orthogonal polynomial approach to estimation of poles of rational functions from data on open curves this paper deals with the problem of finding poles of rational functions from function values on open curves in the complex plane for this problem nara and ando recently proposed an algorithm that reduces the problem to a system of linear equations through contour integration the main aim of this paper is to analyze and improve this algorithm by giving a new interpretation to the algorithm in terms of orthogonal polynomials it is demonstrated that the system of linear equations is not always uniquely solvable and that this difficulty can be remedied by doubling the number of the linear equations moreover to cope with discretization errors caused by numerical integration we introduce new polynomials similar in spirit to discrete orthogonal polynomials which yield an algorithm free from discretization errors

toda equation and special polynomials associated with the garnier system abstract we prove that a certain sequence of functions of the garnier system satisfies toda equation we construct algebraic solutions of the system by the use of toda equation then show that the associated functions are expressed in terms of the universal character which is a generalization of schur polynomial attached to a pair of partitions

an iterative computation of approximations on korobov like spaces this paper treats the multidimensional application of a previous iterative monte carlo algorithm that enables the computation of approximations in l2 the case of regular functions is studied using a fourier basis on periodised functions legendre and tchebychef polynomial bases the dimensional effect is reduced by computing these approximations on korobov like spaces numerical results show the efficiency of the algorithm for both approximation and numerical integration

superconvergence of a class of expanded discontinuous galerkin methods for fully nonlinear elliptic problems in divergence form abstract for fully nonlinear elliptic boundary value problems in divergence form improved error estimates are derived in the frame work of a class of expanded discontinuous galerkin methods it is shown that the error estimate for the discrete flux in l 2 norm is of order k 1 when piecewise polynomials of degree k 1 are used to approximate both potential as well as flux variables then solving a discrete linear elliptic problem in each element locally a suitable post processing of the discrete potential is proposed and it is proved that the resulting post processed potential converges with order of convergence k 2 in l 2 norm by choosing stabilizing parameters appropriately similar results are derived for the expanded hdg methods for nonlinear elliptic problems

a projection method for generalized eigenvalue problems using numerical integration in this paper we propose a method for finding certain eigenvalues of a generalized eigenvalue problem that lie in a given domain of the complex plane the proposed method projects the matrix pencil onto a subspace associated with the eigenvalues that are located in the domain via numerical integration

electrostatic fields without singularities theory algorithms and error analysis the following problems that arise in the computation of electrostatic forces and in the boundary element method are considered given two convex interior disjoint polyhedra in 3 space endowed with a volume charge density which is a polynomial in the cartesian coordinates of r 3 compute the coulomb force acting on them given two interior disjoint polygons in 3 space endowed with a surface charge density which is polynomial in the cartesian coordinates of r 3 compute the normal component of the coulomb force acting on them for both problems adaptive gaussian approximation algorithms are given which for n gaussian points in time o n achieve absolute error o c n for a constant c 1 such a result improves upon previously known best asymptotic bounds this result is achieved by blending techniques from integral geometry computational geometry and numerical analysis in particular integral geometry is used in order to represent the forces as integrals whose kernal is free from singularities

a mixed finite element method for nonlinear elasticity two fold saddle point approach and a posteriori error estimate we extend the applicability of stable mixed finite elements for linear plane elasticity such as peers to a mixed variational formulation of hyperelasticity the present approach is based on the introduction of the strain tensor as a further unknown which yields a two fold saddle point nonlinear operator equation for the corresponding weak formulation we provide the uniqueness of solution for the continuous and discrete schemes and derive the usual cea estimate for the associated error finally a reliable a posteriori error estimate based on the solution of local dirichlet problems and well suited for adaptive computations is also given

a rational function preconditioner for indefinite sparse linear systems this paper introduces a rational function preconditioner for linear systems with indefinite sparse matrices a by resorting to rational functions of a the algorithm decomposes the spectrum of a into two disjoint regions and approximates the restriction of a 1 on these regions separately we show a systematic way to construct these rational functions so that they can be applied stably and inexpensively an attractive feature of the proposed approach is that the construction and application of the preconditioner can exploit two levels of parallelism moreover the proposed preconditioner can be modified at a negligible cost into a preconditioner for a nearby matrix of the form a ci which can be useful in some applications the efficiency and robustness of the proposed preconditioner are demonstrated on a few tests with challenging model problems including problems arising from the helmholtz equation in three dimensions

a posteriori error estimators for the upstream weighting mixed methods for convection diffusion problems abstract in this paper we design a posteriori error estimators for the mixed finite element approximation of the convection diffusion problems with dominant convection which exhibit solutions with internal layers and boundary layers a discontinuous upstream weighting scheme in association with the mixed method is used for the convection term the a posteriori error estimator yields global upper and local lower bounds on the error measured in the l2 norm we present several numerical results to show the performance of the error estimator we find that our estimator is not only reliable and efficient but also robust for various numerical examples

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

an error analysis of two related quadrature methods for computing zeros of analytic functions we present an error analysis for two related quadrature methods the delves lyness method and its modification by kravanja sakurai and van barel for computing all the zeros of an analytic function that lie inside the unit circle we consider the forward as well as the backward approximation error in case the integrals are computed via the trapezoidal rule on the unit circle contrary to the delves lyness method the quadrature error that arises from the zeros located inside the unit circle does not affect the results of the approach of kravanja et al numerical experiments illustrate our main results

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

filon clenshaw curtis rules for a class of highly oscillatory integrals with logarithmic singularities in this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities for these quadrature rules we derive error estimates in terms of n the number of nodes k the rate of oscillations and a sobolev like regularity of the function we prove that the method is not only robust but the error even decreases for fixed n as k increases practical issues about the implementation of the rule are also covered in this paper by a writing down ready to implement algorithms b analysing the numerical stability of the computations and c estimating the overall computational cost we finish by showing some numerical experiments which illustrate the theoretical results presented in this paper

uniform approximation to fractional derivatives of functions of algebraic singularity fractional derivative d qf x 0 1 with g x being a well behaved function we propose a quadrature method for uniformly approximating d q x ag x the present method consists of interpolating g x at abscissae in 0 1 by a finite sum of chebyshev polynomials it is shown that the use of the lower endpoint x 0 as an abscissa is essential for the uniform approximation namely to bound the approximation errors independently of x 0 1 numerical examples demonstrate the performance of the present method

numerical integration of the three dimensional green kernel for an electromagnetic problem in this paper we present a method for the numerical integration of the three dimensional green kernel over two triangles this method is compared with the results obtained by other algorithms the comparison proves the efficiency of the technique in removing the numerical singularity furthermore we provide a numerical algorithm for increasing the computation accuracy without refining the finite element mesh

cubature formulae and orthogonal polynomials abstract the connection between orthogonal polynomials and cubature formulae for the approximation of multivariate integrals has been studied for about 100 yr the article j radon published about 50 yr ago j radon zur mechanischen kubatur monatsh math 52 1948 286 300 has been very influential in this text we describe some of the results that were obtained during the search for answers to questions raised by his article

approximate solution of abel integral equation this paper presents a new stable approximate inversion of abel integral equation by using the taylor expansion of the unknown function abel equation is approximately transformed to a system of linear equations for the unknown function together with its derivatives a desired solution can be determined by solving the resulting system according to cramer s rule this method gives a simple and closed form of approximate abel inversion which can be performed by symbolic computation the nth order approximation is exact for a polynomial of degree up to n abel integral equation is approximately expressed in terms of integrals of input data so the suggested approach is stable for experimental data with random noise an error analysis of this approach is given finally several numerical examples are given to illustrate the accuracy and stability of this method

using implicitly filtered rks for generalised eigenvalue problems the rational krylov sequence rks method can be seen as a generalisation of arnoldi s method it projects a matrix pencil onto a smaller subspace this projection results in a small upper hessenberg pencil as for the arnoldi method rks can be restarted implicitly using the qr decomposition of a hessenberg matrix this restart comes with a projection of the subspace using a rational function in this paper it is shown how the restart can be worked out in practice in a second part it is shown when the filtering of the subspace basis can fail and how this failure can be handled by deflating a converged eigenvector from the subspace using a schur decomposition

a mixed finite element method for a strongly nonlinear second order elliptic problem the approximation of the solution of the first boundary value problem for a strongly nonlinear second order elliptic problem in divergence form by the mixed finite element method is considered existence and uniqueness of the approximation are proved and optimal error estimates in l 2 are established for both the scalar and vector functions approximated by the method error estimates are also derived in l q 2 q

l error estimates for mixed methods for elliptic partial differential equations l infty error estimates of optimal order are derived for the solution of a semilinear second order elliptic problem by mixed methods using raviart thomas elements estimates are also obtained for the associated vector function approximated in the mixed method and for its divergence all the estimates are valid even with the lowest indexed mixed finite elements

a priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations we study the mixed finite element approximation of the second order elliptic problem with gradient nonlinearities in two and three space dimensions existence and uniqueness of the approximate solution are proved and optimal order a priori error estimates in l m omega are obtained also reliable and efficient a posteriori error estimators measured in the l m omega norm are derived numerical examples are provided to illustrate the performance of the proposed estimator

mixed finite element methods for nonlinear second order elliptic problems mixed finite element methods are developed to approximate the solution of the dirichlet problem for the most general quasi linear second order elliptic operator in divergence form existence and uniqueness of the approximation are proved and optimal error estimates in l 2 are demonstrated for both the scalar and vector functions approximated by the method error estimates are also derived in l q 2 leq q leq infty newton s method is presented and analyzed to solve the nonlinear algebraic equations

a reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain the argument principle is a natural and simple method to determine the number of zeros of an analytic functionf z in a bounded domaind n the number of zeros counting multiplicities off z is 1 2 times the change in argf z asz moves along the closed contour d since the range of argf z is a critical point in the computational procedure is to assure that the discretization of d z i i 1 m is such that delta text z i text text z i 1 text arg f z leqq pi discretization control which may violate this inequality can lead to an unreliable algorithm mathematical theorems derived for the discretization of d lead to a completely reliable algorithm to computen this algorithm also treats in an elementary way the case when a zero is on or near the contour d numerical examples are given for the reliable algorithm formulated here and it is pointed out in these examples how inadequate discretization control can lead to failure of other algorithms

hermite pade approximation isomonodromic deformation and hypergeometric integral we develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations we show that a certain duality in hermite s two approximation problems for functions leads to the schlesinger transformations i e transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant since approximants and remainders are described by block toeplitz determinants one can clearly understand the determinantal structure in isomonodromic deformations we demonstrate our method in a certain family of hamiltonian systems of isomonodromy type including the sixth painleve equation and garnier systems particularly we present their solutions written in terms of iterated hypergeometric integrals an algorithm for constructing the schlesinger transformations is also discussed through vector continued fractions

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

on the solution of the abel equation of the second kind by the shifted chebyshev polynomials abstract this paper presents a new approximate method of abel differential equation by using the shifted chebyshev expansion of the unknown function abel differential equation is approximately transformed to a system of nonlinear equations for the unknown coefficients a desired solution can be determined by solving the resulting nonlinear system this method gives a simple and closed form of approximate solution of abel differential equation the solution is calculated in the form of a series with easily computable components the numerical results show the effectiveness of the method for this type of equation comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate

a perturbation result for generalized eigenvalue problems and its application to error estimation in a quadrature method for computing zeros of analytic functions we consider the quadrature method developed by kravanja et al bit 39 4 1999 646 for computing all the zeros of an analytic function that lie inside the unit circle a new perturbation result for generalized eigenvalue problems allows us to obtain a detailed upper bound for the error between the zeros and their approximations to the best of our knowledge it is the first time that such an error estimate is presented for any quadrature method for computing zeros of analytic functions numerical experiments illustrate our results

copula based multivariate input models for stochastic simulation as large scale discrete event stochastic simulation becomes a tool that is used routinely for the design and analysis of stochastic systems the need for input modeling support with the ability to represent complex interactions and interdependencies among the components of multivariate time series input processes is more critical than ever motivated by the failure of independent and identically distributed random variables to represent such input processes a comprehensive framework called vector autoregressive to anything varta has been introduced for multivariate time series input modeling despite its flexibility in capturing a wide variety of distributional shapes we show that varta falls short in representing dependence structures that arise in situations where extreme component realizations occur together we demonstrate that it is possible to extend varta to work for such dependence structures via the use of the copula theory which has been used primarily for random vectors in the simulation input modeling literature for multivariate time series input modeling we show that our copula based multivariate time series input model which includes varta as a special case allows the development of statistically valid fitting and fast sampling algorithms well suited for driving large scale stochastic simulations

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

improvement of convergence of an iterative method for finding polynomial factors of analytic functions in this paper we consider an iterative method for evaluating the coefficients of a monic factor of an analytic function using complex circular arithmetic in a previous paper the authors presented a factoring method that finds a cluster of zeros as a polynomial factor we analyze the convergence behavior of this method and discuss a technique for improving convergence numerical examples illustrate the aspects of the improved method

solution of equations involving analytic functions abstract the location by the method of delves and lyness of the roots of the equation z 0 for analytic is examined new formulas are proposed for calculating the integrals involved in the application of the method these formulas do not require the evaluation of z and are more accurate than some existing ones a highly reliable but not completely foolproof procedure for computing the number of zeros of in a given region is also proposed

mixed finite element methods for quasilinear second order elliptic problems a mixed finite element method is developed to approximate the solution of a quasilinear second order elliptic partial differential equation the existence and uniqueness of the approximation are demonstrated and optimal rate error estimates are derived

numerical algorithms based on analytic function values at roots of unity let f z be an analytic or meromorphic function in the closed unit disk sampled at the n th roots of unity based on these data how can we approximately evaluate f z or f m z at a point z in the disk how can we calculate the zeros or poles of f in the disk these questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations partial differential equations and large scale linear algebra we analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of cauchy integrals we then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension finally we highlight the power of rational in comparison with polynomial approximations for some of these problems

an efficient numerical method for solving abel integral equation in this paper a numerical method for solving abel integral equation is presented we first convert the integral equation to the algebraic equation then we find numerical inversion of laplace transform by power series at last the pade approximants is effectively used to improve the convergence rate and accuracy of the computed series the method is described and illustrated with numerical examples the results reveal that the method is accurate and easy to implement

mixed finite element domain decomposition for nonlinear parabolic problems abstract fully discrete mixed finite element method is considered to approximate the solution of a nonlinear second order parabolic problem a massively parallel iterative procedure based on domain decomposition technique is presented to solve resulting nonlinear algebraic equations robin type boundary conditions are used to transmit information between subdomains the convergence of the iteration for each time step is demonstrated optimal order error estimates are also derived numerical examples are given

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

mixed finite element methods for nonlinear elliptic problems the h p version mixed finite element methods for strongly nonlinear second order elliptic problems are proposed and analyzed existence and uniqueness of the approximate solution are demonstrated using a fixed point argument convergence and stability of the method are proved both with respect to mesh refinement and increase of the degree of the approximating polynomials the analysis is carried out in detail using raviart thomas nedelec spaces as an example numerical results for minimal surface problems are obtained using brezzi douglas marini spaces graphs of the approximate solutions are presented for two different problems

simultaneous factorization of a polynomial by rational approximation in this note we present a numerical method to approximate some relatively prime factors of a polynomial simultaneously our approach gives methods of arbitrary order grau s method carstensen 1992 grau 1971 is obtained as the second order method which is durand kerner s method when we have linear factors for linear factors our approach yields the simultaneous methods introduced in sakurai et al 1991 we prove local convergence and estimate the r order of the total step version as well as the single step version of the methods we derive an algorithm and present numerical examples which confirm the convergence behavior theoretically predicted

analysis of multiscale mortar mixed approximation of nonlinear elliptic equations abstract a multiscale mortar mixed finite element method is established to approximate non linear second order elliptic equations the method is based on non overlapping domain decomposition and mortar finite element methods the existence and uniqueness of the approximation are demonstrated and a priori l 2 error estimates for the velocity and pressure are derived an error bound for mortar pressure is proved convergence estimates of the mortar pressure are based on a linear interface formulation having the discrete pressure dependent coefficient optimal order convergence is achieved on the fine scale by a proper choice of mortar space and polynomial degree of approximation the quadratic convergence of the newton raphson method is proved for the nonlinear algebraic system arising from the mortar mixed formulation of the problem numerical experiments are performed to support theoretic results

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

a multipoint flux mixed finite element method for the compressible darcy forchheimer models abstract we develop a multipoint flux mixed finite element method for the compressible darcy forchheimer models this method is motivated by the multipoint flux approximation method based on the lowest order brezzi douglas marini mixed finite element method and combined with a special quadrature rule that allows for local velocity elimination and leads to a cell centered system for the pressures we develop two approximation methods for the problems the semi discrete method and fully discrete method where the fully discrete method includes the backward euler scheme and the crank nicolson scheme theoretical results show the convergences for both pressure and velocity the numerical experiments show that the convergence rates of the method are in agreement with the theoretical analysis

a user friendly method for computing indefinite integrals of oscillatory functions for indefinite integrals q f x 1 x f t e i t d t x 1 1 torii and the first author hasegawa and torii 1987 developed a quadrature method of clenshaw curtis c c type its improvement was made and combined with sidi s m w transformation by sidi and the first author hasegawa and sidi 1996 to compute infinite oscillatory integrals the improved method per se however has not been elucidated in its attractive features which here we reveal with new results and its detailed algorithm a comparison with a method of c c type for definite integrals q f 1 due to dominguez et al 2011 suggests that a smaller number of computations is required in our method this is achieved by exploiting recurrence and normalization relations and their associated linear system we show their convergence and stability properties and give a verified truncation error bound for a result computed from the linear system with finite dimension for f z analytic on and inside an ellipse in the complex plane z the error of the approximation to q f x of the improved method is shown to be bounded uniformly numerical examples illustrate the stability and performance of the method

a remark on the hankel determinant formula for solutions of the toda equation we consider the hankel determinant formula of the functions of the toda equation we present a relationship between the determinant formula and the auxiliary linear problem which is characterized by a compact formula for the functions in the framework of the kp theory similar phenomena that have been observed for the painleve ii and iv equations are recovered the case of finite lattice is also discussed

a high order accurate scheme for maxwell s equations with a generalized dispersive material model abstract a high order accurate scheme for solving the time domain maxwell s equations with a generalized dispersive material model is described the equations for the electric field are solved in second order form and a general dispersion model is treated with the addition of one or more polarization vectors which obey a set of auxiliary differential equations ade numerical methods are developed for both second order and fourth order accuracy in space and time the equations are discretized using finite differences and advanced in time with a single stage three level space time scheme which remains stable up to the usual explicit cfl restriction as proven using mode analysis because the equations are treated in their second order form there is no need for grid staggering and instead a collocated grid is used composite overlapping grids are used to treat complex geometries with boundary conforming grids and a high order upwind dissipation is added to ensure robust and stable approximations on overlapping grids numerical results in two and three space dimensions confirm the accuracy and stability of the new schemes

a characteristics mixed covolume method for a convection dominated transport problem in this paper we propose a characteristics mixed covolume method for approximating the solution to a convection dominated transport problem the method is a combination of characteristic approximation to handle the convection term in time and mixed covolume method spatial approximation to deal with the diffusion term the velocity and press are approximated by the lowest order raviart thomas mixed finite element space on rectangles the projection of a mixed covolume element is introduced we prove its first order optimal rate of convergence for the approximate velocities in the l 2 norm as well as for the approximate pressures in the l 2 norm

designing rational filter functions for solving eigenvalue problems by contour integration abstract solving nonlinear eigenvalue problems by contour integration requires an effective discretization for the corresponding contour integrals in this paper it is shown that good rational filter functions can be computed using nonlinear least squares optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis the conditions that such an effective filter function should satisfy are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from tensorlab numerical experiments illustrate the validity of this approach

zolotarev quadrature rules and load balancing for the feast eigensolver the feast method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization this relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function we propose improved rational approximants leading to feast variants with faster convergence in particular when using rational approximants based on the work of zolotarev numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues the new approach improves both convergence robustness and load balancing when feast runs on multiple search intervals in parallel

computing integral transforms and solving integral equations using chebyshev polynomial approximations this paper gives a survey of the use of chebyshev polynomials in the computation and the inversion of integral transforms and in the solution of integral equations especially with singular kernels

determinant formula for solutions of the garnier system and pade approximation it is known that a class of special solutions of the garnier system is expressed by a determinant formula in terms of a certain specialization of the schur functions with rectangular shape partitions y yamada showed that such a determinant formula for rational solutions of riccati type can be derived by making use of the pade approximation in this paper we extend yamada s method we derive a determinant formula for transcendental solutions of riccati type by showing that the pade approximation can be utilized in order to construct a schlesinger transformation between isomonodromic deformations in addition we show that this method is effective in generic solutions of the garnier system and derive a determinant structure of them

cubature formulas over the n sphere we examine the method of cartesian product to construct cubature formulae on the unit sphere and obtain a kind of new lobatto formulae furthermore by constraining nodes to lie on orthogonal polynomials an alternative method of producing new cubature formulae on the spherically symmetrical region is also presented finally some new cubature formulae are derived