high order nonlinear initial value problems countably determined we give a numerical approximation of the solution of a high order nonlinear initial value problem by making use of certain properties of an adequate schauder basis

some performances of local bivariate quadratic c 1 quasi interpolating splines on nonuniform type 2 triangulations we investigate spline quasi interpolants defined by c1 bivariate quadratic b splines on nonuniform type 2 triangulations and by discrete linear functionals based on a fixed number of triangular mesh points either in the support or close to the support of such b splines we show they can approximate a real function and its partial derivatives up to an optimal order and we derive local and global upper bounds we also present some numerical and graphical results

multilevel quadratic spline quasi interpolation abstract in this paper we present new approximation schemes based on both quasi interpolation and multilevel methods by using bivariate quadratic b spline functions defined on simple and multiple knot type 2 triangulations improving the classical quasi interpolating spline results we also prove polynomial reproduction optimal approximation order and propose some numerical results and applications

total positivity and its applications preface i spline functions total positivity of the spline kernel and its applications b bojanov total positivity properties of lb splines t lyche l l schumaker total positivity and splines k morken totally positive systems and interpolation by subspaces of generalized splines m sommer h strauss ii matrix theory preservation of total nonnegativity under the hadamard product and related topics j garloff d g wagner vertex implications for totally nonnegative under the hadamard product and related topics j garloff on factorizations of totally positive matrices j garloff on factorizations of totally positive matrices m gasca j m pena iii geometric modeling total positivity and optimal bases j m carnicer j m pena total positivity and the shape of curves t n t goodman tchebycheff curves m l mazure h pottmann iv probability and mathematical biology analytical and probabilistic methods for totally positive matrices g s goodman some inequalities of total positivity in pure and applied mathematics s karlin z ziegler new directions and problems of multivariate total positivity motivated by genetic recombination probabilities s karlin v approximation theory tchebycheff spaces and total positivity j m carnicer j m pena cebysev and weak cebysev systems r a zalik vi complex analysis zero sets of entire fourier transforms of multiply positive density functions o m katkova vii statistics total nonnegativity of moment matrices and its application to e optimal designs in polynomial regression b heiligers analysis of regression dependence in 2 xn bivariate distributions and some applications in contingency tables s kasala m b rao viii real analysis total positivity and nonlinear analysis r q jia solutions of certain functional equations s karlin alternance and snakes in the nonlinear finite moment problem k tatalian ix combinatorics the applications of total positivity to combinatorics and conversely f brenti x integral equations spectral properties of totally positive kernels and matrices a pinkus index

accurate solution of dense linear systems part i algorithms in rounding to nearest we investigate how extra precise accumulation of dot products can be used to solve ill conditioned linear systems accurately for a given p bit working precision extra precise evaluation of a dot product means that the products and summation are executed in 2p bit precision and that the final result is rounded into the p bit working precision denote by u 2 p the relative rounding error unit in a given working precision we treat two types of matrices first up to condition number u 1 and second up to condition number u 2 for both types of matrices we present two types of methods first for calculating an approximate solution and second for calculating rigorous error bounds for the solution together with the proof of non singularity of the matrix of the linear system in the first part of this paper we present algorithms using only rounding to nearest in part ii we use directed rounding to obtain better results all algorithms are given in executable matlab code and are available from my homepage

the index of an algebraic variety let k be the field of fractions of a henselian discrete valuation ring mathcal o k let xk k be a smooth proper geometrically connected scheme admitting a regular model x mathcal o k we show that the index xk k of xk k can be explicitly computed using data pertaining only to the special fiber xk k of the model x

product integration in the presence of a singularity this paper considers the approximate evaluation of int a b k x f x dx where k is a fixed lebesgue integrable function by quadrature rules of the form sigma i 0 m n w in f x in normally rules of this kind are used only for smooth functions f and any singularities are incorporated into k here however we allow f to have a singularity either at an endpoint or in the interior general convergence properties are established for a wide class of product integration rules more detailed results are established for the class of rules which are exact if f belongs to a specified family of piecewise polynomials

which spaces for design we determine the largest class of spaces of sufficient regularity which are suitable for design in the sense that they do possess blossoms it is the class of all spaces containing constants of which the spaces derived under differentiation are quasi extended chebyshev spaces i e they permit hermite interpolation taylor interpolation excepted it is also the class of all spaces which possess bernstein bases or of all spaces for which any associated spline space does possess a b spline basis note that blossoms guarantee that such bases are normalised totally positive bases they even are the optimal ones

approximating the finite hilbert transform via a companion of ostrowski s inequality for function of bounded variation and applications the finite hilbert transform plays an important role in scientific and engineering computing by using a companion of ostrowski s inequality for function of bounded variation we give some new approximations of the finite hilbert transform which may have the better error bounds than the known results obtained via ostrowski type inequality some numerical experiments are also presented

on the convergence of spline product quadratures for cauchy principal value integrals abstract in a recent paper this journal 1990 the authors proposed product integration formulas for the numerical evaluation of the cauchy principal value integral f1 1u x f x x dx based on cubic spline interpolation of f and obtained convergence results for functions f ck 1 1 k 1 2 or 3 in this report the same rules are considered and their convergence is investigated for a larger class of functions f an error bound and some uniform convergence results are established in the case of equally spaced quadrature nodes for functions f satisfying a holder condition of order on 1 1 0

least squares methods for solving partial differential equations by using bezier control points abstract we discuss the least squares methods for solving partial differential equations on arbitrary polygon domain by using bezier control points it uses triangular bezier patches of degree n with c k continuity to approximate the exact solution of partial differential equations numerical examples are provided to illustrate the proposed method is flexible

growth factors of pivoting strategies associated with neville elimination neville elimination is a direct method for solving linear systems several pivoting strategies for neville elimination including pairwise pivoting are analyzed bounds for two different kinds of growth factors are provided finally an approximation of the average normalized growth factor associated with several pivoting strategies is computed and analyzed using random matrices

spline approximation for cauchy principal value integrals abstract in this paper we prove the uniform convergence of some quadrature formulas based on spline approximation for cauchy principal value integrals of the type f b a w x f k x x y d x k 0 1 and we present some numerical applications in particular we apply our rules to the well known prandtl s integral equation

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

efficient quadrature rules based on spline quasi interpolation for application to iga bems abstract we propose a new class of quadrature rules for the approximation of weakly and strongly singular integrals based on the spline quasi interpolation scheme introduced in mazzia and sestini 2009 these integrals in particular occur in the entries of the stiffness matrix coming from isogeometric boundary element methods iga bems the presented formulas are efficient since they combine the locality of any spline quasi interpolation scheme with the capability to compute the modified moments for b splines i e the weakly or strongly singular integrals of such functions no global linear system has to be solved to determine the quadrature weights but just local systems whose size linearly depends on the adopted spline degree the rules are preliminarily tested in their basic formulation i e when the integrand is defined as the product of a singular kernel and a continuous function g then such basic formulation is compared with a new one specific for the approximation of the singular integrals appearing in the iga bem context where a b spline factor is explicitly included in g such a variant requires the usage of the recursive spline product formula given in morken 1991 and it is useful when the ratio between g and its b spline factor is smooth enough

an algorithm based on q i modified splines for singular integral models abstract in this paper we propose an algorithm supporting an approximation using quasi interpolatory q i splines for the numerical solution of integro differential equations with cauchy singular kernel some different choices of parameters defining the numerical model are analyzed and compared in view of the algorithm efficiency

quasi interpolatory splines based on schoenberg points by using the schoenberg points as quasi interpolatory points we achieve both generality and economy in contrast to previous sets which achieve either generality or economy but not both the price we pay is a more complicated theory and weaker error bounds although the order of convergence is unchanged applications to numerical integration are given and numerical examples show that the accuracy achieved using the schoenberg points is comparable to that using other sets

non symbolic algorithms for the inversion of tridiagonal matrices a representation for the entries of the inverse of general tridiagonal matrices is based on the determinants of their principal submatrices it enables us to introduce through the linear recurrence relations satisfied by such determinants a simple algorithm for the entries of the inverse of any tridiagonal nonsingular matrix reduced as well as unreduced the numerical approach is preserved here without invoking the symbolic computation for tridiagonal diagonally dominant matrices a scaling transformation on the recurrences allows us to give another algorithm to avoid overflow and underflow

characterizations of rectangular totally and strictly totally positive matrices abstract an n m real matrix a is said to be totally positive strictly totally positive if every minor is nonnegative positive in this paper we study characterizations of these classes of matrices by minors by their full rank factorization and by their thin qr factorization

a numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions abstract in this paper we use hat basis functions to solve the system of fredholm integral equations sfies and the system of volterra integral equations svies of the second kind this method converts the system of integral equations into a linear or nonlinear system of algebraic equations also we consider the order of convergence of the method and show that it is o h 2 application of the method on some examples show its accuracy and efficiency

a computational method for solving a class of two dimensional volterra integral equations the use of the geometric series theorem together with schauder bases in certain banach spaces allows us to design a numerical algorithm in order to solve an important type of two dimensional linear integral equations a study of the convergence and error is presented here all the calculations can be easily implemented and the efficiency of this method will be shown with numerical results

conditioning and accurate computations with pascal matrices a result on the ill conditioning of pascal matrices is proved however it is shown that bidiagonal factorizations of pascal matrices can be applied to perform computations with high relative accuracy

generalized oscillatory matrices we define a new class of generalized oscillatory matrices shortly go matrices over a noncommutative ring with identity and a positive part similarly to the classical case this class consists of square generalized totally nonnegative matrices gtn matrices of which some power is generalized totally positive using the previously defined ordering of invertible gtn matrices we study in particular so called basic go matrices which form in a sense atoms of this ordering

numerical integration based on a multilevel quartic quasi interpolation operator quasi interpolation is very important in the study of the scattered data approximation numerical integration and numerical solutions of differential equations in this paper we proposed a multilevel quasi interpolation scheme by using b spline basis functions on quartic spline space moreover the proposed scheme is applied to the numerical integration of two dimensional singular integrals some numerical results with contrast to the existing methods are provided

accurate eigenvalues of certain sign regular matrices we present a new o n 3 algorithm for computing all eigenvalues of certain sign regular matrices to high relative accuracy in floating point arithmetic the accuracy and cost are unaffected by the conventional eigenvalue condition numbers a matrix is called sign regular when the signs of its nonzero minors depend only of the order of the minors the sign regular matrices we consider are the ones which are nonsingular and whose kth order nonzero minors are of sign 1 k k 1 2 for all k this class of matrices can also be characterized as nonsingular totally nonnegative matrices with columns in reverse order we exploit a characterization of these particular sign regular matrices as products of nonnegative bidiagonals and the reverse identity we arrange the computations in such a way that no subtractive cancellation is encountered thus guaranteeing high relative forward accuracy

numerical solution of second kind fredholm integral equations system by using a taylor series expansion method in this paper we present a taylor series expansion method for a second kind fredholm integral equations system with smooth or weakly singular kernels this method reduce the system of integral equations to a linear system of ordinary differential equation after constructing boundary conditions this system reduce to a system of equations that can be solved easily with any of the usual methods finally we show the efficiency of the method using some numerical examples

a quadrature rule of interpolatory type for cauchy integrals abstract we construct a quadrature rule of interpolatory type based on a trigonometric interpolation for cauchy integrals we obtain the same error bound o ann as hasegawa and torii math comp 56 1991 741 754 numerical examples are included to substantiate the performance of the present method

a note on matrices with maximal growth factor for neville elimination neville elimination is a direct method for the solution of linear systems of equations with advantages for some classes of matrices and in the context of pivoting strategies for parallel implementations the growth factor is an indicator of the numerical stability of an algorithm in the literature bounds for the growth factor of neville elimination with some pivoting strategies have appeared in this work we determine all the matrices such that the minimal upper bound of the growth factor of neville elimination with those pivoting strategies is reached

a univariate quasi multiquadric interpolationwith better smoothness in this paper we propose a multilevel univariate quasi interpolation scheme usingmultiquadric basis it is practical as it does not require derivative values of the function being interpolated it has a higher degree of smoothness than the original level 0 formula as it allows a shape parameter c o h our level 1 quasi interpolation costs o email protected n flops to set up it preserves strict convexity and monotonicity when c o h we prove the proposed scheme converges with a rate of o h 2 email protected h furthermore if both a and are relatively small compared with email protected the convergence rate will increase we verify numerically that c h is a good shape parameter to use for our method hence we need not find the optimal parameter for all test functions both convergence speed and error are optimized for c between 0 5h and 1 5h our method can be generalized to a multilevel scheme we include the numerical results for the level 2 scheme the shape parameter of the level 2 scheme can be chosen between 2h to 3h

numerical integration based on approximating splines abstract convergence results are proved for product integration rules based on approximating splines these results are both for bounded and unbounded integrands pointwise and uniform convergence results are proved for sequences of cauchy principal values of these approximating splines

spline collocation for singular integro differential equations over 0 1 this paper analyses the convergence of spline collocation methods for singular integro differential equations over the interval 0 1 as trial functions we utilize smooth polynomial splines the degree of which coincides with the order of the equation depending on the choice of collocation points we obtain sufficient and even necessary conditions for the convergence in sobolev norms we give asymptotic error estimates and some numerical results

embedding problems over large fields in this paper we study galois theoretic properties of a large class of fields a class which includes all fields satisfying a universal local global principle for the existence of rational points on varieties as applications we give a positive answer to a long standing conjecture which originates in an unpublished note of roquette and asserts that the absolute galois group of a countable hilbertian pac field is profinite free see f j problem 24 41 secondly we give new evidence for the conjecture of shafarevich which asserts that the absolute galois group of qab is profinite free finally one of the most interesting applications of the theory we develop here is the insight in the galois structure of the totally eg adic numbers

on the evaluation of cauchy principal value integrals by rules based on quasi interpolating splines abstract in this paper we consider the numerical evaluation of one dimensional cauchy principal value integrals of the form 1 1 k x f x x d x 1 by rules obtained by subtracting out the singularity and then applying quadrature formulas based on quasi interpolating splines

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

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

symmetric functions and the vandermonde matrix this work deduces the lower and the upper triangular factors of the inverse of the vandermonde matrix using symmetric functions and combinatorial identities the l and u matrices are in turn factored as bidiagonal matrices the elements of the upper triangular matrices in both the vandermonde matrix and its inverse are obtained recursively the particular value x i 1 q qi 1 in the indeterminates of the vandermonde matrix is investigated and it leads to q binomial and q stirling matrices it is also shown that q stirling matrices may be obtained from the pascal matrix

product integration of singular integrands using quasi interpolatory splines we consider product integrals of the form kf 11k x f x dx where kf l1 but f is unbounded in the integration interval we shall approximate f by a sequence of quasi interpolatory splines qnf involving only function values and study the convergence of i kqnf to i kf it turns out that we can ignore an endpoint singularity but must in general avoid an interior singularity

numerical approaches for systems of volterra fredholm integral equations in this work we introduce two numerical methods for solving systems of volterra fredholm integral equations in the nonlinear case we suggest a fixed point method where the iterations are perturbed in a suitable way according to a schauder basis in the banach space of continuous functions c a b 2 in the linear case we propose a collocation method based on a particular class of approximating functions in both methods convergence analysis and or low computational cost are analysed taking into account the properties of the basis under consideration numerical results confirm the theoretical study

product integration of piecewise continuous integrands based on cubic splineinterpolation at equally spaced nodes in this paper we consider the approximate evaluation of int limits a b k x f x dx wherek x is a fixed lebesgue integrable function by product formulas of the form sum limits i 0 n w i f x i based on cubic spline interpolation of the functionf generally whenever it is possible product quadratures incorporate the bad behaviour of the integrand in the kernelk here however we allowf to have a finite number of jump discontinuities in a b convergence results are established and some numerical applications are given for a logarithmic singularity structure in the kernel

on the construction of local quadratic spline quasi interpolants on bounded rectangular domains in this paper local bivariate c 1 spline quasi interpolants on a criss cross triangulation of bounded rectangular domains are considered and a computational procedure for their construction is proposed numerical and graphical tests are provided

spline product quadrature rules for cauchy singular integrals abstract in this paper product quadrature rules based on cubic spline interpolation are obtained for the numerical evaluation of cauchy singular integrals and an error bound is proposed some numerical examples that test the performance of the rules are given

on the convergence of closed interpolatory integration rules based on the zeros of gegenbauer polynomials abstract the gegenbauer polynomials c n x are orthogonal with respect to the weight function 1 x2 1 2 it is known that the interpolatory integration rules based on the zeros of 1 x2 c n 2 x converge for all riemann integrable functions for 1 2 4 this is shown to hold also for 1 2 1 2

numerical evaluation of cauchy principal value integrals with singular integrands convergence results are proved for sequences of interpolatory integration rules for cauchy principal value integrals of the form fk x f x x a dx 1 a 1 when f x is singular at a point a 7 i and the singularity is ignored or avoided

product integration based on hermite feje r interpolation abstract the use of generalized hermite fejer interpolation polynomials for product integration is studied algorithms for implementing this method are developed and convergence results are proved comparisons are made between this new approach and the classical approach based on lagrange interpolation polynomials it turns out that integration rules based on hermite fejer interpolation converge in more cases but in general more slowly than those based on lagrange interpolation when the latter converge

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

polynomial least squares fitting in the bernstein basis the problem of polynomial least squares fitting in which the usual monomial basis is replaced by the bernstein basis is considered the coefficient matrix of the overdetermined system to be solved in the least squares sense is then a rectangular bernstein vandermonde matrix in order to use the method based on the qr decomposition of a the first stage consists of computing the bidiagonal decomposition of the coefficient matrix a starting from that bidiagonal decomposition an algorithm for obtaining the qr decomposition of a is the applied finally a triangular system is solved by using the bidiagonal decomposition of the r factor of a some numerical experiments showing the behavior of this approach are included

growth factor and expected growth factor of some pivoting strategies several definitions of growth factors for gaussian elimination are compared some new pivoting strategies intermediate between partial pivoting and rook pivoting are introduced for random matrices an approximation of the average normalized growth factor associated with several pivoting strategies is computed and analyzed a stationary behavior of the expected growth factors of the new pivoting strategies is observed bounds for the growth factors of these pivoting strategies are provided it is also shown that partial pivoting by columns produces small growth factors for matrices appearing in practical observations and for which the growth factors produced by partial pivoting are very large

a new b spline collocation method for singular integro differential equations of higher orders abstract two numerical schemes based on b spline collocation method are presented for approximate solution of the systems of singular integro differential equations of higher orders appearing in fracture problems with steigmann ogden surface energy it is shown that the resulting systems of linear equations are well conditioned and the numerical examples are in good agreement with known solutions although this paper is concerned with the numerical solution to the specific equations the proposed method can be easily generalized to a large class of singular integro differential equations

about some numerical approaches for mixed integral equations abstract in this paper we tackle on mixed volterra fredholm integral equations as in linear as in non linear cases to the aim to obtain numerical solutions of these models the authors propose in the linear case the direct collocation method using a p order quasi interpolating spline class and in the nonlinear case the fixed point method based on polynomial approximation built by schauder tensor bases the advantages of both methods are outlined and their convergence is studied numerical results confirm the theoretical statements

solving system of volterra fredholm integral equations with bernstein polynomials and hybrid bernstein block pulse functions this work approximates the unknown functions based on the bernstein polynomials and hybrid bernstein block pulse functions in conjunction with the collocation method for the numerical solution of system of fredholm volterra integral equations in both methods the system of integral equations is approximated by the gauss quadrature formula with respect to the legendre weight function the proposed methods reduce the system of integral equations to a system of algebraic equations that can be easily solved by any usual numerical methods moreover the convergence analysis of these algorithms will be shown by preparing some theorems numerical experiments are presented to show the superiority and efficiency of proposed methods in comparison with some other well known methods

bidiagonal factorizations and quasi oscillatory rectangular matrices abstract a real matrix a of size m n is called totally nonnegative totally positive if all its minors are nonnegative positive a variant of the neville elimination process is studied in relation to the existence of a totally nonnegative elementary bidiagonal factorization of a the class of quasi oscillatory rectangular matrices which in the square case contains the oscillatory matrices is introduced and a characterization of this class of matrices by incorporating bidiagonal factorization is showed

product integration of singular integrands based on cubic spline interpolation at equally spaced nodes in this paper the convergence of product integration rules based on cubic spline interpolation at equally spaced nodes with not a knot end condition is investigated for integrand functions with a interior or endpoint singularity in the integration interval

the numerical evaluation of a 2 d cauchy principal value integral arising in boundary integral equation methods in this paper we consider the problem of computing 2 d cauchy principal value integrals of the form

direct and iterative methods for the numerical solution of mixed integral equations abstract in this paper we analyze and compare two classical methods to solve volterra fredholm integral equations the first is a collocation method the second one is a fixed point method both of them are proposed on a particular class of approximating functions precisely the first method is based on a linear spline class approximation and the second one on schauder linear basis we analyze some problems of convergence and we propose some remarks about the peculiarities and adaptability of both methods numerical results complete the work

numerical evaluation of cauchy principal value integrals based on local spline approximation operators abstract in this paper a local spline approximation method defined for any function l 1 1 1 is applied to evaluate cauchy principal value integrals and convergence results with an error bound are established some comparisons with other methods and numerical examples are given

modeling of a curvilinear planar crack with a curvature dependent surface tension an approach to modeling fracture incorporating interfacial mechanics is applied to the example of a curvilinear plane strain crack the classical neumann boundary condition is augmented with curvature dependent surface tension it is shown that the considered model eliminates the integrable crack tip stress and strain singularities of order 1 2 present in the classical linear fracture mechanics solutions and also leads to the sharp crack opening that is consistent with empirical observations unlike for the case of a straight crack for a general curvilinear crack some components of the stresses and the derivatives of the displacements may still possess weaker singularities of a logarithmic type generalizations of the present study that lead to complete removal of all crack tip singularities including logarithmic are the subject of a future paper

comparing pivoting strategies for almost strictly sign regular matrices abstract in this paper some properties of two determinant pivoting for neville elimination are presented in particular we consider a zero increasing property and we show an optimal normwise growth factor comparisons with other pivoting strategies for neville elimination and with gaussian elimination with partial pivoting of almost strictly sign regular matrices are performed numerical examples are included

elastic surface substrate interactions a theory for three dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described the gurtin murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance this modification yields a model that can be used to describe equilibrium deformations in the presence of compressive surface stress fields an associated variational theory is given and material symmetry considerations are discussed the theory is illustrated by examples

a numerical algorithm for singular perturbation problems exhibiting weak boundary layers abstract a class of singularly perturbed two point boundary value problems bvp for second order ordinary differential equations is considered here to avoid the numerical difficulties in the solution to these problems we divide the domain into two subdomains the first bvp is a layer domain problem and the second bvp is a regular domain problem error estimates are derived for the numerical solution numerical examples are provided in support of the proposed method

on the numerical solution of the generalized prandtl equation using variation diminishing splines convergence results are proved for cauchy principal value integrals of the schoenberg variation diminishing splines and its first derivative the use of such splines in the numerical solution of the prandtl and generalized prandtl integral equations is proposed a nystrom type method and a modified nystrom method are used and compared computationally

bivariate quartic spline spaces and quasi interpolation operators in this paper we study two bivariate quartic spline spaces s 4 3 2 d m n 2 and s 4 2 3 d m n 2 and present two classes of quasi interpolation operators in the two spaces respectively some results on the operators are given

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

a collection of examples where neville elimination outperforms gaussian elimination neville elimination is an elimination procedure alternative to gaussian elimination it is very useful when dealing with totally positive matrices for which nice stability results are known here we include examples most of them test matrices used in matlab which are not totally positive matrices where neville elimination outperforms gaussian elimination

ramification theory for varieties over a perfect field for an l adic sheaf on a variety of arbitrary dimension over a perfect field we define the swan class measuring the wild ramification as a 0 cycle class supported on the ramification locus we prove a lefschetz trace formula for open varieties and a generalization of the grothendieck ogg shararevich formula using the swan class

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

neville elimination a study of the efficiency using checkerboard partitioning abstract it is well known that checkerboard partitioning can exploit more concurrency than striped partitioning because the matrix computation can be divided among more processors than in the case of striping in this work we analyze the performance of neville method when a checkerboard partitioning is used focusing on the special case of block cyclic checkerboard partitioning this method is an alternative to gaussian elimination and it has been proved to be very useful for some classes of matrices such as totally positive matrices the performance of this parallel system is measured in terms of the efficiency the fraction of time for which a processor is usefully employed which in our model is close to one when the optimum block size is used also we have executed our algorithms on a parallel pc cluster observing that both efficiencies theoretical and empirical are quite similar

a matricial description of neville elimination with applications to total positivity abstract the neville elimination process used by the authors in some previous papers in connection with totally positive matrices is studied in detail in the case of nonsingular matrices a wide class of matrices is found where neville elimination has a lower computational cost than gauss elimination finally some new characterizations are obtained for strictly totally positive and nonsingular totally positive matrices in terms of their neville elimination and that of their inverses

on rational fixed points of finite group actions on the affine space consider a finite l group acting on the affine space of dimension n over a field k whose characteristic differs from l we prove the existence of a fixed point rational over k in the following cases the field k is p special for some prime p different from its characteristic the field k is perfect and fertile and n 3

product integration of logarithmic singular integrands based on cubic splines abstract this paper is concerned with the practical evaluation of the product integral 1 1 f x k x d x for the case when k x in x 1 1 and f is bounded in 1 1 the approximation is a quadrature rule where the weights w n n i are chosen to be exact when f is given by a linear combination of a chosen set of functions n j in this paper the functions n j are chosen to be cubic b splines an error bound for product quadrature rules based on cubic splines is provided examples that test the performance of the product quadrature rules for different choices of the function are given a comparison is made with product quadrature rules based on first kind chebyshev polynomials

biorthogonal systems for solving volterra integral equation systems of the second kind with the aid of biorthogonal systems in adequate banach spaces the problem of approximating the solution of a system of nonlinear volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically

on the stability of exponential fitting bdf algorithms we present bdf type formulas capable of the exact integration with only round off errors of differential equations whose solutions are linear combinations of an exponential with parameter l and ordinary polynomials for l 0 new formulas reduces to the classical bdf formulas plots of their 0 stability regions in terms of l are provided plots of their regions absolute stability that include all the negative real axis are provided numerical examples shows the efficiency of the proposed codes specially when we are integrating stiff oscillatory problems