pade approximants and noise rational functions abstract we continue a study of pade approximants pa for a series perturbed by random noise this time we consider more general rational functions we begin with the simple case of a sum of two geometric series and then show how these considerations can be extended to a general rational function we do not study the most general case but rather concentrate on demonstrating how our results for geometric series extend to new situations encountered when a general rational function is considered we show that froissart doublets are a universal feature and we construct an analog of the froissart polynomial introduced in the earlier paper

bandwidth truncation for chebyshev polynomial and ultraspherical chebyshev galerkin discretizations of differential equations the petrov galerkin ultraspherical polynomial chebyshev polynomial discretization of the highest derivative of a differential equation is a diagonal matrix the same is true for fourier galerkin discretizations nevertheless the spectral discretizations of simple problems like u x x q x u f x are usually dense matrices the villain is the multiplication matrix the galerkin representation of a term like q x u x unfortunately this part of the galerkin matrix is dense however if the ode coefficient q x has a chebyshev or fourier series that converges much more rapidly than u x then it is possible to realize great cost savings at no loss of accuracy by truncating the full n n galerkin matrix to a banded matrix where the bandwidth m n one of our themes is that when the spectral series for q x and u x have similar rates of convergence as is almost universal when a nonlinear equation is linearized for a newton krylov iteration such accuracy lossless truncation is impossible nonlinearity is but one of many causes of this sort of solution coefficient equiconvergence when bandwidth truncation is possible though our second theme is to show that a modest amount of floating point operations and memory can be saved by an unsymmetric truncation in which the number of elements retained to the left of the main diagonal is roughly double the number kept to the right our second improvement is to replace the m term spectral series for q x by its m 2 m 2 chebyshev pade rational approximation this sometimes allow one to halve the matrix bandwidth reducing the linear algebra costs by a factor of four

spurious poles in pade approximation in the theory of pade approximation locally uniform convergence has been proved only for special classes of functions for much larger classes convergence in capacity has been shown to hold true the reason for one type of convergence to hold true but the other one not can be found in poles of the approximants that may occur apparently anywhere in the complex plane because of their unwanted nature and since they do not correspond to singularities of the function f to be approximated these poles are called spurious the denominators of pade approximants satisfy orthogonality relations and consequently the location and distribution of spurious poles depend on properties of the orthogonality relations in the present paper the possibility of spurious poles is studied from the perspective of these orthogonal relations

convergence analysis of the chebyshev legendre spectral method for a class of fredholm fractional integro differential equations abstract in this paper we propose and analyze a spectral chebyshev legendre approximation for fractional order integro differential equations of fredholm type the fractional derivative is described in the caputo sense our proposed method is illustrated by considering some examples whose exact solutions are available we prove that the error of the approximate solution decays exponentially in l 2 norm

about a numerical method of successive interpolations for functional hammerstein integral equations a new and robust numerical method for hammerstein functional integral equations is proposed the convergence and the numerical stability of the method are mathematically proved and tested on some examples

on rational functions without froissart doublets in this paper we consider the problem of working with rational functions in a numeric environment a particular problem when modeling with such functions is the existence of froissart doublets where a zero is close to a pole we discuss three different parameters which allow one to monitor the absence of froissart doublets for a given general rational function these include the euclidean condition number of an underlying sylvester type matrix a parameter for determining coprimeness of two numerical polynomials and bounds on the spherical derivative we show that our parameters sharpen those found in a previous paper by two of the authors

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

existence of fractional integral equation with hysteresis the existence of solutions to an integral equation of fractional order with hysteresis has been established a fixed point theorem due to schauder is the main tool in carrying out our proof

on the single operation worst case time complexity of the disjoint set union problem we give an algorithm for the disjoint set union problem within the class of algorithms defined by tarjan which has o log n log log n single operation time complexity in the worst case also we define a class of algorithms for the disjoint set union problem which includes the class of algorithms defined by tarjan we prove that any algorithm from this class has at least omega log n log log n single operation time complexity in the worst case

robust pade approximation via svd pade approximation is considered from the point of view of robust methods of numerical linear algebra in particular the singular value decomposition this leads to an algorithm for practical computation that bypasses most problems of solution of nearly singular systems and spurious pole zero pairs caused by rounding errors for which a matlab code is provided the success of this algorithm suggests that there might be variants of pade approximation that are pointwise convergent as the degrees of the numerator and denominator increase to infty unlike traditional pade approximants which converge only in measure or capacity

new representations of pade pade type and partial pade approximants pade approximants are rational functions with a denominator which does not vanish at zero and whose series expansion match a given series as far as possible these approximants are usually written under a rational form in this paper we will show how to write them also under two different barycentric forms and under a partial fraction form depending on free parameters according to the choice of these parameters pade type approximants can be obtained under a barycentric or a partial fraction form

pade approximations in noise filtering resume we first recall the results on the distribution of poles and zeros of pade approximations to noisy taylor series around a point we then report the numerical experiment on noisy data coming from inelastic electron scattering cross sections on rare gases as functions of the momentum transfer at fixed energy we apply the thiele algorithm pade approximations of type ii to those data which are known to belong to an analytic function in a cut plane from to 0 in this case the poles and zeros split into three families 1 i doublet pole zero at a distance of the order of the computer roundoff 2 ii doublet pole zero at a distance of the order of the noise in the experimental data 3 iii remaining poles and zeros compatible with the analytic structure of the experimentally measured function by throwing away the first two families of poles and zeros noisy ones we were able to obtain an excellent filtering of the data compatible with the analytic structure of the analyzed function

two nonlinear lower bounds for on line computations the following lower bounds for on line computation are proved 1 simulating two tape nondeterministic machines by one tape machines requires n log n time 2 simulating k tape deterministic machines by machines with k pushdown stores requires n log1 k 1 n time

pade type rational and barycentric interpolation in this paper we consider the particular case of the general rational hermite interpolation problem where only the value of the function is interpolated at some points and where the function and its first derivatives agree at the origin thus the interpolants constructed in this way possess a pade type property at 0 numerical examples show the interest of the procedure the interpolation procedure can be easily modified to introduce a partial knowledge on the poles and the zeros of the function to approximated a strategy for removing the spurious poles is explained a formula for the error is proved in the real case applications are given

common factor detection and estimation the paper introduces a noniterative maximum likelihood ml based approach for the detection and estimation of the greatest common divisor of two given polynomials the coefficients of which are assumed to have a normal distribution with a consistently estimable covariance matrix it then goes on to show how this approach can be used to derive a noniterative ml method for estimating the orders and parameters of difference equation models with autoregressive residuals numerical examples are included to illustrate the performance of the proposed methodology

stable polefinding and rational least squares fitting via eigenvalues a common way of finding the poles of a meromorphic function f in a domain where an explicit expression of f is unknown but f can be evaluated at any given z is to interpolate f by a rational function pq such that r i f i at prescribed sample points i i 1l and then find the roots of q this is a two step process and the type of the rational interpolant needs to be specified by the user many other algorithms for polefinding and rational interpolation or least squares fitting have been proposed but their numerical stability has remained largely unexplored in this work we describe an algorithm with the following three features 1 it automatically finds an appropriate type for a rational approximant thereby allowing the user to input just the function f 2 it finds the poles via a generalized eigenvalue problem of matrices constructed directly from the sampled values f i in a one step fashion and 3 it computes rational approximants p q in a numerically stable manner in that p p q q f with small p q at the sample points making it the first rational interpolation or approximation algorithm with guaranteed numerical stability our algorithm executes an implicit change of polynomial basis by the qr factorization and allows for oversampling combined with least squares fitting through experiments we illustrate the resulting accuracy and stability which can significantly outperform existing algorithms

pade approximants for finite time ruin probabilities in this article we investigate pade approximants of hyper exponential to base on the first moments and the matrix exponential representation of pade approximated function an explicit formula is given for the laplace transform in the time to calculate finite time ruin probabilities of classical cramer lundberg model this formula generalizes the ultimate ruin probabilities formula of asmussen and rolski 1991 12 to illustrate this formula several numerical examples with different values u are given

on the functional integral equations of mixed type and integro differential equations of fractional orders the existence of monotonic solution for the functional integral equation of the mixed typex t g t email protected 1 0k 1 t s f 1s 0 sk 2 s t f 2 t x t email protected be proved in l 1 0 1 such kind of equations plays a significant role in many branches of mathematical physics engineering and economics on the other hand our considered equation are the general form of hammerstein and volterra integral equations see 4 and 5 as an application we prove the existence of solution for the initial value problem of the fractional order integro differential equation dx t dt g t email protected 1 0k 1 t s f 1 s d bx s ds email protected 0 1

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

model reduction of large scale systems by least squares abstract in this paper we introduce an approximation method for model reduction of large scale dynamical systems this is a projection which combines aspects of the svd and krylov based reduction methods this projection can be efficiently computed using tools from numerical analysis namely the rational krylov method for the krylov side of the projection and a low rank smith type iteration to solve a lyapunov equation for the svd side of the projection for discrete time systems the proposed approach is based on the least squares fit of the r 1 th column of a hankel matrix to the preceding r columns where r is the order of the reduced system the reduced system is asymptotically stable matches the first r markov parameters of the full order model and minimizes a weighted h 2 error the method is also generalized for moment matching at arbitrary interpolation points application to continuous time systems is achieved via the bilinear transformation numerical examples prove the effectiveness of the approach the proposed method is significant because it combines guaranteed stability and moment matching together with an optimization criterion

evaluating and approximating fir filters an approach based on functions of matrices most of the controllers designed for systems with timedelays involve finite impulse response fir filters in their structure in particular prediction based controllers for systems with input output delays we present theory algorithms and software to evaluate the frequency response to approximate the filters by stable linear timeinvariant systems and to compute their induced cal l 2 gain the approach is based on the theory of functions of matrices in the approach pole zero cancelations in frequency domain representations are explicitly taken into account

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

the rkfit algorithm for nonlinear rational approximation the rkfit algorithm outlined in m berljafa and s ga ttel generalized rational krylov decompositions with an application to rational approximation siam j matrix anal appl 2015 is a krylov based approach for solving nonlinear rational least squares problems this paper puts rkfit into a general framework allowing for its extension to nondiagonal rational approximants and a family of approximants sharing a common denominator furthermore we derive a strategy for the degree reduction of the approximants as well as methods for their conversion to partial fraction form for the efficient evaluation and root finding we also discuss commons and differences of rkfit and the popular vector fitting algorithm a matlab implementation of rkfit is provided and numerical experiments including the fitting of a mimo dynamical system and an optimization problem related to exponential integration demonstrate its applicability

on pointers versus addresses summary what is the cost of random access to memory we address this fnndamental problem by studying the simulation of random addressing by a machine which lacks it a pointer machine the model we define allows the use of a data type of our choice a ram program of time i and space s can be simulated in o t log a time using a tree however this is not an obvious lower bound since a high level data type may allow us to encode the data in a more economic way our major contribution is the formalization of incompressibility for general data types the definition extends a similar property 01 strings that underlies the theory of kolmogorov complexity the main theorem states that for all incompressible data types an q t log s lower bound holds incompressibility trivially holds for strings but is harder to prove for a powerlul data type we prove incompressibility for the real numbers with a set of primitives which includes all functions which are continuously differentiable except on a countable closed set this may be the richest set of operations considered in a lower bound proof the prooi relies on the implicit functions theorem and baire s category theorem we also show that the integers with arithmetic x and lz zj any boolean operations and left shift are incompressible the inclusion of right shift reverses the situation and we obtain an o tq s upper bound where s is a functional inverse of ackermann s function

functional approach to data structures and its use in multidimensional searching we establish new upper bounds on the complexity of multidimensional searching our results include in particular linear size data structures for range and rectangle counting in two dimensions with logarithmic query time more generally we give improved data structures for rectangle problems in any dimension in a static as well as a dynamic setting several of the algorithms we give are simple to implement and might be the solutions of choice in practice central to this paper is the nonstandard approach followed to achieve these results at its root we find a redefinition of data structures in terms of functional specifications

fast algorithms for finding nearest common ancestors we consider the following problem given a collection of rooted trees answer on line queries of the form what is the nearest common ancester of vertices x and y we show that any pointer machine that solves this problem requires omega log log n time per query in the worst case where n is the total number of vertices in the trees on the other hand we present an algorithm for a random access machine with uniform cost measure and a bound of omega log n on the number of bits per word that requires o 1 time per query and o n preprocessing time assuming that the collection of trees is static for a version of the problem in which the trees can change between queries we obtain an almost linear time and linear space algorithm

an algorithm for computing a pade approximant with minimal degree denominator in this paper a new definition of a reduced pade approximant and an algorithm for its computation are proposed our approach is based on the investigation of the kernel structure of the toeplitz matrix it is shown that the reduced pade approximant always has nice properties which the classical pade approximant possesses only in the normal case the new algorithm allows us to avoid the appearance of froissart doublets induced by computer roundoff in the non normal pade table

lower bounds for dynamic algebraic problems we consider dynamic evaluation of algebraic functions matrix multiplication determinant convolution fourier transform etc in the model of reif and tate i e if f x1 xn y1 ym is an algebraic problem we consider serving online requests of the form change input xi to value v or what is the value of output yi we present techniques for showing lower bounds on the worst case time complexity per operation for such problems the first gives lower bounds in a wide range of rather powerful models for instance history dependent algebraic computation trees over any infinite subset of a field the integer ram and the generalized real ram model of ben amram and galil using this technique we show optimal n bounds for dynamic matrix vector product dynamic matrix multiplication and dynamic discriminant and an n lower bound for dynamic polynomial multiplication convolution providing a good match with reif and tate s o nlogn upper bound we also show linear lower bounds for dynamic determinant matrix adjoint and matrix inverse and an n lower bound for the elementary symmetric functions the second technique is the communication complexity technique of miltersen nisan safra and wigderson which we apply to the setting of dynamic algebraic problems obtaining similar lower bounds in the word ram model the third technique gives lower bounds in the weaker straight line program model using this technique we show an log n 2 log log n lower bound for dynamic discrete fourier transform technical ingredients of our techniques are the incompressibility technique of ben amram and galil and the lower bound for depth two superconcentrators of radhakrishnan and ta shma the incompressibility technique is extended to arithmetic computation in arbitrary fields 2001 elsevier science

pade approximants and noise a case of geometric series abstract we study effects on pade approximants pa of a random noise added to coefficients of a power series numerical experiments performed during the last 30 years have shown that such noise results in the appearance of the so called froissart doublets fd pairs of zeros and poles separated by a distance of a scale of the noise for the first time we can prove that this effect actually takes place for geometric series we can also show what happens with noise induced poles or zeros that do not form fd we construct a polynomial that has as its roots mean positions of fd coefficients of this polynomial are expressed by random perturbations of coefficients of the series we study numerically this polynomial for low orders to show where in these orders fd coalesce numerical study of this polynomial to see a behaviour of fd for large orders is much easier than a study of the entire pa we also express a deviation of a pole of pas approximating the true pole of 1 1 z from 1 by perturbations of coefficients and show numerically that when order of pas increases this deviation drops down

analytic solution of the seir epidemic model via asymptotic approximant abstract an analytic solution is obtained to the seir epidemic model the solution is created by constructing a single second order nonlinear differential equation in ln s and analytically continuing its divergent power series solution such that it matches the correct long time exponential damping of the epidemic model this is achieved through an asymptotic approximant barlow et al 2017 in the form of a modified symmetric pade approximant that incorporates this damping the utility of the analytical form is demonstrated through its application to the covid 19 pandemic

dynamic normal forms and dynamic characteristic polynomial we present the first fully dynamic algorithm for computing the characteristic polynomial of a matrix in the generic symmetric case our algorithm supports rank one updates in o n2logn randomized time and queries in constant time whereas in the general case the algorithm works in o n2klogn randomized time where k is the number of invariant factors of the matrix the algorithm is based on the first dynamic algorithm for computing normal forms of a matrix such as the frobenius normal form or the tridiagonal symmetric form the algorithm can be extended to solve the matrix eigenproblem with relative error 2 b in additional o nlog2nlogb time furthermore it can be used to dynamically maintain the singular value decomposition svd of a generic matrix together with the algorithm the hardness of the problem is studied for the symmetric case we present an n2 lower bound for rank one updates and an n lower bound for element updates