Tables of integrals

Computational geometry

Mathematica--A System for Doing Mathematics by Computer.

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

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Methods of Numerical Integration

We discuss and compare two greedy algorithms that compute discrete versions of Fekete-like points for multivariate compact sets by basic tools of numerical linear algebra. The first gives the so-called approximate Fekete points by QR factorization with column pivoting of Vandermonde-like matrices. The second computes discrete Leja points by LU factorization with row pivoting. Moreover, we study the asymptotic distribution of such points when they are extracted from weakly admissible meshes.

/pdf/computing-multivariate-fekete-and-leja-points-by-numerical-2u7ijbb306.pdf

Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

We propose a numerical method (implemented in Matlab) for computing approximate Fekete points on compact multivariate domains. It relies on the search of maximum volume submatrices of Vandermonde matrices computed on suitable discretization meshes, and uses a simple greedy algorithm based on QR factorization with column pivoting. The method gives also automatically an algebraic cubature formula, provided that the moments of the underlying polynomial basis are known. Numerical tests are presented for the interval and the square, which show that approximate Fekete points are well suited for polynomial interpolation and cubature.

https://www.math.unipd.it/~marcov/pdf/boh.pdf

Computing approximate Fekete points by QR factorizations of Vandermonde matrices

We have implemented in Matlab a Gauss-like cubature formula over convex, nonconvex or even multiply connected polygons. The formula is exact for polynomials of degree at most 2n-1 using N∼mn 2 nodes, m being the number of sides that are not orthogonal to a given line, and not lying on it. It does not need any preprocessing like triangulation of the domain, but relies directly on univariate Gauss–Legendre quadrature via Green’s integral formula. Several numerical tests are presented.

/pdf/product-gauss-cubature-over-polygons-based-on-green-s-22rnp1u2n9.pdf

Product Gauss cubature over polygons based on Green’s integration formula

Using the concept of Geometric Weakly Admissible Meshes (see §2 below) together with an algorithm based on the classical QR factorization of matrices, we compute efficient points for discrete multivariate least squares approximation and Lagrange interpolation.

/pdf/geometric-weakly-admissible-meshes-discrete-least-squares-1q6xpngycq.pdf

Geometric weakly admissible meshes, discrete least squares approximations and approximate Fekete points

https://www.math.unipd.it/~marcov/pdf/splinecub_elsart_jcam.pdf

Alvise Sommariva

Papers

Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

Computing approximate Fekete points by QR factorizations of Vandermonde matrices

Product Gauss cubature over polygons based on Green’s integration formula

Geometric weakly admissible meshes, discrete least squares approximations and approximate Fekete points

Gauss-Green cubature and moment computation over arbitrary geometries