similarities of the integral pade approximants ii in this paper we show further similarities of the integral pade approximant the similarity of the integral pade approximant with the functional pade approximant and the improved functional epsilon algorithms is demonstrated the purpose of demonstrating the similarity between the integral pade approximant and the functional pade approximant is to verify our conjecture which is the integral pade approximant and the hybrid functional pade approximant produce identical rational functions finally we demonstrate the improvement of the classical functional epsilon algorithms which also has a relationship with the integral pade approximant

a generalization of the e algorithm abstract salam a on a generalization of the algorithm journal of computational and applied mathematics 46 1993 455 464 the algorithm is a quite powerful algorithm for accelerating the convergence of some classes of sequences it was generalized in two different ways some years ago in this paper a determinantal formula for its second generalization is obtained then its kernel the set of sequences which are transformed into a constant sequence is given as the solution of a difference equation

the polynomial approximation in the finite element method dipartimento di matematica pura e applicata universitd di padova via belzoni 7 1 35131 padova italy received 20 october 1992 abstract in the analysis of a finite element method fem we can describe the shape of a given element by a set of elementary functions known as shape functions the approaches describing these functions are quite different ones in the plane x 1 x2 these functions are a product of lagrangian polynomials when the coordinate system can be chosen with the axes parallel to the sides of the element otherwise a system of barycentric coordinates sometimes called area coordinates could be introduced the aim of this paper is the description and the representation of shape functions when the element has trianoular shape the simplest the representation has been done by using two algorithmic schemes neville aitken and de casteljau for these schemes we have deduced very important properties keywords barycentric coordinates shape functions neville aitken scheme de casteljau scheme characteristic space

the just noticeable difference of center time and clarity index in large reverberant spaces just noticeable difference jnd values are available for most acoustical parameters currently used in practice however they have been determined with reference to conditions typically encountered in concert halls and in rooms for speech covering a range of reverberation times t spanning from 0 5 s to 2 s when reverberation gets longer the relationship between measured parameters describing acoustic clarity may change significantly and subjective perception might also be different the proposed research investigates the influence of reverberation time on jnd for clarity measures taking into account three reference cases having t values varying from 2 s to 6 s measured b format impulse responses were properly modified to introduce the desired changes and then auralized with two music motifs for presentation on a 4 channel playback system listening tests based on paired comparisons were carried out to determine subjective limens the results proved to be independent of music motifs and showed that jnd in the clarity index is almost independent of t while jnd in the center time is significantly related to t and can be assumed as the 8 5 of the reference t s value

a derivation of extrapolation algorithms based on error estimates in this paper we shall emphasize the role played by error estimates and annihilation difference operators in the construction of extrapolation processes it is shown that this approach leads to a unified derivation of many extrapolation algorithms and related devices to general results about their kernels and that it opens the way to many new algorithms their convergence and acceleration properties could also be studied within this framework

extrapolation methods for vector sequences this is an expository paper that describes and compares five methods for extrapolating to the limit or anti limit of a vector sequence without explicit knowledge of the sequence generator the methods are the minimal polynomial extrapolation mpe studied by cabay and jackson mesina and skelboe the reduced rank extrapolation rre of eddy which we show to be equivalent to mesina s version of mpe the vector and scalar versions of the epsilon algorithm vea sea introduced by wynn and extended by brezinski and gekeler and the topological epsilon algorithm tea of brezinski we cover the derivation and error analysis of iterated versions of the algorithms as applied to both linear and nonlinear problems and we show why these versions tend to converge quadratically we also present samples from extensive numerical testing that has led us to the following conclusions a tea in spite of its role as a theoretical link between the polynomial type and the epsilon type methods has no practical appl

extended procedures for extrapolation to the limit in this paper new procedures for the extrapolation to the limit of slowly convergent sequences and functions are proposed they are based on the notions of error estimates and annihilation operators we obtain generalizations of the discrete and confluent e transformation which are the most general sequence and function transformations known so far it is shown that many transformations studied in the literature are included in our formalism particular cases of these procedures are discussed then several extensions of the e algorithm are given finally the procedure q is applied to our procedures to produce new procedures for extrapolation to the limit

a schur complement approach to a general extrapolation algorithm this paper is devoted to a schur complement approach to the e transformation which is the most general scalar sequence transformation known so far for accelerating the convergence a new derivation of known results on schur complements is given in the first part of the paper then schur complements and their properties are used to obtain various interpretations of the e transformation the recursive rules of the e algorithm for its implementation are also recovered new results on its kernel are derived and issues on its convergence are discussed this approach can be extended to the vector case thus leading to new vector sequence transformations

sound propagation and energy relations in churches the results of an acoustic survey carried out in a group of italian churches differing in style typology and location were used in order to study how the acoustic energy varies inside this kind of space the effect of different architectural elements on sound propagation was investigated by means of three dimensional impulse responses measured using a b format microphone with sweep signals side chapels columns and trussed roofs appeared to scatter the reflections so that the purely diffuse exponential sound decay begins after a time interval which grows with the source receiver distance and with the complexity of the church the results of the measurements were then compared with predictions given by existing theoretical models to check their accuracy in particular a model previously proposed by the authors for a specific type of romanesque churches was further refined taking into account the new findings and making some simplifications its application to the wider sample of churches under analysis showed that strength clarity and center time can be predicted with reasonable accuracy

perceptual evaluation of measures of spectral variance in many applications it is desirable to achieve a signal that is as close as possible to ideal white noise one example is in the design of an artificial reverberator whereby there is a need for its lossless prototype output from an impulse input to be perceptually white as much as possible the ljung box test the drouiche test and the wiener entropy also called the spectral flatness measure are three well known methods for quantifying the similarity of a given signal to ideal white noise in this paper listening tests are conducted to measure the just noticeable difference jnd on the perception of white noise which is the jnd between ideal gaussian white noise and noise with a specified deviation from the flat spectrum this paper reports the jnd values using one of these measures of whiteness which is the ljung box test this paper finds considerable disagreement between the ljung box test and the other two methods and shows that none of the methods is a significantly better predictor of listeners perception of whiteness this suggests a need for a whiteness test that is more closely correlated to human perception in many applications it is desirable to achieve a signal that is as close as possible to ideal white noise one example is in the design of an artificial reverberator whereby there is a need for its lossless prototype output from an impulse input to be perceptually white as much as possible the ljung box test the drouiche test and the wiener entropy also called the spectral flatness measure are three well known methods for quantifying the similarity of a given signal to ideal white noise in this paper listening tests are conducted to measure the just noticeable difference jnd on the perception of white noise which is the jnd between ideal gaussian white noise and noise with a specified deviation from the flat spectrum this paper reports the jnd values using one of these measures of whiteness which is the ljung box test this paper finds considerable disagreement between the ljung box test and the other two methods and shows that none of the methods is a significantly better predictor of listen

recursive interpolation extrapolation and projection abstract the recursive projection algorithm derived in a previous paper is related to several well known methods of numerical analysis such as the conjugate gradient method rosen s method and henrici s it is connected with the general interpolation problem with extrapolation methods with orthogonal projection on a subspace and with fourier expansions several other connections and applications are presented

anderson acceleration for fixed point iterations this paper concerns an acceleration method for fixed point iterations that originated in work of d g anderson j assoc comput mach 12 1965 pp 547 560 which we accordingly call anderson acceleration here this method has enjoyed considerable success and wide usage in electronic structure computations where it is known as anderson mixing however it seems to have been untried or underexploited in many other important applications moreover while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities this method has received relatively little attention from these communities over the years a recent paper by h fang and y saad numer linear algebra appl 16 2009 pp 197 221 has clarified a remarkable relationship of anderson acceleration to quasi newton secant updating methods and extended it to define a broader anderson family of acceleration methods in this paper our goals are to shed additional light on anderson acceleration and to draw further attention to its usefulness as a general tool we first show that on linear problems anderson acceleration without truncation is essentially equivalent in a certain sense to the generalized minimal residual gmres method we also show that the type 1 variant in the fang saad anderson family is similarly essentially equivalent to the arnoldi full orthogonalization method we then discuss practical considerations for implementing anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications

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

some results about vector extrapolation methods and related fixed point iterations abstract in this work we consider polynomial extrapolation methods for vector sequences we give a unified presentation of these methods and a new recursive algorithm for the implementation of the modified minimal polynomial extrapolation mmpe method properties operation counts and storage requirements of the algorithm will be given the second part of the work is devoted to the application of these methods for solving systems of nonlinear equations a sufficient condition for quadratic convergence is given

the effects of different test methods on the just noticeable difference of clarity index for musica the just noticeable differences jnds of room acoustics metrics are necessary for research and design of performing arts venues the goal of this work was to evaluate the effects of different testing methods on the measured jnd of clarity index for music c80 an initial study was conducted to verify the findings of other published works that the c80 jnd is approximately 1 db as currently listed in iso 3382 2009 international organization for standardization switzerland 2009 however the results suggested a higher value in the second study the effects of using two variations of the method of constant stimuli were examined where one variation required the subjects to evaluate the pair of signals by listening to each of them in their entirety while the second approach allowed the participants to switch back and forth in real time more consistent results were obtained with the latter variation and the results indicated a c80 jnd greater than 1 db in the final study an extensive training period using the first variation was required based on the second study and the data were collected using the second variation the analysis revealed that for the conditions used in this study concert hall and chamber music hall that the c80 jnd is approximately 3 db

regularized nonlinear acceleration we describe a convergence acceleration technique for generic optimization problems our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method the weights in this average are computed via a simple linear system whose solution can be updated online this acceleration scheme runs in parallel to the base algorithm providing improved estimates of the solution on the fly while the original optimization method is running numerical experiments are detailed on classical classification problems

nonlinear dimensionality reduction by topologically constrained isometric embedding many manifold learning procedures try to embed a given feature data into a flat space of low dimensionality while preserving as much as possible the metric in the natural feature space the embedding process usually relies on distances between neighboring features mainly since distances between features that are far apart from each other often provide an unreliable estimation of the true distance on the feature manifold due to its non convexity distortions resulting from using long geodesics indiscriminately lead to a known limitation of the isomap algorithm when used to map non convex manifolds presented is a framework for nonlinear dimensionality reduction that uses both local and global distances in order to learn the intrinsic geometry of flat manifolds with boundaries the resulting algorithm filters out potentially problematic distances between distant feature points based on the properties of the geodesics connecting those points and their relative distance to the boundary of the feature manifold thus avoiding an inherent limitation of the isomap algorithm since the proposed algorithm matches non local structures it is robust to strong noise we show experimental results demonstrating the advantages of the proposed approach over conventional dimensionality reduction techniques both global and local in nature

other manifestations of the schur complement abstract many series and sequence transformations used in numerical analysis are defined as ratios of determinants they are implemented by recursive algorithms based on determinantal identities the aim of this paper is to study the connections of these transformations and algorithms with the schur complement of a matrix the schur s formula is extended to the vector case thus providing the same treatment for vector sequence transformations and the corresponding recursive algorithms thanks to these connections particular rules for avoiding division by zero or numerical instability are obtained for these algorithms some fixed point methods and continued fractions also fit in this framework

ideal membership in polynomial rings over the integers we will reproduce a proof using hermann s classical method in section 3 below note that the computable character of this bound reduces the question of whether fo g fi fn for given fj g f x to solving an enormous system of linear equations over f hence in this way one obtains a naive algorithm for solving the ideal membership problem for f x provided f is given in some explicitly computable manner later buchberger in his ph d thesis 1965 introduced the important concept of a gr bner basis and gave an algorithm for deciding ideal membership for f x which is widely used today see e g 6 the doubly exponential nature of above is essentially unavoidable as a family of examples due to mayr and meyer 27 shows in fact they prove that ideal membership for q x is exponential space hard the amount of space needed by any algorithm to decide ideal membership for q x or z x grows exponentially in the size of the input if we restrict to o n ofa special form often dramatic

about henrici s transformation for accelerating vector sequences abstract the aim of this paper is to study henrici s transformation for accelerating a vector sequence a convergence acceleration theorem is proved and two different implementations are given the paper ends with some numerical examples

acceleration of convergence of vector sequences a general approach to the construction of convergence acceleration methods for vector sequences is proposed using this approach one can generate some known methods such as the minimal polynomial extrapolation the reduced rank extrapolation and the topological epsilon algorithm and also some new ones some of the new methods are easier to implement than the known methods and are observed to have similar numerical properties the convergence analysis of these new methods is carried out and it is shown that they are especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iteratively a stability analysis is also given and numerical examples are provided the convergence and stability properties of the topological epsilon algorithm are likewise given

comparison of nine methods to estimate ear canal stimulus levels the reliability of nine measures of the stimulus level in the human ear canal was compared by measuring the sensitivity of behavioral hearing thresholds to changes in the depth of insertion of an otoacoustic emission probe four measures were the ear canal pressure the eardrum pressure estimated from it and the pressure measured in an ear simulator with and without compensation for insertion depth the remaining five quantities were derived from the ear canal pressure and the thevenin equivalent source characteristics of the probe forward pressure initial forward pressure the pressure transmitted into the middle ear eardrum sound pressure estimated by summing the magnitudes of the forward and reverse pressure integrated pressure and absorbed power two sets of behavioral thresholds were measured in 26 subjects from 0 125 to 20 khz with the probe inserted at relatively deep and shallow positions in the ear canal the greatest dependence on insertion depth was for transmitted pressure and absorbed power the measures with the least dependence on insertion depth throughout the frequency range best performance included the depth compensated simulator eardrum forward and integrated pressures among these forward pressure is advantageous because it quantifies stimulus phase

normative thresholds in the 8 to 20 khz range as a function of age using a prototype high frequency audiometer auditory thresholds in the 8 to 20 khz range were obtained from 240 subjects ranging in age from 10 60 years these measurements were obtained in interest of developing a normative database for frequencies above 8 khz and to evaluate intersubject variability as a function of age an analysis of variance anova revealed significant effects of frequency age and sex and a significant frequency by age interaction the largest changes in sensitivity with age occurred between 40 to 59 years below approximately 15 khz the intersubject variability of threshold estimates increased as a function of both age and frequency further analysis revealed that the age related changes in variability were related to absolute thresholds rather than to age per se when data are converted to db hl relative to the youngest group tested the region of maximum hearing loss shifts to lower frequencies with increasing age and threshold shifts with age are greatest in the 13 to

hybrid vector transformations we present in this paper hybrid and semi hybrid vector extrapolation methods for a vector sequence transformation we will define the generalized residual which allows us to introduce a new hybrid vector transformation when solving linear systems this new transformation reduces to the hybrid procedure defined by brezinski and redivo zaglia 1994 which is a generalization of the smoothing technique in the second part of this paper we define a new composite transformation called semi hybrid when applied to linearly generated sequences we will show that the residual polynomial of the semi hybrid transformation is the product of two residual polynomials

determinantal representations for the j transformation the iterative j transformation homeier h h h 1993 some applications of nonlinear convergence accelerators int j quantum chem 45 545 562 is of similar generality as the well known e algorithm brezinski c 1980 a general extrapolation algorithm numer math 35 175 180 h vie t 1979 generalized neville type extrapolation schemes bit 19 204 213 the properties of the j transformation were studied recently in two companion papers homeier h h h 1994a a hierarchically consistent iterative sequence transformation numer algo 8 47 81 homeier h h h 1994b analytical and numerical studies of the convergence behavior of the j transformation j comput appl math to appear in the present contribution explicit determinantal representations for this sequence transformation are derived the relation to the brezinski walz theory brezinski c walz g 1991 sequences of transformations and triangular recursion schemes with applications in numerical analysis j comput appl math 34 361 383 is discussed overholt s process overholt k j 1965 extended aitken acceleration bit 5 122 132 is shown to be a special case of the j transformation consequently explicit determinantal representations of overholt s process are derived which do not depend on lower order transforms also families of sequences are given for which overholt s process is exact as a numerical example the euler series is summed using the j transformation the results indicate that the j transformation is a very powerful numerical tool

solution of integral equations using generalised inverse function valued pade approximants i abstract neumann series solutions for the linear integral equation x 1 1 0 1 x y y d y converge for small values of we use generalised inverse function valued pade approximants to accelerate the convergence of the neumann series and to estimate the singular value of these estimates are found to compare very favourably with those from the fredholm determinant method and the pade approximant method

a contour integral representation of linear extrapolation methods extrapolation methods are well known to be very efficient tools for the acceleration of the convergence of certain sequences of numbers or functions cf 2 3 4 8 9 in this note we present a representation of linear extrapolation procedures in terms of complex contour integrals for the proof we make use of a complete characterization of these procedures as linear functionals which is itself of some interest and can be found for example in 2 or 8

vector orthogonal polynomials and matrix series abstract existence and uniqueness of the vector orthogonal polynomials associated to f z n 0cznanx0 are studied where cn c x0 c d a a square d d matrix a necessary and sufficient condition for the existence and uniqueness of the vector polynomial psd is given when a is a regular matrix the vector orthogonal polynomials associated to the resolvent of the matrix a are studied and the eigenvalues of a are computed by using the vector qd algorithm an initialization of that algorithm is given finally the normality of the pade table of the function f z ezax0 is evoked

sequences of transformations and triangular recursion schemes with applications in numerical analysis abstract in many fields of numerical analysis there appear transformations of the form t k v v k i v k i v t i when v varies a sequence of transformations is obtained this approach covers for example the e and algorithms the recursion formulae for b splines bernstein polynomials and orthogonal polynomials pade approximants the divided difference scheme and projection methods in this paper it will be proved that such transformations can be written as a ratio of determinants and can be recursively computed by a triangular recursion scheme the reciprocal of these results also holds furthermore we will show that t v k can be represented in terms of a complex contour integral throughout the paper we will study several examples in some detail and it will turn out that the application of our general theory leads to interesting new results in the special cases among others we will derive a new determinantal representation formula for b splines a recurrence relation for generalized bernstein polynomials a generalization of the e algorithm and we will prove that the algorithm can be represented as a quotient of determinants

hearing thresholds for pure tones above 16 khz hearing thresholds for pure tones between 16 and 30khz were measured by an adaptive method the maximum presentation level at the entrance of the outer ear was about 110db spl to prevent the listeners from detecting subharmonic distortions in the lower frequencies pink noise was presented as a masker even at 28khz threshold values were obtained from 3 out of 32 ears no thresholds were obtained for 30khz tone between 20 and 28khz the threshold tended to increase rather gradually whereas it increased abruptly between 16 and 20khz

reference hearing thresholds in an extended frequency range as a function of age the iso 7029 2000 standard defines normative hearing thresholds h db hearing level as a function of age y years given by h y 18 2 up to 8 khz the purpose of this study was to determine reference thresholds above 8 khz hearing thresholds were examined using pure tone audiometry over the extended frequency range 0 125 16 khz and the acquired values were used to specify the optimal approximation of the dependence of hearing thresholds on age a sample of 411 otologically normal men and women 16 70 years of age was measured in both ears using a high frequency audiometer and sennheiser hda 200 headphones the coefficients of quadratic linear polynomial and power law approximations were calculated using the least squares fitting procedure the approximation combining the square function h y 18 2 with a power law function h y 18 1 5 both gender independent was found to be the most appropriate coefficient was determined at frequencies of 9 khz 0 021 10 khz 0 024 11 2 khz 0 029 and coefficient at frequencies of 12 5 khz 0 24 14 khz 0 32 16 khz 0 36 the results could be used to determine age dependent normal hearing thresholds in an extended frequency range and to normalize hearing thresholds when comparing participants differing in age

a multi rate decay model to predict energy based acoustic parameters in churches multi rate decays are sometimes observed in room acoustics appearing when markedly different volumes are coupled together and resulting in nonlinear decay curves such behavior appears in several churches at the very beginning of the decay process although in conditions which cannot be explicitly referred to as coupling phenomena consequently multi rate exponential decays may be suitable to model energy distribution in this group of buildings providing a more elegant and easily applicable set of equations in place of a previously defined linear model used to adapt barron s revised theory the paper shows that the multi rate approach ensures ease of calculation without significant loss in accuracy in predicting energy based acoustic parameters

asymptotic expansions and acceleration of convergence for higher order iteration processes we analyze the convergence behavior of sequences of real numbers x n which are defined through an iterative process of the formx n t x n 1 wheret is a suitable real function it will be proved that under certain mild assumptions ont these numbersx n possess an asymptotic error expansion where the type of this expansion depends on the derivative oft in the limit point xi mathop lim limits n to infty x n this generalizes a result of g meinardus 6 it is well known that the convergence of sequences which possess an asymptotic expansion can be accelerated significantly by application of a suitable extrapolation process we introduce two types of such processes and study their main properties in some detail in addition we analyze practical aspects of the extrapolation and present the results of some numerical tests as we shall see even the convergence of newton s method can be accelerated using the very simple linear extrapolation process

the simplified topological varepsilon algorithms for accelerating sequences in a vector space when a sequence of numbers is slowly converging it can be transformed into a new sequence which under some assumptions converges faster to the same limit one of the best known sequence transformations is the shanks transformation which can be recursively implemented by the varepsilon algorithm of wynn this transformation and this algorithm have been extended in two different ways to a sequence of elements of a topological vector space e in this paper we present new algorithms for implementing these topological shanks transformations they no longer require the manipulation of elements of the algebraic dual space e of e nor do they use the duality product inside the rules of the algorithms they need the storage of fewer elements of e and the stability is improved they also allow us to prove convergence and acceleration results for some types of sequences various applications involving sequences of vectors or matrices show the interest of the new algorithms

analysis of some vector extrapolation methods for solving systems of linear equations in this paper we consider some vector extrapolation methods for solving nonsymmetric systems of linear equations when applied to sequences generated linearly these methods namely the minimal polynomial extrapolation mpe and the reduced rank extrapolation rre are krylov subspaces methods and are respectively equivalent to the method of arnoldi and to the gcr and gmres by considering the geometrical aspect of these methods we derive new expressions for their residual norms and give a relationship between them this allows us to compare the two methods using this new approach we will show that for nonsingular skew symmetric matrices the gmres stagnates every two iterations and the restarted version gmres m m ge 2 is always convergent finally the incomplete forms are considered and some convergence results are given

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem