Numerical Integration Of Analytic Functions Research Articles

Decomposition and conformal mapping techniques for the quadrature of nearly singular integrals

Gauss–Legendre quadrature, Clenshaw–Curtis quadrature and the trapezoid rule are powerful tools for numerical integration of analytic functions. For nearly singular problems, however, these standard methods become unacceptably slow. We discuss and generalize some existing methods for improving on these schemes when the location of the nearby singularity is known. We conclude with an application to some nearly singular surface integrals that arise in three-dimensional viscous fluid flow.

A quadrature rule of Lobatto-Gaussian for numerical integration of analytic functions

<p style='text-indent:20px;'>A novel quadrature rule is formed combining Lobatto six point transformed rule and Gauss-Legendre five point transformed rule each having precision nine. The mixed rule so formed is of precision eleven. Through asymptotic error estimation the novelty of the quadrature rule is justified. Some test integrals have been evaluated using the mixed rule and its constituents both in non-adaptive and adaptive modes. The results are found to be quite encouraging for the mixed rule which is in conformation with the theoretical prediction.</p>

A mixed quadrature rule of modified Birkhoff-Young rule and SM2(f) rule for the numerical integration of analytic functions*

A mixed quadrature rule of modified Birkhoff-Young rule and SM2(f) rule for the numerical integration of analytic functions*

A generalized Birkhoff–Young quadrature formula

A generalized (4n + 1)-point Birkhoff–Young quadrature of interpolatory type with the maximal degree of precision for numerical integration of analytic functions is derived. An explicit form of the node polynomial of such kind of quadratures is obtained. Special cases and an example are presented.

A Mixed quadrature rule for numerical integration of analytic functions by using fejer and gaussian quadrature rules

In this paper 5-point Fejer’s second rule has been modified for computing integrals involving analytic functions on a line. This rule has been mixed with Gaussian 3-point rule to form a mixed quadrature rule. The relative efficiencies of this rule with respect to its constituent rules and other previously developed mixed rules have been numerically verified on test integrals. Asymptotic error estimate of this rule has been determined.

Generalized quadrature formulae for analytic functions

A kind of generalized quadrature formulae of maximal degree of precision for numerical integration of analytic functions is considered. Precisely, a general weighted quadrature of Birkhoff–Young type with 4n+3 nodes and degree of precision 6n+5 is studied. Its nodes are characterized by an orthogonality relation and a general numerical method for their computation is given. Special cases and numerical results are also included.

A generalized Birkhoff–Young–Chebyshev quadrature formula for analytic functions

A generalized N -point Birkhoff–Young quadrature of interpolatory type, with the Chebyshev weight, for numerical integration of analytic functions is considered. The nodes of such a quadrature are characterized by an orthogonality relation. Some special cases of this quadrature formula are derived.

On the evaluation of correction terms in Gauss–Legendre quadrature

In the numerical integration of analytic functions, the singularities of the integrand affect the rate of convergence of the quadrature. This convergence can be improved significantly by adding the residue correction terms for the poles of the integrand. But this needs the evaluation of the basis function and its corresponding second kind function with complex arguments. We indicate a simple and accurate method to evaluate the correction term involving the basis and its second kind functions in the case of Gauss–Legendre quadrature. This approach does not call for the evaluation of the hypergeometric functions.

Compound Birkhoff‐Young rule for numerical integration of analytic functions

A compound rule for the numerical integration of an analytic function along a directed line segment in the complex plane, basing on the Birkhoff‐Young 5‐point formula has been formulated. The rule has been tested in case of some examples and the convergence of the computed approximate values to the exact values of the integrals has been asserted.

The error in numerical integration of analytic functions

The error in numerical integration of analytic functions

Asymptotic estimates of the errors in the numerical integration of analytic functions

An expression in terms of a contour integral for the remainder term in a numerical integration formula of a general type is obtained. Because the function whose integral is under consideration is allowed to have singularities at the end points of the interval of integration special emphasis is placed on the behaviour of the contour at these points. It is then shown how the corresponding contour integral form of the remainder term in some of the well-known integration formulae may be derived. The contour integral in conjunction with known asymptotic expressions for part of the integrand is used firstly to examine the convergence properties of certain quadrature schemes (Chapter III) and secondly to estimate the error in approximating to a real integral by a quadrature sum (Chapter IV). In conclusion (Chapter V) we discuss some of the problems which are to be subjects of further research.

Numerical integration of analytic functions

A method of numerical integration is presented which is simple, versatile and accurate. It is particularly valuable for integrals with complicated end point singularities.

Rough and ready error estimates in Gaussian integration of analytic functions

Two expressions are derived for use in estimating the error in the numerical integration of analytic functions in terms of the maximum absolute value of the function in an appropriate region of regularity. These expressions are then specialized to the case of Gaussian integration rules, and the resulting error estimates are compared with those obtained by the use of tables of error coefficients.