2 edition of **new construction of Chebyshev quadrature formulae** found in the catalog.

new construction of Chebyshev quadrature formulae

Kestutis Е alkauskas

- 371 Want to read
- 38 Currently reading

Published
**1966**
by University of Calgary, Dept. of Mathematics in Calgary
.

Written in English

- Equations, Quadratic,
- Chebyshev polynomials

**Edition Notes**

Bibliography: leaf 5

Statement | by K. Salkauskas |

Series | University of Calgary. Dept. of Mathematics. Research paper -- no. 1 |

The Physical Object | |
---|---|

Pagination | 5 l. ; |

ID Numbers | |

Open Library | OL14548674M |

— Gaussian quadrature uses good choices of x i nodes and ω i weights. • Exact quadrature formulas: Let F k be the space of degree k polynomials — A quadrature formula is exact of degree k if it correctly integrates each function in F k — Gaussian quadrature formulas use n points and are exact of degree 2n−1 8File Size: KB. of general real Kronrod extensions with multiple nodes of some standard Gauss-ian quadrature formulas with multiple nodes for the generalized Chebyshev and Gori-Micchelli weight functions, in order to construct some new generalizations of quadrature formulas for the Fourier coe cients by using the important theorem.

“On computing rational Gauss-Chebyshev quadrature formulas” Mathematics of Computation. ; – Waldvogel J. “Fast Construction of the Fejér and Clenshaw-Curtis Quadrature Rules” BIT Numerical Mathematics. ; 46 (1)– Weideman JAC, Laurie DP. “Quadrature rules based on partial fraction expansions” Numerical Cited by: Perform Gauss Chebyshev quadrature. This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for Chebyshev C polynomials.

Gaussian quadrature 5 Proof that the weights are positive Consider the following polynomial of degree 2n-2 where as above the are the roots of the polynomial. Since the degree of f(x) is less than 2n-1, the Gaussian quadrature formula involving the weights and nodes obtained from applies. Since for j not equal to i, we haveFile Size: KB. If the function values have random error, then it is of practical interest that the quadrature formula Q n has a small variance Var(Q n). The question of Cited by:

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New Quadrature Formulas Based on the Zeros of the Chebyshev Polynomials of the Second Kind SHIJUN YANG AND XINGHUA WANG ZIMP & Department of Mathematics Zhejiang University HangzhouZhejiang, P.R. China [email protected] (Received June ; accepted April ) Abstract-The aim of this work is to construct a new quadrature.

The aim of this work is to construct a new quadrature formula based on the divided differences of the integrand at points -1, 1 and the zeros of the n th Chebyshev polynomial of the second kind. The interesting thing is that this quadrature rule is closely related to the well-known Gauss-Turán quadrature formula and includes a recent result obtained by A.K.

Varma and E. Cited by: 1. Convergence of the quadrature formulae for functions with singularities In this section we examine the convergence of the interpolatory quadrature formulae () and (), having as nodes the zeros of any one of the four Chebyshev polynomials of degree n, for functions with a monotonic singularity at -1 or by: The aim of this work is to construct a new quadrature formula for Fourier-Chebyshev coefficients based on the divided differences of the integrand at points-1, 1 and the zeros of the nth Chebyshev polynomial of the second kind.

The interesting thing is that this quadrature rule is closely related to the well-known Gauss-Turán quadrature formula and Author: Shi-jun Yang. OPTIMAL WEIGHTED CHEBYSHEV-TYPE QUADRATURE:FORMULAS (I) L. A:NDERSO~(2).

GAUTSCItI(2) ABSTRACT - A weighted quadrature formula is called of Chebyshev type if it has equal coefficients and real (but not necessarily distinct) nodes.

Among such quadrature rules we construct an optimal one, i. Rocky Mountain J. Math. Vol Number 2 (), New linearization formulae for the products of Chebyshev polynomials of third and fourth kindsCited by: 7.

GAUSS-CHEBYSHEV QUADRATURE FORMULAE special case considered in this paper is made and agreement between the formulae is. found (§4). In this paper an explicit procedure is presented that demonstrates the derivation of the Gauss-Chebyshev rule for finite-part integrals from the corresponding Cauchy integral quadrature formulae.

Several examples in Section 5 will illustrate this. Quadrature formulas We ﬁnally arrive at the main purpose of this paper, which is the construction of rational Gauss-Chebyshev quadrature formulas. The nodes in these formulas are the zeros of ϕ(i) n (x).

of the actual construction of the quadrature formula. If the e ort involved in computing the nodes and weights is too large, the e ciency gained by superior accuracymaybetotallylost.

Thenodesandweightsinthe(polynomial)Gauss-Chebyshev formula are explicitly known and have very simple expressions, butFile Size: KB. A~7 quadrature rule (2), (3), on the other hand, with only real nodes, will be referred to as a quadrature formula.

Such a quadrature rule, there- fore, need not have algebraic degree of exactness n, in fact, need not even (n). Chapter 3 Quadrature Formulas There are several di erent methods for obtaining the area under an unknown curve f(x) based on just values of that function at given points.

During our investigations in this class we will look at the following main categories for numerical integration: 1. Newton-Cotes formulasFile Size: KB. New Quadrature Formulas from Conformal Maps Article in SIAM Journal on Numerical Analysis 46(2) January with 20 Reads How we measure 'reads'.

Chebyshev polynomials crop up in virtually every area of numerical analysis, and they hold particular importance in recent advances in subjects such as orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods.

Yet no book dedicated to Chebyshev polynomials has be. All the zeros x 2 m,i, i = 1(1)2 m, of the Chebyshev polynomials T 2 m (x), m = 0(1)n, are found recursively just by taking n2 n-1 real square roots. For interpolation or integration of ƒ(x), given ƒ(x 2 m,i), only x 2 m,i is needed to calculate (a) the (2 m - 1)-th degree Lagrange interpolation polynomial, and (b) the definite integral over [-1, 1], either with or without the weight Author: E SalzerHerbert.

In this paper we give some new bounds for the general three-point quadrature for- mulae of Euler type using Theorem 1, Theorem 2 and the general three-point quadrature formulae recently published. parameters of quadrature rules for n≤8 can be found in the book Katsikadelis [8, pp.

In a general Gaussian quadrature formula the nodes x k and the weights A k in (3) must be selected so that R n(f)=0 for each f ∈P2n−1.

In that case, the nodes x k are zeros of the monic orthogonal polynomial π n(w;x) and the corresponding Author: Gradimir V. Milovanović, Tomislav Igić, Novica Tončev.

type. The construction of the quadrature formulas is based on the modification of discrete vortices method and linear spline interpolation over the finite interval [−1, 1]. They proved that the constructed quadrature formula converges for any singular point x not coinciding with the end points of the interval [1, 1].

Discrete Chebyshev-Gauss expansion (Hesthaven et al., ) In the continuous L2 0, space, we define the inner product and L2 -norm as 0 uv utvtdt, and u uu, 1/2 for uv L, 0, For the discrete expansion, using the Chebyshev-Gauss quadrature formula, the discrete inner product and norm on 0, is defined byFile Size: 1MB.

in the Chebyshev points of the ﬂrst or second kind does not suﬁer from the Runge phenomenon ([19], pp. ), which makes it much better than the interpolant in equally spaced points, and the accuracy of the approximation can improve remarkably fast when the number of.

Chebyshev approximation We can make use of these nice orthogonality relations to make an approximation of an arbitrary function of x in the interval [-1,1] by calculating the coefficients cj at the N zero's xk of the N-th Chebyshev polynomial: Then, the function is represented exactly at those N values of x and approximated at other x by cj= 2.

rates of convergence for analytic integrands. For example, one of the formulas presented is proved to converge 50% faster than Gauss quadrature for functions analytic in an ε-neighborhood of [−1,1].

Key words. Gauss quadrature, Clenshaw–Curtis quadrature, spectral methods, conformal mapping AMS subject classiﬁcations. 65D32, 30CCHEBYSHEV1_RULE is a MATLAB program which generates a specific Gauss-Chebyshev type 1 quadrature rule, based on user input. The rule is written to three files for easy use as input to other programs.

The Gauss Chevbyshev type 1 quadrature rule is used as follows.Chapter 8 Integration Using Chebyshev Polynomials case of Gauss–Chebyshev quadrature, where particularly simple proce- The idea of Gauss quadrature is to ﬁnd that formula () that gives an exact result for all polynomials of as high a degree as possible.

If J n File Size: KB.