t is additive if it is a subgroup of spl zopf sub 2 sup spl alpha spl times spl zopf sub 4 sup spl beta the punctured code by deleting a spl zopf sub 2 coordinate if there is one gives a perfect additive code 1 perfect additive codes were completely characterized and by using that characterization we compute the possible parameters spl alpha spl beta rank and dimension of the kernel for extended 1 perfect additive codes a very special case is that of extended 1 perfect spl zopf sub 4 linear codes

ultraspherical stieltjes polynomials and gauss kronrod quadrature behave nicely for lambda 0 we show that the zeros of the stieltjes polynomials associated with the ultraspherical weight function with parameter lambda 0 are real and simple and except for two of them belong to 1 1 we also prove that they strictly interlace with the zeros of the corresponding ultraspherical polynomials consequently a gauss kronrod quadrature formula with two nodes outside the interval 1 1 is obtained we prove that the coefficients of such quadrature rules are positive finally an asymptotic representation of stieltjes polynomials which converges uniformly on the whole interval 1 1 is provided

on gauss kronrod quadrature formulae of chebyshev type we prove that there is no positive measure da on a b such that the corresponding gauss kronrod quadrature formula is also a chebyshev quadrature formula the same is true if we consider measures of the form do t a t dt where w t is even on a symmetric interval a a and the gauss kronrod formula is required to have equal weights only for n even we also show that the only positive and even measure da t da t on 1 1 for which the gauss kronrod formula has all weights equal if n 1 or has the form i f t da t w j t to l w 2 fil wi l r f for all n 2 is the chebyshev measure of the first kind doc t t2 ll2dt

a family of gauss kronrod quadrature formulae etude d une famille de formules de quadrature de gauss kronrad formules explicites des poids