Quadrature 3/8 rule

Maurits, N.M., Altevogt, P., Evers, O.A., Fraaije, J.G.E.M. Simple numerical quadrature rules for Gaussian Chain polymer density functional calculations in 3D and implementation on parallel platforms. Comput. Theor. Polymer Sci. 6 (1996) 1-8. 

Problem formulations [ 1-3 ] for designing lead-generation library under different constraints belong to a class of combinatorial resource allocation problems, which have been widely studied. They arise in many different applications such as minimum distortion problems in data compression (11), facility location problems (12), optimal quadrature rules and discretization of partial differential equations (13), locational optimization problems in control theory (9), pattern recognition (14), and neural networks... 

The use of a finite-basis expansion to represent the continuum is reminiscent of the use of quadratures to represent an integration. Heller, Reinhardt and Yamani (1973) showed that use of the Laguerre basis (5.56) is equivalent to a Gaussian-type quadrature rule. The underlying orthogonal polynomials were shown by Yamani and Reinhardt (1975) to be of the Pollaczek (1950) class. 

After moment transformation and applying the quadrature rule the model is given by... 

Recursive monotone stable (RMS) formulas due to FLR [4] constitutes a nice family of quadrature rules with increasing precision, which... 

Combining properly the Ninomiya and the FLR schemes requires a selection criterion to decide at each stage during the integration process whether the current interval is bisected (Ninomiya s scheme) or a higher order rule is applied (the FLR scheme). To this end, usually the ratio of the error estimates of two quadrature rules is used [16]. 

Using a quadrature rule in order to approximate the integral yields ... 

The recursive relation is the most important property for constructive and computational use of orthogonal polynomials. In fact, as will be shown below, knowledge of the recursion coefficients allows the zeros of the orthogonal polynomials to be computed, and with them the quadrature rule. Therefore the calculation of the coefficients of this three-term recurrence relation is of paramount importance. The recursive relationship in Eq. (3.5) generates a sequence of monic polynomials that are orthogonal with respect to the weight function... 

Golub, G. H. Welsch, J. H. 1969 Calculation of Gauss quadrature rules. Mathematics of Computation 23, 221-230. 

Customarily, in developing a quadrature rule to approximate the integral... 

To say it in other words, without loosing accmacy of the finite difference scheme, the number of nodes of the optimal grid along each direction can be reduced to just a few ones, if the steps are arranged in a specific way. It can be considered as an extension of the concept of the Gaussian quadrature rule for the numerical integration to the finite differences. 

No readily useful analytical solutions are available for the system (7) and we resort to an expansion of the amplitude functions in a basis set. The discrete variable representation is a convenient means and we chose to employ a localized basis associated with the Lobatto quadrature rule. It is convenient then to choose units and displacement such that the interval [r,p] equals the standard one, [-1,1]. A basis of n+1 Lagrange interpolation functions is defined from the Legendre polynomial Pniq) as follows... 

In this appendix we describe a stencil algorithm which avoids many of the drawbacks of quadrature rules used in classical lattice models, while the extra computational cost is modest. The derivation consists of finding a unique and optimal set of stencil coefficients for a convolution with a Gaussian kernel, adapted to the special case of off-lattice density functional calculations. Stencil coefficients are the multipliers of the function values at corresponding grid points. 

The amount of work involved in the double radial integrals is substantial, so it is desirable to minimize it. We do this by two means. First of all, we use efficient Gaussian quadrature rules to evalute the integrals this minimizes the number of terms in the quadrature sums. Secondly we note that many values of R, and lead to geometries which have no physical importance so that... 

It is mentioned that integrals of/(x) can be calculated from this solution expansion and a suitable quadrature rule. In the current context the discrete Fourier transform is convenient due to the following properties. For a function/(x, t) being periodic in space and given on the interval 0 < x < 27t and the space grid being formed by N points with space step Ax = 2-k/N, the DFT is defined by [206] ... 

By approximating the distribution function by the truncated series expansion (12.310) and using the Gaussian quadrature rule (12.311), the -th moments (9.133) can be approximated as ... 

In many applications it is desirable to generate Gauss-type quadrature rules with preassigned abscissas, e.g. for solving boundary value problems. For the Gauss-Lob atto-Jacobi quadratures, we prescribe go = -1 and gp = 1. A matrix J must thus be constructed such that Amin (7) = -1 and Amax(T) = 1. This implies that a polynomial pp+i(g) must be determined so that ... 

It is noticed that a quadrature rule is applied in (12.477) and (12.478). In order to reduce time consuming operations, the quadrature points are normally chosen the same as the collocation points in the approximation of the norm integrals of the least-squares method. The quadrature points and the collocation points are defined at the same locations when both type of points are determined as the roots of the same type of orthogonal polynomial of the same order. In this case [f] = fj coincides with [f] =flQ. 

In order to obtain a system of algebraic equations, the PBE (12.399) and the boundary conditions must be transformed into a discrete form. In spectral methods, the solution function is approximated in terms of a polynomial solution function expansion (12.408). The differentiation of the discrete solution approximation were presented as (12.410) and (12.411). The PBE (12.399) is an integro-differential equation. Thus, appropriate quadrature rules are required for the numerical solution. Integral approximations can be presented on the form ... 

Golub GH, Welsch JH (1969) Calculation of Gauss quadrature rules. Math Comput 23 221-... 

By construction, Lebedev quadrature rules exactly integrate all spherical harmonics up to a certain degree on the surface of a sphere. The 6, 38, 86 and 194 point rules are exact for all spherical harmonics up to 3rd, 9th, 15th and 23rd degree, respectively. 

Clearly, in the stochastic collocation technique, unlike in the Galerkin s method, one does not require transforming the original equations into any other form. Instead, the focus is on evaluating the multidimensional integrals. An inspection of Eq. 30 reveals that these integrals can be evaluated using suitable quadrature rules. 

A tensor product of one-dimensional quadrature point set is used as the collocation point set in a product grid formula. Let us consider a one-dimensional quadrature rule... 

The above two Gauss quadrature rules are non-nested in nature. More discussions on nested quadrature rules will be discussed later. 

As it was seen in the previous section, numerical evaluation of multidimensional integrals using quadrature rules is not computationally feasible for high dimensions. In such situations, Monte Carlo simulations(MCS) provide a method for... 

The Smolyak s quadrature rule enables creating a grid of collocation points in a multi-dimensional space with a minimal number of points. Let/(x) be the function to be integrated over the /-dimensional domain Q. Let the smooth function / (x) be defined in [0,1] —> (H. For 1-dimensional case, i.e., when d = 1, the smooth function / (x) can be approximated using the interpolation formula... 

Typically, the quadrature formulae that can be used are either of the non-nested t3q>e such as the Gauss-Legendre and Gauss-Hermite quadrature rules or nested t)q)e. A brief discussion on a couple of nested quadrature rules are presented next. 

Clenshaw-Curtis Quadrature The Clenshaw-Curtis quadrature rule is widely used for generating sparse grids. It uses the roots of Chebyshev polynomials as the nodes. For any choice of m,- >1, the nodes are given by... 

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