Monomial cubature rules since “Stroud”: a compilation

Numerical path following

In this paper we present a general, theoretical foundation for the construction of cubature formulae to approximate multivariate integrals. The focus is on cubature formulae that are exact for certain vector spaces of polynomials. Our main quality criteria are the algebraic and trigonometric degrees. The constructions using ideal theory and invariant theory are outlined. The known lower bounds for the number of points are surveyed and characterizations of minimal cubature formulae are given. We include references to all known minimal cubature formulae. Finally, some methods to construct cubature formulae illustrate the previously introduced concepts and theorems.

Constructing cubature formulae: the science behind the art

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A Bibliography of Publications about the Fortran Programming Language: Part 1: 1956{1980

In this paper, we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons. These quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required. We have used this algorithm for the construction of symmetric and non-symmetric quadrature rules over convex and concave polygons. While in the case of symmetric quadratures our results are comparable to available rules, the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions. To demonstrate the efficiency of the new quadrature rules, we have tested them for the integration of rational polygonal shape functions over a regular hexagon. For a relative error of 10−8 in the computation of stiffness matrix entries, one needs at least 198 evaluation points when the region is partitioned, whereas 85 points suffice with our quadrature rule. Copyright © 2009 John Wiley & Sons, Ltd.

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Generalized Gaussian quadrature rules on arbitrary polygons

A method of constructing twelve-point cubature formulas with polynomial precision seven and nineteen-point and eighteen-point cubature formulas with polynomial precision nine is given for planar regions and weight functions, which are symmetric in each variable. The nodes are computed as common zeros of a set of linearly independent orthogonal polynomials.

Construction of Cubature Formulas of Degree Seven and Nine Symmetric Planar Regions, Using Orthogonal Polynomials

A comparison of non-linear equation solvers

We propose a modification of Newton's method for computing multiple roots of systems of analytic equations. Under mild assumptions the iteration converges quadratically. It involves certain constants whose product is a lower bound for the multiplicity of the root. As these constants are usually not known in advance, we devise an iteration in which not only an approximation for the root is refined, but also approximations for these constants. Numerical examples illustrate the effectiveness of our approach.

A modification of Newton's method for analytic mappings having multiple zeros

A method of constructing 19-point cubature formulas with degree of exactness 9 is given for two-dimensional regions and weight functions which are symmetric in each variable. For some regions, e.g., the square and the circle, these formulas can be reduced to 18-point formulas.

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Cubature formulas of degree nine for symmetric planar regions

Abstract A new method is described for the construction of cubature formulae of degree 4k − 3 for two-dimensional symmetric regions. This method is a generalisation of the T-method. Some formulae of degree 5, 9, 13 and 17 for the square, the circle and the entire plane are constructed.

Anny Haegemans

Papers

Construction of Cubature Formulas of Degree Seven and Nine Symmetric Planar Regions, Using Orthogonal Polynomials

A comparison of non-linear equation solvers

A modification of Newton's method for analytic mappings having multiple zeros

Cubature formulas of degree nine for symmetric planar regions

Construction of fully symmetric cubature formulae of degree 4k-3 for fully symmetric planar regions