pseudo skeleton approximations with better accuracy estimates abstract we propose a priori accuracy estimates for low rank matrix approximations that use just a small number of the rows and columns this number is greater than the approximation rank unlike the existing methods of pseudo skeleton approximation but the estimates are more accurate than previously known ones this paper generalizes the results of 12 13

an automatic quadrature for cauchy principal value integrals an automatic quadrature is presented for computing cauchy principal value integrals q f c faf t t c dt a c b for smooth functions f t after subtracting out the singularity we approximate the function f t by a sum of chebyshev polynomials whose coefficients are computed using the ftt the evaluations of q f c for a set of values of c in a b are efficiently accomplished with the same number of function evaluations numerical examples are also given

gaussian quadrature formulae of the third kind for cauchy principal value integrals basic properties and error estimates abstract let n 1 f be the polynomial of degree n 1 interpolating the function f at the points x1 x2 xn with pn xi 0 i e at the nodes of the classical gaussian quadrature formula for the numerical approximation of the cauchy principal value integral 1 1 x x 1 d x with 1 1 and f c1 1 1 we present the quadrature formula qn 1g3 given by q n 1 g3 1 n 1 x n 1 x d x 1 n 1 1 we show that this quadrature formula does not have the disadvantages of the other two well known quadrature formulae based on the same set of nodes in particular we prove that the sequence based on the same set of nodes in particular we prove that the sequence qn 1g3 f converges to the true value of the integral uniformly for all 1 1 we give estimates for the error term furthermore we state some relations connecting the present quadrature formula to the previously introduced formulae

uniform approximation to finite hilbert transform of oscillatory functions and its algorithm abstract for the finite hilbert transform of oscillatory functions q f c 1 1 f x e i x x c d t with a smooth function f and real 0 for c 1 1 in the sense of cauchy principal value or for c 1 of hadamard finite part we present an approximation method of clenshaw curtis type and its algorithm interpolating f by a polynomial p n of degree n and expanding in terms of the chebyshev polynomials with o n log n operations by the fft we obtain an approximation q p n c q f c we write q p n c as a sum of the sine and cosine integrals and an oscillatory integral of a polynomial of degree n 1 we efficiently evaluate the oscillatory integral with a combination of authors previous method and keller s method for f z analytic on the interval 1 1 in the complex plane z the error of q p n c is bounded uniformly with respect to c and numerical examples illustrate the performance of our method

uniform approximations to finite hilbert transform and its derivative interpolatory integration rules of numerical stability are presented for approximating cauchy principal value p v integrals 11 f t t c dt and hadamard finite part f p integrals 11f t t c 2 dt 1 c 1 respectively for a given smooth function f t present quadrature rules consist of interpolating f t at abscissae in the interval of integration 1 1 except for the pole c where neither the function value f c nor its derivative f c is required followed by subtracting out the singularities we demonstrate that the use of both endpoints 1 as abscissae in interpolating f t is essential for uniformly approximating the integrals namely for bounding the approximation errors independently of the values of c in fact for the f p integrals the use of double abscissae at both endpoints 1 as well as simple abscissae in 1 1 enables the uniform approximations while the use of simple abscissae at both endpoints 1 and those in 1 1 is sufficient for the p v integrals these facts suggest that finite hilbert transforms p v integrals and their derivatives f p integrals with varied values of c could be approximated efficiently with the same number of abscissae respectively some numerical examples are given

sinc nystro m method for numerical solution of a dominant system of cauchy singular integral equations given on a piecewise smooth contour a numerical method is proposed for obtaining an approximate solution to a dominant system of cauchy singular integral equations dscsies taken over a piecewise smooth contour l consisting of a finite number of arcs l j first the dscsies are reduced to a system of fredholm integral equations of the second kind which are then solved approximately by nystrom s method based on a sinc quadrature rule it is shown that the error of an n point approximation converges to zero at the rate o e cn 1 2 as n to infty provided that the coefficients of the dscsies are analytic in neighborhoods d j of each arc l j and satisfy a lipschitz condition in each d j

a high order solver for the heat equation in 1d domains with moving boundaries we describe a fast high order accurate method for the solution of the heat equation in domains with moving dirichlet or neumann boundaries and distributed forces we assume that the motion of the boundary is prescribed our method extends the work of greengard and strain comm pure appl math xliii 1990 pp 949 963 our scheme is based on a time space chebyshev pseudo spectral collocation discretization which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with green s function for the heat equation we present numerical results that exhibit up to eighth order convergence rates assuming n time steps and m spatial discretization points the evaluation of the solution of the heat equation at the same number of points in space time requires mathcal o n m log m work thus our scheme can be characterized as fast that is it is work optimal up to a logarithmic factor

uniform approximation to cauchy principal value integrals with logarithmic singularity abstract an approximation of clenshaw curtis type is given for cauchy principal value integrals of logarithmically singular functions i f c 1 1 f x log x c x c d x c 1 1 with a given function f using a polynomial p n of degree n interpolating f at the chebyshev nodes we obtain an approximation i p n c i f c we expand p n in terms of chebyshev polynomials with o n log n computations by using the fast fourier transform our method is efficient for smooth functions f for which p n converges to f fast as n grows and so simple to implement this is achieved by exploiting three term inhomogeneous recurrence relations in three stages to evaluate i p n c for f z analytic on the interval 1 1 in the complex plane z the error of the approximation i p n c is shown to be bounded uniformly using numerical examples we demonstrate the performance of the present method

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

numerical integration in the presence of an interior singularity abstract the convergence of sequences of integration rules applied to an improperly integrable function with an interior singularity is investigated the rules studied include the gauss jacobi rules and the interpolatory integration rules based on the zeros of 1 x 2 m p n 2m x m 0 1 for certain values of and the results are then applied to the study of the convergence of hunter s method for cauchy principal value integrals

an algorithm based on the fft for a generalized chebyshev interpolation an algorithm for a generalized chebyshev interpolation procedure increasing the number of sample points more moderately than doubling is pre sented the fft for a real sequence is incorporated into the algorithm to enhance its efficiency numerical comparison with other existing algorithms is given

interleaving schemes for multidimensional cluster errors we present two dimensional and three dimensional interleaving techniques for correcting two and three dimensional bursts or clusters of errors where a cluster of errors is characterized by its area or volume correction of multidimensional error clusters is required in holographic storage an emerging application of considerable importance our main contribution is the construction of efficient two dimensional and three dimensional interleaving schemes the proposed schemes are based on t interleaved arrays of integers defined by the property that every connected component of area or volume t consists of distinct integers in the two dimensional case our constructions are optimal they have the lowest possible interleaving degree that is the resulting t interleaved arrays contain the smallest possible number of distinct integers hence minimizing the number of codewords required in an interleaving scheme in general we observe that the interleaving problem can be interpreted as a graph coloring problem and introduce the useful special class of lattice interleavers we employ a result of minkowski dating back to 1904 to establish both upper and lower bounds on the interleaving degree of lattice interleavers in three dimensions for the case t spl equiv 0 mod 6 the upper and lower bounds coincide and the minkowski lattice directly yields an optimal lattice interleaver for t spl ne 0 mod 6 we construct efficient lattice interleavers using approximations of the minkowski lattice

corrigendum generalized chebyshev interpolation and its application to automatic quadrature math comp 41 1983 no 164 537 553 mr0717701 84m 65037 by hasegawa t torii and i ninomiya

asymptotic expansions and fast computation of oscillatory hilbert transforms in this paper we study the asymptotics and fast computation of the one sided oscillatory hilbert transforms of the form begin aligned h f t e i omega t x int nolimits 0 infty e i omega t frac f t t x dt quad omega 0 quad x ge 0 end aligned where the bar indicates the cauchy principal value and f is a real valued function with analytic continuation in the first quadrant except possibly a branch point of algebraic type at the origin when x 0 the integral is interpreted as a hadamard finite part integral provided it is divergent asymptotic expansions in inverse powers of omega are derived for each fixed x ge 0 which clarify the large omega behavior of this transform we then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms depending on the position of x we classify our discussion into three regimes namely x mathcal o 1 or x gg 1 0 x ll 1 and x 0 numerical experiments show that the convergence of the proposed methods greatly improve when the frequency omega increases some extensions to oscillatory hilbert transforms with bessel oscillators are briefly discussed as well

tt cross approximation for multidimensional arrays abstract as is well known a rank r matrix can be recovered from a cross of r linearly independent columns and rows and an arbitrary matrix can be interpolated on the cross entries other entries by this cross or pseudo skeleton approximation are given with errors depending on the closeness of the matrix to a rank r matrix and as well on the choice of cross in this paper we extend this construction to d dimensional arrays tensors and suggest a new interpolation formula in which a d dimensional array is interpolated on the entries of some tt cross tensor train cross the total number of entries and the complexity of our interpolation algorithm depend on d linearly so the approach does not suffer from the curse of dimensionality we also propose a tt cross method for computation of d dimensional integrals and apply it to some examples with dimensionality in the range from d 100 up to d 4000 and the relative accuracy of order 10 10 in all constructions we capitalize on the new tensor decomposition in the form of tensor trains tt decomposition

two dimensional error pattern correcting codes two dimensional 2d cyclic codes are presented which correct any single occurrence of known 2d error patterns within a 2d array of bits applications for this type of codes include storage and display devices the code construction begins with a generation of distinct syndrome sets for all targeted 2d error patterns a method to refine the syndrome sets is then presented for making each syndrome set to contain distinct members thereby guaranteeing full correction capability for the given list of known error patterns using an example construction the effectiveness of the proposed coding approach is demonstrated versus the maximum distance separable mds random error correcting code and known 2d burst correcting codes for a 2d intersymbol interference isi channel that yields a few dominant but relatively large error patterns

minimum distance analysis of a certain class of 2 d isi channels we perform a minimum distance analysis of a class of two dimensional intersymbol interference isi channels applicable to multitrack magnetic recording and orthogonal frequency division multiplex transmission systems exact minimum distance for a wide class of isi responses is derived the fundamental analytical technique is to transform the channel into an equivalent minimum phase channel the results improve upon the prior work of soljanin and georghiades

a sinc quadrature rule for hadamard finite part integrals a sinc quadrature rule is presented for the evaluation of hadamard finite part integrals of analytic functions integration over a general are in the complex plane is considered special treatment is given to integrals over the interval 1 1 theoretical error estimates are derived and numerical examples are included

approximating cauchy type singular integral by an automatic quadrature scheme an automatic quadrature scheme is developed for the approximate evaluation of the product type indefinite integral q f x y c x y r t k t c f t dt 1 x y 1 1

error correction of multidimensional bursts we present several methods and constructions to generate binary codes for correction of a multidimensional cluster error whose shape can be a box error a lee sphere error or an error with an arbitrary shape our codes have very low redundancy close to optimal and a large range of parameters of arrays and clusters our main results are summarized as follows 1 a construction of two dimensional codes capable to correct a rectangular error with considerably more flexible parameters from previously known constructions this construction is easily generalized for d dimensions 2 a novel method based on d colorings of the d dimensional space for constructing d dimensional codes correcting a d dimensional cluster error of various shapes 3 a transformation of the d dimensional space into another d dimensional space in a way that a d dimensional lee sphere is transformed into a shape located in a d dimensional box of a relatively small size 4 applying the coloring method to correct more efficiently a two dimensional error whose shape is a lee sphere 5 a construction of d dimensional codes capable to correct a d dimensional cluster error of size b in which the number of erroneous positions is relatively small compared to b 6 we present a code which corrects a d dimensional arbitrary cluster error with relatively small redundancy

subset selection for matrices abstract we consider the question of how to delete m k rows from a matrix x r m n so that the resulting matrix a r k n is as non singular as possible bounds for the singular values of a are derived which decrease only algebraically with m and n in addition a number of applications where subset selection is necessary are examined

singleton type bounds for blot correcting codes consider the transmission of codewords over a channel which introduces dependent errors thinking of two dimensional codewords such errors can be viewed as blots of a particular shape on the codeword for such blots of errors the combinatorial metric was introduced by gabidulin 1971 and it was shown that a code with distance d in combinatorial metric can correct d 2 blots we propose an universal singleton type upper bound on the rate r of a blot correcting code with the distance d in arbitrary combinatorial metric the rate is bounded by r spl les 1 d 1 d where d is the maximum possible distance between two words in this metric

the classical collocation method for singular integral equations first of all a method of collocation which we call classical collocation is described for the approximate solution of complete singular integral equations with cauchy kernel taken over the arc 1 1 secondly we demonstrate that under reasonable conditions the approximate solutions converge to the solution of the original equation

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

a comparison of some quadrature methods for approximating cauchy principal value integrals cauchy principal value integrals are evaluated by the imt quadrature scheme which like the tanh quadrature scheme is essentially a trapezoidal scheme after making a transformation of the variable of integration numerical results for some test problems demonstrate that the imt scheme is superior to the tanh scheme while both these methods are comparable to or even better than the standard methods like the gaussian or the chebyshev schemes in terms of accuracy and simplicity

extracting a basis with fixed block inside a matrix abstract given u an n m matrix of rank n whose columns are denoted by u j j m several authors have already considered the problem of finding a subset 1 m such that u i i span r n and tr i u i u i t 1 is minimized in this paper we generalize this problem by selecting arbitrary rank matrices instead of rank 1 matrices another generalization is considering the same problem while allowing a part of the matrix to be fixed the methods of selection employed develop into algorithms

computing cauchy principal value integrals using a standard adaptive quadrature we investigate the possibility of fast accurate and reliable computation of the cauchy principal value integrals a b f x x 1 d x a b using a standard adaptive quadrature in order to properly control the error tolerance for the adaptive quadrature and to obtain a reliable estimation of the approximation error we research the possible influence of round off errors on the computed result as the numerical experiments confirm the proposed method can successfully compete with other algorithms for computing such type integrals moreover the presented method is very easy to implement on any system equipped with a reliable adaptive integration subroutine

efficient algorithms for computing a strong rank revealing qr factorization given anm n matrixm withm n it is shown that there exists a permutation fi and an integer k such that the qr factorization myi q ak ckbk reveals the numerical rank of m the k k upper triangular matrix ak is well conditioned ilckll2 is small and bk is linearly dependent on ak with coefficients bounded by a low degree polynomial in n existing rank revealing qr rrqr algorithms are related to such factorizations and two algorithms are presented for computing them the new algorithms are nearly as efficient as qr with column pivoting for most problems and take o ran2 floating point operations in the worst case

analysis of facial motion patterns during speech using a matrix factorization algorithm this paper presents an analysis of facial motion during speech to identify linearly independent kinematic regions the data consists of three dimensional displacement records of a set of markers located on a subject s face while producing speech a qr factorization with column pivoting algorithm selects a subset of markers with independent motion patterns the subset is used as a basis to fit the motion of the other facial markers which determines facial regions of influence of each of the linearly independent markers those regions constitute kinematic eigenregions whose combined motion produces the total motion of the face facial animations may be generated by driving the independent markers with collected displacement records

a practical algorithm for computing cauchy principal value integrals of oscillatory functions abstract a new automatic quadrature scheme is proposed for evaluating cauchy principal value integrals of oscillatory functions 1 1 f x exp i x x 1 dx 1 1 r the desired approximation is obtained by expanding the function f in the series of chebyshev polynomials of the first kind and then by constructing the indefinite integral for a properly modified integrand to overcome the singularity the method is proved to converge uniformly with respect to both and for any function f satisfying max 1 x 1 f x

piecewise polynomial quadratures for cauchy singular integrals in this paper we propose piecewise polynomial methods for the approximation of cauchy principal value integrals and develop a simple efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise polynomial quadratures we present two examples to illustrate the advantages of these quadratures versus the gauss jacobi quadratures

a note on subset selection for matrices in an earlier paper the authors examined the problem of selecting rows of a matrix so that the resulting matrix is as non singular as possible however the proof of the key result in that paper is not constructive in this note we give a constructive proof for that result in addition we examine a case where as non singular as possible means maximizing a determinant and provide a new bound and a constructive proof for this case also

on solving singular integral equations via a hyperbolic tangent quadrature rule on propose un schema de resolution des equations integrales singulieres base sur une regle de quadrature a tangente hyperbolique l equation integrale est reduite a un systeme d equations lineaires apres quadrature et collocation

quadrature rule for indefinite integral of algebraic logarithmic singular integrands an automatic quadrature method is presented for approximating the indefinite integral of functions having algebraic logarithmic singularities q x y c f email protected x yf t t c alog t c dt 1 1 within a finite range 1 1 for some smooth function f t that is approximated by a finite sum of chebyshev polynomials we expand the given indefinite integral in terms of chebyshev polynomials by using auxiliary algebraic logarithmic functions present scheme approximates the indefinite integral q x y c f uniformly namely bounds the approximation error independently of the value c as well x and y this fact enables us to evaluate the integral transform q x y c f with varied values of x y and c efficiently some numerical examples illustrate the performance of the present quadrature scheme

roundoff errors in the problem of computing cauchy principal value integrals we investigate the possibility of fast accurate and reliable computation of the cauchy principal value integrals mathrm p int a b f x x tau 1 dx a tau b using standard adaptive quadratures in order to properly control the error tolerance for the adaptive quadrature and to obtain a reliable estimation of the approximation error we research the possible influence of round off errors on the computed result as the numerical experiments confirm the proposed method can successfully compete with other algorithms for computing such type integrals moreover the presented method is very easy to implement on any system equipped with a reliable adaptive integration subroutine

error bounds for gaussian quadrature of analytic functions for gaussian quadrature rules over a finite interval applied to analytic or meromorphic functions we develop error bounds from contour integral representations of the remainder term as in previous work on the subject we consider both circular and elliptic contours in contrast with earlier work however we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus we succeed in answering this question for a large class of weight distributions including all jacobi weights when the contour is a circle in the more difficult case of elliptic contours we can settle the question for certain special jacobi weight distributions with parameters pm frac 1 2 and we provide empirical results for more general jacobi weights we further point out that the kernel of the error functional at any complex point outside the interval of integration can be evaluated accurately and efficiently by a recursive procedure the same procedure is useful also t

a sinc quadrature subroutine for cauchy principal value integrals a subroutine is presented for the evaluation of cauchy principal value integrals based on a truncated sum of a sinc function series with the function evaluated at shifted sinc points the subroutine returns an approximation to the integral together with estimates of the relative and the absolute errors numerical results demonstrate that the new method works well on analytic functions with end point singularities

hilbert and hadamard transforms by generalized chebyshev expansion abstract an automatic quadrature is presented for approximating hadamard finite part fp integrals of a smooth function with a double pole singularity within the range of integration the quadrature rule is derived from the differentiation of an approximation to a cauchy principal value integral or the hilbert transform the approximation to the fp integral is represented as a function of the value of pole by using chebyshev polynomials of the second kind since the error can be estimated independently of the value of pole a set of integrals for a set of values of pole can be efficiently approximated to a required tolerance with the same number of function evaluations numerical examples are also included to illustrate the performance of the methods

sinc nystrom method for numerical solution of one dimensional cauchy singular integral equation given on a smooth arc in the complex plane on developpe une methode numerique basee sur les fonctions sinc pour la solution approchee d une equation integrale singuliere de cauchy a une dimension sur un arc regulier du plan complexe

an error analysis of the cauchy principal value integral evaluation by the imt scheme we provide an error analysis for the cauchy principal value integral evaluation by the imt method for analytic functions we get the discretization error e d in terms of a contour integral and we derive an asymptotic estimate for this error integral which shows that e d o e n 2 n 3 4 we give two numerical examples where the asymptotically estimated errors are compared with known actual values

application of a modified fft to product type integration abstract an automatic integration scheme is proposed for evaluating the so called product type indefinite integral q k y x k t t dt 1 x y where t is assumed to be a smooth function and k t are some singular or badly behaved functions typical examples for k t are 1n t c t c 1 cauchy principal value 1 t c and ei t 1 the function t is approximated by a truncated chebyshev series pn t of degree n whose coefficients are efficiently computed using the fft the approximation qn k to the integral q k is given by q k pn the sequence pn t is recursively generated until the required tolerance for the integral is satisfied to enhance the efficiency of the automatic quadrature the degree n is increased more slowly than doubling which is usually the case the evaluations of qn k q k pn for a set of x y c can be efficiently made by using recurrence relations for the singular kernels k t above numerical examples for the algebraic singular kernel k t t c 1 are included

generalized chebyshev interpolation and its application to automatic quadrature a generalized chebyshev interpolation procedure increasing a fixed number of sample points at a time is developed and analvzed it is incorporated into an efficient automatic quadrature scheme of clenshaw curtis tvpe numerical examples indicate that the present method is efficient not onlv for well behaved functions but for those w ith discontinuous low order derivatives by virtue of adequate error estimation as wvell as saving of sample points

on the convergence of interpolatory type quadrature rules for evaluating cauchy integrals the aim of this work is to analyse the stability and the convergence for the quadrature rule of interpolatory type based on the trigonometric approximation for the discretization of the cauchy principal value integrals 11 f t d we prove that the quadrature rule has almost optimal stability property behaving in the form o logn 1 sin2x x cos t using this result we show that the rule has an exponential convergence rate when the function f is differentiable enough when f possesses continuous derivatives up to order p 0 and the derivative f p t satisfies holder continuity of order we can also prove that the rule has the convergence rate of the form o a b log n n2 np p where is as small as we like a and b are constants depending only on x

rectangular maximum volume submatrices and their applications abstract we introduce a definition of the volume of a general rectangular matrix which is equivalent to an absolute value of the determinant for square matrices we generalize results of square maximum volume submatrices to the rectangular case show a connection of the rectangular volume with an optimal experimental design and provide estimates for a growth of coefficients and an approximation error in spectral and chebyshev norms three promising applications of such submatrices are presented recommender systems finding maximal elements in low rank matrices and preconditioning of overdetermined linear systems the code is available online

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

on selecting a maximum volume sub matrix of a matrix and related problems given a matrix email protected r m x n n vectors in m dimensions we consider the problem of selecting a subset of its columns such that its elements are as linearly independent as possible this notion turned out to be important in low rank approximations to matrices and rank revealing qr factorizations which have been investigated in the linear algebra community and can be quantified in a few different ways in this paper from a complexity theoretic point of view we propose four related problems in which we try to find a sub matrix email protected r m x k of a given matrix email protected r m x n such that i s m a x c the largest singular value of c is minimum ii s m i n c the smallest singular value of c is maximum iii k c email protected m a x c s m i n c the condition number of c is minimum and iv the volume of the parallelepiped defined by the column vectors of c is maximum we establish the np hardness of these problems and further show that they do not admit ptas we then study a natural greedy heuristic for the maximum volume problem and show that it has approximation ratio 2 o k l o g k our analysis of the greedy heuristic is tight to within a logarithmic factor in the exponent which we show by explicitly constructing an instance for which the greedy heuristic is 2 w k from optimal when a has unit norm columns a related problem is to select the maximum number of vectors with a given volume we show that if the optimal solution selects k columns then greedy will select w k logk columns providing a logk approximation

a method for the practical evaluation of the hilbert transform on the real line abstract we investigate a method for the numerical evaluation of the weighted hilbert transform over the entire real line exp x 2 f x x t 1 d x where the integral is taken in the cauchy principal value sense the method is based on a combination of two polynomial approximation processes in this way we obtain a procedure that is numerically stable and efficient the algorithm can be implemented in a fast way and existing well known software packages may be used from the point of view of theoretical error bounds the method is competitive generalizations to other approximation methods are also possible

a sinc hunter quadrature rule for cauchy principal value integrals a sine function approach is used to derive a new hunter type quadrature rule for the evaluation of cauchy principal value integrals of cer tain analytic functions integration over a general arc in the complex plane is considered special treatment is given to integrals over the interval 1 1 it is shown that the quadrature error is of order 0 e where n is the number of nodes used and where c is a positive constant which is indepen dent of n an application of the rule to the approximate solution of cauchy singular integral equations is also discussed numerical examples are included to illustrate the performance of the rule

fast monte carlo algorithms for matrices i approximating matrix multiplication motivated by applications in which the data may be formulated as a matrix we consider algorithms for several common linear algebra problems these algorithms make more efficient use of computational resources such as the computation time random access memory ram and the number of passes over the data than do previously known algorithms for these problems in this paper we devise two algorithms for the matrix multiplication problem suppose a and b which are m times n and n times p respectively are the two input matrices in our main algorithm we perform c independent trials where in each trial we randomly sample an element of 1 2 ldots n with an appropriate probability distribution cal p on 1 2 ldots n we form an m times c matrix c consisting of the sampled columns of a each scaled appropriately and we form a c times n matrix r using the corresponding rows of b again scaled appropriately the choice of cal p and the column and row scaling are crucial features of the algorithm when these are chosen judiciously we show that cr is a good approximation to ab more precisely we show that left ab cr right f o left a right f left b right f sqrt c where cdot f denotes the frobenius norm i e a 2 f sum i j a ij 2 this algorithm can be implemented without storing the matrices a and b in ram provided it can make two passes over the matrices stored in external memory and use o c m n p additional ram to construct c and r we then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm in addition we present a model the pass efficient model in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets in this model the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm the input matrices may be presented in any order of the entries and not just row or column order as is the case in many applications where e g the data has been written in by multiple agents in addition the input matrices may be presented in a sparse representation where only the nonzero entries are written

error pattern correcting cyclic codes tailored to a prescribed set of error cluster patterns a new class of cyclic codes is discussed which is highly tailored to a prescribed set of dominant error cluster patterns the cyclic code construction is based on a generator polynomial that produces a distinct syndrome set for each error pattern in the target set by tailoring the generator polynomial specifically to the set of dominant error patterns the code becomes highly effective in handling single and multiple occurrences of dominant error patterns at a very high code rate a list decoding strategy based on a set of test word error events is developed for the proposed codes which efficiently utilizes both the algebraic information from the captured syndrome and the reliability measures provided by the local correlators matched to the dominant error patterns by forcing a decoder to correct a single pattern event for each test input word multiple decoders running in parallel on the list of test words can effectively correct multiple error pattern occurrences within the channel detector output word

quasioptimality of maximum volume cross interpolation of tensors we consider a cross interpolation of high dimensional arrays in the tensor train format we prove that the maximum volume choice of the interpolation sets provides the quasioptimal interpolation accuracy that differs from the best possible accuracy by the factor which does not grow exponentially with dimension for nested interpolation sets we prove the interpolation property and propose greedy cross interpolation algorithms we justify the theoretical results and measure speed and accuracy of the proposed algorithm with numerical experiments

convergence results for piecewise linear quadratures for cauchy principal value integrals etablissement des conditions de convergence simple et uniforme des quadratures lineaires par morceaux pour les integrales a valeur principale de cauchy

uniform approximations to cauchy principal value integrals of oscillatory functions this paper presents an interpolatory type integration rule for the numerical evaluation of cauchy principal value integrals of oscillatory integrands 1 1e i w xf x email protected where 1 t 1 for a given smooth function f x the proposed method is constructed by interpolating f x at practical chebyshev points and subtracting out the singularity a numerically stable procedure is obtained and the corresponding algorithm can be implemented by fast fourier transform the validity of the method has been demonstrated by several numerical experiments

fast monte carlo algorithms for matrices ii computing a low rank approximation to a matrix in many applications the data consist of or may be naturally formulated as an m times n matrix a it is often of interest to find a low rank approximation to a i e an approximation d to the matrix a of rank not greater than a specified rank k where k is much smaller than m and n methods such as the singular value decomposition svd may be used to find an approximation to a which is the best in a well defined sense these methods require memory and time which are superlinear in m and n for many applications in which the data sets are very large this is prohibitive two simple and intuitive algorithms are presented which when given an m times n matrix a compute a description of a low rank approximation d to a and which are qualitatively faster than the svd both algorithms have provable bounds for the error matrix a d for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively in the first algorithm c columns of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling then it is shown that from c tc approximations to the top singular values and corresponding singular vectors may be computed from the computed singular vectors a description d of the matrix a may be computed such that mathrm rank d le k and such that left a d right xi 2 le min d mathrm rank d le k left a d right xi 2 poly k 1 c left a right 2 f holds with high probability for both xi 2 f this algorithm may be implemented without storing the matrix a in random access memory ram provided it can make two passes over the matrix stored in external memory and use o cm c 2 additional ram the second algorithm is similar except that it further approximates the matrix c by randomly sampling r rows of c to form a r times c matrix w thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a 2 f both algorithms take time which is polynomial in k 1 epsilon and log 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n our bounds improve previously published results with respect to the rank parameter k for both the frobenius and spectral norms in addition the proofs for the error bounds use a novel method that makes important use of matrix perturbation theory the probability distribution over columns of a and the rescaling are crucial features of the algorithms which must be chosen judiciously

linear time cur approximation of bem matrices in this paper we propose linear time cur approximation algorithms for admissible matrices obtained from the hierarchical form of boundary element matrices we propose a new approach called geometric sampling to obtain indices of most significant rows and columns using information from the domains where the problem is posed our strategy is tailored to boundary element methods bem since it uses directly and explicitly the cluster tree containing information from the problem geometry our cur algorithm has precision comparable with low rank approximations created with the truncated qr factorization with column pivoting qrcp and the adaptive cross approximation aca with full pivoting which are quadratic cost methods when compared to the well known linear time algorithm aca with partial pivoting we show that our algorithm improves in general the convergence error and overcomes some cases where aca fails we provide a general relative error bound for cur approximations created with geometrical sampling finally we evaluate the performance of our algorithms on traditional bem problems defined over different geometries

on a certain quadrature formula traduction anglaise d un article en japonais publie en aout 1970 dans la serie kokyuroku vol 91 pp 82 118

error event characterization on 2 d isi channels in this paper we analyze the distance properties of two dimensional 2 d intersymbol interference isi channels in particular the 2 d partial response class 1 pr1 channel which is an extension of the one dimensional 1 d pr1 channel the minimum squared euclidean distance of this channel is proved to be 4 and a complete characterization of the squared euclidean distance 4 error events is provided as for 1 d channels we can construct error state diagrams for 2 d channels to help characterize error events we propose an efficient error event search algorithm operating on the error state diagram that is applicable to any 2 d channel

a quadrature rule for weighted cauchy integrals abstract using a variable transformation t cos y for weighted cauchy integrals we develop an automatic quadrature method based on trigonometric interpolation we show that error bounds are independent of the location of poles

numerical integration of functions with poles near the interval of integration abstract an automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole or poles of order 2 i e a double pole on the real axis or a complex conjugate pair of double poles near the interval of integration the present scheme is based on product integration rules of interpolatory type using function values of the abscissae only in the interval of integration the integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the chebyshev polynomials the fast fourier transform fft technique is used to generate efficiently the sequence of the finite chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error numerical examples are included to illustrate the performance of the method