An algorithm for one-and two-dimensional quadrature rules

An algorithm is presented which implements one- and two-dimensional quadrature formulas on finite domainsD under the assumption that the function to be integrated, ?(t), is not completely known. We suppose that ? (t) is the product of two functions,w (t) andf (t), such thatw (t) is not given in analytical form but his moments are known in the regionD, whilef (t) is completely given. Moreover error bounds are obtained for these formulas, which appear to be satisfactory in many actual cases.     

Modified compound quadrature rules for strongly singular integrals

Computing - We show that the error term of every modified compound quadrature rule for Cauchy principal value integrals with degree of exactnesss is of optimal order of magnitude in the classesC k...     

Error estimates for Gauss-Chebyshev and Clenshaw Curtis quadrature formulas

For functionsf analytic in a circle we obtain bounds for the Chebyshev-Fourier coefficients off. These results are then used to obtain bounds for the errors of Gauss-Chebyshev quadratures, and the quadrature formula of Clenshaw and Curtis. Two examples are given to illustrate the error bounds.     

Error exponents for product convolutional codes

An upper bound on the error probability (first error event) of product convolutional codes over a memoryless binary symmetric channel, and the resulting error exponent are derived. The error exponent is estimated for two decoding procedures. It is shown that, for both decoding methods, the error probability exponentially decreasing with the constraint length of product convolutional codes can be attained with nonexponentially increasing decoding complexity. Both estimated error exponents are similar to those for woven convolutional codes with outer and inner warp.     

Some quadrature rules for Galerkin methods using B-spline basis functions

Quadrature rules are obtained for the evaluation of integrals resulting from the solution of quasilinear parabolic equations using the Galerkin method with B-spline basis functions.     

Error bounds for polynomial tensor product interpolation

We provide estimates for the maximum error of polynomial tensor product interpolation on regular grids in ${\mathbb{R}^d}$ . The set of partial derivatives required to form these bounds depends on the clustering of interpolation nodes. Also bounds on the partial derivatives of the error are derived.     

Gauss-Jacobi quadrature rules for n-simplex with applications to finite element methods

For numerical integrations over the n-simplex, the Gauss-Jacobi quadrature operators are presented definitively. It is demonstrated in finite-element applications that the Gauss-Jacobi quadratures could be highly effective under the isoparametric mapping as well as under the semi-radial singularity mapping.     

On the convergence of some quadrature rules for Cauchy principal-value and finite-part integrals

In this paper sufficient conditions are derived to ensure the convergence of the Elliott and Hunter types of quadrature rules for the evaluation of weighted Cauchy principal-value integrals of the form: The simultaneous convergence in the interval (?1, 1) of both quadratures was established for a class of Hölder-continuous functionsf(f∈H _μ ). Corrections of some previous statements on the subject of convergence of such quadratures are also included. Moreover, a simple derivation of the Hunter and Elliott types of quadrature rules for the evaluation of the derivative of thep-th-order of the abovestated integral was given and sufficient conditions for the convergence of the Hunter-type quadrature were obtained. Thus, the convergence of this integral was ensured for functionsf such thatf ^(p) ∈H _μ .     

A trace bound for a general square matrix product

Estimates of bounds on the solutions of Lyapunov and Riccati equations are important for analysis and synthesis of linear systems. In this paper, we propose new trace bounds for the product of two general matrices. The key point for removing the restriction of symmetry is to replace eigenvalues partly by singular values in the equation of bounds. The results obtained are valid for both symmetric and nonsymmetric cases and give tighter bounds in certain cases     

Error bound for the generalized complementarity problem with analytic functions

In this paper, we consider the global error bound for the generalized complementarity problem (GCP) with analytic functions. Based on the new technique, we establish computable global error bound under milder conditions, which refines the previously known results.     

Error analysis and minimum bound method for atmospheric remote sensing

In this study, we present an error analysis for Tikhonov regularization in a semi-stochastic setting. The analysis is carried out in such a way that it can be applied to any kind of inverse problem in atmospheric remote sensing. A method for selecting the optimal regularization parameter relying on the minimization of an estimator of the bound of the error between the first iterate and the exact solution is also discussed. Numerical simulations are performed for NO₂ retrieval from SCIAMACHY limb scatter measurements.     

A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions

We present a numerical algorithm for the construction of efficient, high-order quadratures in two and higher dimensions. Quadrature rules constructed via this algorithm possess positive weights and interior nodes, resembling the Gaussian quadratures in one dimension. In addition, rules can be generated with varying degrees of symmetry, adaptable to individual domains. We illustrate the performance of our method with numerical examples, and report quadrature rules for polynomials on triangles, squares, and cubes, up to degree 50. These formulae are near optimal in the number of nodes used, and many of them appear to be new.     

Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method

New Gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method. For discontinuous functions, we construct Gauss-like quadrature rules over arbitrarily-shaped elements in two dimensions without the need for partitioning the finite element. A point elimination algorithm is used in the construction of the quadratures, which ensures that the final quadratures have minimal number of Gauss points. For weakly singular integrands, we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately. Numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques.     

A hybrid of sequential rules and collaborative filtering for product recommendation

Customers’ purchase behavior may vary over time. Traditional collaborative filtering (CF) methods make recommendations to a target customer based on the purchase behavior of customers whose preferences are similar to those of the target customer; however, the methods do not consider how the customers’ purchase behavior may vary over time. In contrast, the sequential rule-based recommendation method analyzes customers’ purchase behavior over time to extract sequential rules in the form: purchase behavior in previous periods ⇒ purchase behavior in the current period. If a target customer’s purchase behavior history is similar to the conditional part of the rule, then his/her purchase behavior in the current period is deemed to be the consequent part of the rule. Although the sequential rule method considers the sequence of customers’ purchase behavior over time, it does not utilize the target customer’s purchase data for the current period. To resolve the above problems, this work proposes a novel hybrid recommendation method that combines the segmentation-based sequential rule method with the segmentation-based KNN-CF method. The proposed method uses customers’ RFM (Recency, Frequency, and Monetary) values to cluster customers into groups with similar RFM values. For each group of customers, sequential rules are extracted from the purchase sequences of that group to make recommendations. Meanwhile, the segmentation-based KNN-CF method provides recommendations based on the target customer’s purchase data for the current period. Then, the results of the two methods are combined to make final recommendations. Experiment results show that the hybrid method outperforms traditional CF methods.     

Error estimation in a balanced norm for a convection-diffusion problem with two different boundary layers

The \(\varepsilon \) -weighted energy norm is the natural norm for singularly perturbed convection-diffusion problems with exponential layers. But, this norm is too weak to recognise features of characteristic layers. We present an error analysis in a differently weighted energy norm—a balanced norm—that overcomes this drawback.     

A polynomial time algorithm for GapCVPP in l1 norm

This paper concerns the hardness of approximating the closest vector in a lattice with preprocessing in l1norm,and gives a polynomial time algorithm for GapCVPPγin l1norm with gapγ=O(n/log n).The gap is smaller than that obtained by simply generalizing the approach given by Aharonov and Regev.The main technical ingredient used in this paper is the discrete Laplace distribution on lattices which may be of independent interest.     

Quadrature rules for the integration of the product of two oscillatory functions with different frequencies

We construct quadrature rules for the efficient computation of the integral of a product of two oscillatory functions y₁(x) and y₂(x), where , and the functions f_i,j(x) are smooth. The weights are evaluated by the exponential fitting technique of Ixaru [Comput. Phys. Comm. 105 (1997) 1-19], which is now extended to cover the case of two frequencies. We give a numerical illustration on how the new rules compare for accuracy with the one-frequency dependent rules and with the classical ones.