Gaussian Rules Research Articles

A meta-cognitive sequential learning algorithm for neuro-fuzzy inference system

In this paper, we present a meta-cognitive sequential learning algorithm for a neuro-fuzzy inference system, referred to as, ‘Meta-Cognitive Neuro-Fuzzy Inference System’ (McFIS). McFIS has two components, viz., a cognitive component and a meta-cognitive component. The cognitive component employed is a Takagi–Sugeno–Kang type-0 neuro-fuzzy inference system. A self-regulatory learning mechanism that controls the learning process of the cognitive component, by deciding what-to-learn, when-to-learn and how-to-learn from sequential training data, forms the meta-cognitive component. McFIS realizes the above decision by employing sample deletion, sample reserve and sample learning strategy, respectively. The meta-cognitive component use the instantaneous error of the sample and spherical potential of the rule antecedents to select the best training strategy for the current sample. Also, in sample learning strategy, when a new rule is added the rule consequent is assigned such that the localization property of Gaussian rule is fully exploited. The performance of McFIS is evaluated on four regression and eight classification problems. The performance comparison shows the superior generalization performance of McFIS compared to existing algorithms.

Gaussian interval quadrature rule for exponential weights

Interval quadrature formulae of Gaussian type on R and R+ for exponential weight functions of the form w(x)=exp(−Q(x)), where Q is a continuous function on its domain and such that all algebraic polynomials are integrable with respect to w, are considered. For a given set of nonoverlapping intervals and an arbitrary n, the uniqueness of the n-point interval Gaussian rule is proved. The results can be applied also to corresponding quadratures over (−1,1). An algorithm for the numerical construction of interval quadratures is presented. Finally, in order to illustrate the presented method, two numerical examples are done.

A high-order Nyström discretization scheme for boundary integral equations defined on rotationally symmetric surfaces

A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in R3 is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a surface to a sequence of BIEs defined on a generating curve for the surface. It can handle loads that are not necessarily rotationally symmetric. Nyström discretization is used to discretize the BIEs on the generating curve. The quadrature is a high-order Gaussian rule that is modified near the diagonal to retain high-order accuracy for singular kernels. The reduction in dimensionality, along with the use of high-order accurate quadratures, leads to small linear systems that can be inverted directly via, e.g., Gaussian elimination. This makes the scheme particularly fast in environments involving multiple right hand sides. It is demonstrated that for BIEs associated with the Laplace and Helmholtz equations, the kernel in the reduced equations can be evaluated very rapidly by exploiting recursion relations for Legendre functions. Numerical examples illustrate the performance of the scheme; in particular, it is demonstrated that for a BIE associated with Laplace’s equation on a surface discretized using 320,800 points, the set-up phase of the algorithm takes 1min on a standard laptop, and then solves can be executed in 0.5s.

Gaussian quadrature rules with exponential weights on (−1, 1)

We study the behavior of some “truncated” Gaussian rules based on the zeros of Pollaczek-type polynomials. These formulas are stable and converge with the order of the best polynomial approximation in suitable function spaces. Moreover, we apply these results to the related Lagrange interpolation process and to prove the stability and the convergence of a Nystrom method for Fredholm integral equations of the second kind. Finally, some numerical examples are shown.

On the Direct Evaluation of Surface Integral Equation Impedance Matrix Elements Involving Point Singularities

The direct evaluation method tailored to the 4-D singular integrals over vertex adjacent triangles, arising in the first-kind and second-kind Fredholm surface integral equation formulations, is presented. A combination of singularity cancellation, reordering of the integrations, and one analytical integration results in 3-D integrals of sufficiently smooth functions, allowing a straightforward computation by standard Gaussian rules. Numerical results demonstrate that the uncertainty about the accuracy of the impedance matrix elements associated to the interaction of vertex adjacent triangles is safely removed.

A method to evaluate the Hilbert transform on (0, +∞)

The author proposes a method to approximate the Hilbert transform on the real positive semiaxis by a suitable Lagrange interpolating polynomial. The method employs truncated Gaussian rules and uses the interlacing properties of the zeros of generalized Laguerre polynomials. The error estimate in a weighted uniform norm is proved and some numerical tests show the efficacy of the proposed procedure.

An adaptive numerical integration algorithm for polygons

In this paper an adaptive numerical integration algorithm for polygons is constructed by cubatures based on finite quadrilateral elements, with particular reference to an 8-node spline finite element. Some numerical examples are given to evaluate the performance of the proposed algorithm, combined with several cubatures with accuracy of order two. The results are also compared with those obtained by two cubatures based on triangulation and a recent method based on the Green's integral formula and Gaussian rules.

Product Gauss quadrature rules vs. cubature rules in the meshless local Petrov–Galerkin method

A crucial point in the implementation of meshless methods such as the meshless local Petrov–Galerkin (MLPG) method is the evaluation of the domain integrals arising over circles in the discrete local weak form of the governing partial differential equation. In this paper we make a comparison between the product Gauss numerical quadrature rules, which are very popular in the MLPG literature, with cubature formulas specifically constructed for the approximation of an integral over the unit disk, but not yet applied in the MLPG method, namely the spherical, the circularly symmetrical and the symmetric cubature formulas. The same accuracy obtained with 64×64 points in the product Gauss rules may be obtained with symmetric quadrature formulas with very few points.

On certain symmetric strong distributions, two-point Padé approximation and related quadratures

Let ϕ be a c-inversive strong distribution as defined in [A. Sri Ranga, E.X.L. de Andrade, J.H. McCabe, Some consequences of a symmetry in strong distributions, J. Math. Anal. Appl. 193 (1) (1995) 158–168]. In this paper, two-point Padé approximants, both with free and prescribed poles, related to the distribution ϕ are analyzed. In particular, the existence of c-inversive rational approximants to the Stieltjes transform of ϕ is studied, in order to make computations in an advantageous way. An application to numerical quadratures is also given, and several examples applying these Gauss-type quadrature formulas in the case of integrands which can be well approximated by Laurent polynomials are displayed, showing better results than the corresponding for the usual Gaussian rules.

Complex Gaussian quadrature of oscillatory integrals

We construct and analyze Gauss-type quadrature rules with complex- valued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behaviour, suitable for the application of Gaussian rules with non-standard weight functions. The results differ from those in previous research in the sense that the constructed rules are asymptotically optimal, i.e., among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand.

D(1±) AND Ds(1±) MESONS SPECTROSCOPY IN GAUSSIAN SUM RULES

The Gaussian sum rules (GSRs) for the D(1±) and Ds(1±) mesons current are obtained by means of Laplacian transformation. Based on it, the GSRs for the mass and the coupling constants of D(1±) and Ds(1±) mesons are worked out. Using the standard input of quantum chromodynamics nonperturbative parameters, the corresponding curves and tables are presented. The results are in accordance well with the experimental data, and more well than those from the Borel sum rule approach.

An evaluation of Clenshaw–Curtis quadrature rule for integration w.r.t. singular measures

This work is devoted to the study of quadrature rules for integration with respect to (w.r.t.) general probability measures with known moments. Automatic calculation of the Clenshaw–Curtis rules is considered and analyzed. It is shown that it is possible to construct these rules in a stable manner for quadrature w.r.t. balanced measures. In order to make a comparison Gauss rules and their stable implementation for integration w.r.t. balanced measures are recalled. Convergence rates are tested in the case of binomial measures.

On quadrature formulae based on derivative collocation

The recent article “A new type of weighted quadrature rules and its relation to orthogonal polynomials” by Masied-Jamei [M. Masjed-Jamei, A new type of weighted quadrature rules and its relation to orthogonal polynomials, Appl. Math. Comput. 188 (2007) 154–165] introduces quadrature rules based on the evaluation of the derivative(s) of the integrand function rather the function itself. The approach appears useful when a number of derivatives, including the integrand, vanish at a point λ, leading to increased order of accuracy compared to standard Gaussian rules. It is also shown by Masjed-Jamei (2007) how the nodes and weights of the resulting quadrature formula relate to nodes and weights of standard Gaussian quadratures applied to a weight function w to be determined by solving a specific system of integral equations. We give here an explicit expression for w and provide strategies for the practical computation of the quadrature nodes and weights. Additional comments on the examples used by Masjed-Jamei (2007) as well as a generalization involving multiple λ’s, are also included.

On the Discriminant Analysis in the 2-Populations Case

The empirical Bayes Gaussian rule, which in the normal case yields good values of the probability of total error, may yield high values of the maximum probability error. From this point of view the presented modified version of the classification rule of Broffitt, Randles and Hogg appears to be superior. The modification included in this paper is termed as a WR method, and the choice of its weights is discussed. The mentioned methods are also compared with the K nearest neighbours classification rule.

A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.

Sparse Inverse Covariance Estimates for Hyperspectral Image Classification

Classification of remotely sensed hyperspectral images calls for a classifier that gracefully handles high-dimensional data, where the amount of samples available for training might be very low relative to the dimension. Even when using simple parametric classifiers such as the Gaussian maximum-likelihood rule, the large number of bands leads to copious amounts of parameters to estimate. Most of these parameters are measures of correlations between features. The covariance structure of a multivariate normal population can be simplified by setting elements of the inverse covariance matrix to zero. Well-known results from time series analysis relates the estimation of the inverse covariance matrix to a sequence of regressions by using the Cholesky decomposition. We observe that discriminant analysis can be performed without inverting the covariance matrix. We propose defining a sparsity pattern on the lower triangular matrix resulting from the Cholesky decomposition, and develop a simple search algorithm for choosing this sparsity. The resulting classifier is used on four different hyperspectral images, and compared with conventional approaches such as support vector machines, with encouraging results

Some remarks on the numerical computation of integrals on an unbounded interval

An account of the error and the convergence theory is given for Gauss–Laguerre and Gauss–Radau–Laguerre quadrature formulae. We develop also truncated models of the original Gauss rules to compute integrals extended over the positive real axis. Numerical examples confirming the theoretical results are given comparing these rules among themselves and with different quadrature formulae proposed by other authors (Evans, Int. J. Comput. Math. 82:721–730, 2005; Gautschi, BIT 31:438–446, 1991).

A special Gaussian rule for trigonometric polynomials

Abram Haimovich Turetzkii [Uchenye Zapiski, 1 (149) (1959), 31-55 (translation in English in East J. Approx. 11 (2005), 337-359)] considered interpolatory quadrature rules which have the following form $\int_0^{2\pi}f(x)w(x)d x\approx \sum_{\nu=0}^{2n}w_\nu f(x_\nu)$, and which are exact for all trigonometric polynomials of degree less than or equal to $n$. Maximal trigonometric degree of exactness of such quadratures is $2n$, and such kind of quadratures are known as quadratures of Gaussian type or Gaussian quadratures for trigonometric polynomials. In this paper we prove some interesting properties of a special Gaussian quadrature with respect to the weight function $w_m(x)=1+\sin mx$, where $m$ is a positive integer.

Approximation of multiple integrals by simple integrals involving periodic functions

In this paper, we consider problems of numerical integration of fast oscillatory functions of one variable, obtained by using α‐dense curves and approximating multiple integrals. Using first, periodic and regular α‐dense curves we propose a trapezoidal formula for calculating the periodic integrand obtained. Then, we consider the simple integrals as integrals with weight. We propose a method to evaluate the moments of the weight function. This allows us to build a recurrent formula for the orthogonal polynomials family and to use a Gaussian rule to estimate the simple integral. Finally, we adapt the Filon's method, consisting in evaluating the Fourier coefficients of a function, to the oscillatory integrand obtained by using reducing transformations.

Highly accurate solutions of Motz's and the cracked beam problems

For Motz's problem and the cracked beam problem, the collocation Trefftz method is used to seek their approximate solutions uN=∑i=0NDiri+(1/2)cos(i+(1/2))θ, where Di are the expansion coefficients. The high-order Gaussian rules and the central rule are used in the algorithms, to link the collocation method and the least squares method, and to provide exponential convergence rates of the obtained solutions. Compared with the solutions in the previous literature, our Motz's solutions are more accurate and the leading coefficient D0 using the Gaussian rule with six nodes arrives at 17 significant (decimal) digits. Similarly for the cracked beam problem, the collocation Trefftz method also provides the highly accurate solutions, and D0 with 17 significant digits by the Gaussian rules. This papers proves that when the rules of quadrature involved have the relative errors less than three quarters, the solution form the collocation Trefftz method may converge exponentially. Such an analysis supports the collocation Trefftz method to become theoretically the most accurate method for Motz's and the cracked beam problems.