Anny Haegemans

Algorithm 824: CUBPACK : a package for automatic cubature; framework description

CUBPACK aims to offer a collection of re-usable code for automatic n-dimensional (n ≥ 1) numerical integration of functions over a collection of regions, i.e., quadrature and cubature. The current version allows this region to consist of a union of n-simplices and n-parellellepids. The framework of CUBPACK is described as well as its user interface. The functionality of several well known routines is embedded. New features include integration algorithms using the ε-algorithm for extrapolation for regions other than triangles and the implementation of a new type of subdivision for 3-cubes.

The GBQ -algorithm for constructing start systems of homotopies for polynomial systems

Homotopy methods have become a standard tool for the computation of all solutions of a polynomial system. This paper concerns the solution of deficient polynomial systems which appear to be typical in many engineering applications. The

An efficient and reliable algorithm for computing the singular subspace of a matrix, associated with its smallest singular values

GBQ

Construction of cubature formulas of degree eleven for symmetric planar regions, using orthogonal polynomials

-algorithm presented consists of two parts: the computation of a generalized Bezout number

A comparison of non-linear equation solvers

GB

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

and the construction of a multi-homogeneous start system Q. The approach generalizes m-homogenization into multihomogenization. It can also be regarded as a generalization “towards” the random product homotopy, however, without making assumptions on the coefficients of the polynomials in the system. As is illustrated in the examples, symmetric polynomial systems also can be solved more efficiently.

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

Abstract In this paper, an improved algorithm PSVD for computing the singular subspace of a matrix corresponding to its smallest singular values is presented. As only a basis of the desired singular subspace is needed, the classical Singular Value Decomposition (SVD) algorithm can be modified in three ways. First, the Householder transformations of the bidiagonalization need only to be applied on the base vectors of the desired singular subspace. Second, the bidiagonal must only be partially diagonalized and third, the convergence rate of the iterative diagonalization can be improved by an appropriate choice between QR and QL iteration steps. An analysis of the operation counts, as well as computational results, show the relative efficiency of PSVD with respect to the classical SVD algorithm. Depending on the gap, the desired numerical accuracy and the dimension of the desired subspace, PSVD can be three times faster than the classical SVD algorithm while the same accuracy can be maintained. The new algorithm can be successfully used in total least squares applications, in the computation of the null space of a matrix and in solving (non) homogeneous linear equations. Based on PSVD a very efficient and reliable algorithm is also derived for solving nonhomogeneous equations.

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

SummaryA method of constructing 28-point, 26-point and 25-point cubature formulas with polynomial precision 11 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.

Another step forward in searching for cubature formulae with a minimal number of knots for the square

Abstract A comparative study of 10 FORTRAN and ALGOL programs for solving non-linear equations with one unknown, without using derivatives, was made. This paper gives the results and conclusions of the study.

Automatic computation of knots and weights of cubature formulae for circular 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.