New Formulas of Numerical Quadrature Using Spline Interpolation

Journal of Computational and Applied Mathematics, 2012

Two new classes of quadrature formulas associated to the BS Boundary Value Methods are discussed. The first is of Lagrange type and is obtained by directly applying the BS methods to the integration problem formulated as a (special) Cauchy problem. The second descends from the related BS Hermite quasi-interpolation approach which produces a spline approximant from Hermite data assigned on meshes with general distributions. The second class formulas is also combined with suitable finite difference approximations of the necessary derivative values in order to define corresponding Lagrange type formulas with the same accuracy.

Numerische Mathematik, 2000

Using a method based on quadratic nodal spline interpolation, we define a quadrature rule with respect to arbitrary nodes, and which in the case of uniformly spaced nodes corresponds to the Gregory rule of order two, i.e. the Lacroix rule, which is an important example of a trapezoidal rule with endpoint corrections. The resulting weights are explicitly calculated, and Peano kernel techniques are then employed to establish error bounds in which the associated error constants are shown to grow at most linearly with respect to the mesh ratio parameter. Specializing these error estimates to the case of uniform nodes, we deduce non-optimal order error constants for the Lacroix rule, which are significantly smaller than those calculated by cruder methods in previous work, and which are shown here to compare favourably with the corresponding error constants for the Simpson rule.

Journal of Computational and Applied Mathematics, 1975

A method is described for the interpolation of N arbitrarily given data points using fifth degree polynomial spline functions. The interpolating spline is built from a set of basis functions belonging to the fifth degree smooth Hermite space. The resulting algebraic system is symmetric and bloc-tridiagonal. Its solution is calculated using a direct inversion method, namely a block-gaussian elimination without pivoting. Various boundary conditions are provided for independently at each end point. The stability of the algorithm is examined and some examples are given of experimental convergence rates for the interpolation of elementary analytical functions. A listing is given of the two FORTRAN subroutines INSPL5 and SPLIN5 which form the algorithm.

2020

In this paper quadratic and quartic B-splines were used for reconstruction of an approximating function, where the integral values of successive subintervals were used instead of function values at the knots. After introducing integro quadratic and quartic interpolation a comparison was done between them through presenting numerical examples. The interpolation errors for quadratic and quartic integro interpolation are studied. Numerical results illustrate the efficiency and effectiveness of the new interpolation methods.

2010

Journal of Computational and Applied Mathematics, 2018

We propose a new class of quadrature rules for the approximation of weakly and strongly singular integrals, based on the spline quasi-interpolation scheme introduced in Mazzia and Sestini (2009). These integrals in particular occur in the entries of the stiffness matrix coming from Isogeometric Boundary Element Methods (IgA-BEMs). The presented formulas are efficient, since they combine the locality of any spline quasi-interpolation scheme with the capability to compute the modified moments for B-splines, i.e. the weakly or strongly singular integrals of such functions. No global linear system has to be solved to determine the quadrature weights, but just local systems, whose size linearly depends on the adopted spline degree. The rules are preliminarily tested in their basic formulation, i.e. when the integrand is defined as the product of a singular kernel and a continuous function g. Then, such basic formulation is compared with a new one, specific for the approximation of the singular integrals appearing in the IgA-BEM context, where a B-spline factor is explicitly included in g. Such a variant requires the usage of the recursive spline product formula given in Mørken (1991), and it is useful when the ratio between g and its B-spline factor is smooth enough.

2011

Kampus Binawidya Pekanbaru (28293) we discuss and do some analysis o., .ruorllilrrr"to*"" based on interpolation, midpoint, trapezoidal rule and Simpson rule. We end up with some new formulas, which are not mentioned in numerical analysis textbooks. The strategy we discuss, in terms of pedagogy, illuminate how research on mathematics can be carried out.

In this paper we introduce different algorithm for reconstruction of a one dimensional function from its zero crossings. However, none of them is stable and computable in real time. An algorithm for computing the cubic spline interpolation coefficients for polynomials is presented in this paper. The matrix equation involved is solved analytically so that numerical inversion of the coefficient matrix is not required. For f(t) = í µí±¡ í µí±š , a set of constants along with the degree of polynomial m are used to compute the coefficients so that they satisfy the Interpolation constraints but not necessarily the derivative constraints. Then, another matrix equation is solved analytically to take care of the derivative constraints. The results are combined linearly to obtain the unique solution of the original matrix equation. This algorithm is tested and verified numerically for various examples. 1. Introduction In the mathematical field of numerical analysis, interpolation is a method of constructing new data point within the range of a discrete set of known data points. In a more informal language, interpolation means a guess at what happens between two values already known. In Engineering and Environmental sciences application, data collected from the field are usually discrete, therefore a more analytically controlled function that fits the field data is desirable and the process of estimating the outcomes in between these desirable data points is achieved through interpolation. There are two main uses of interpolation. The first use is in reconstructing the function (í µí±¥) when it is not given explicitly and/or only the values of (í µí±¥) and certain order derivatives at a set of points, called nodes are known. Secondly interpolation is used to replace the function (í µí±¥) by an interpolating polynomial (í µí±¥) so that many common operations like the determination of roots, differentiation, integration etc. which are intended for the function (í µí±¥) may be performed using the interpolate (í µí±¥). Spline interpolation is a form of interpolation where the interpolate is a special type of piecewise polynomial called spline. Spline interpolation is often preferred over polynomial interpolation because the interpolation error can be made small even when using low degree polynomials for the spline. Spline interpolation avoids the problem of Runge's phenomenon, in which oscillation occurs between points when interpolating using high degree polynomials. Cubic Spline interpolation is a special case of spline interpolation that is used very often to avoid the problem of Runge's phenomenon. This method gives an interpolating polynomial that is smoother and has smaller error than other interpolating polynomials such as Lagrange polynomial and Newton polynomial.

Numerical quadrature

The numerical integration of polynomial functions is one of the most interesting processes for numerical calculus and analyses, and represents thus, a compulsory step especially in finite elements analyses. Via the Gauss quadrature, the users concluded a great inconvenience that is processing at certain points which not required the based in finite element method points for deducting the form polynomials constants. In this paper, the same accuracy and efficiency as the Gauss quadrature extends for the numerical integration of the polynomial functions, but as such at the same points and nods have chosen for the determination of the form polynomials. Not just to profit from the values of the polynomials at those points and nods, but also from their first derivatives, the chosen points positions are arbitrary and the resulted deducted formulas are therefore different, as will be presented bellow and implemented.