A corrected quadrature formula and applications

Ujević, Nenad; Roberts, A. J.

doi:10.21914/anziamj.v45i0.499

Cited by 32 publications

(15 citation statements)

References 4 publications

(8 reference statements)

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“…Applied to numerical integration, the error of the MSR is of sixth order in grid spacing (see Ujevic and Roberts [19], Corollary 2). The purpose of the present note is to investigate the impact of the O3R and MSR modified quadrature formulas on the fast Fourier transform approximation of probability functions.…”

Section: Numerical Integration Quadrature Rules and Improvementmentioning

confidence: 99%

“…For example, an optimal 3-point quadrature (O3Q) formula of closed type, similar in simplicity to the Simpson rule, has been revealed by Ujevic [18]. Another attempt to improve Simpson's rule is due to Ujevic and Roberts [19]. They have proposed a modified Simpson rule (MSR), which takes additionally the first derivative of the approximating integrand at the two end-points into account.…”

Section: Introductionmentioning

confidence: 99%

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Improved FFT approximations of probability functions based on modified quadrature rules

Hürlimann¹

2013

imf

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In the last ten years, the traditional Simpson quadrature rule for numerical integration has been improved to an optimal 3-point quadrature formula and a modified Simpson rule that takes additionally the first derivative of the approximated integrand at the two end-points of integration into account. The impact of the improved quadrature rules on the fast Fourier transform (FFT) approximation of probability density functions with known characteristic functions is discussed. The quality of the approximations is measured with a discrete version of the total variation distance. Numerical examples suggest that a discrete total variation distance of approximately 0.25 is the best possible attainable value across all the considered FFT approximations.Mathematics Subject Classification: 41A55, 62E17, 65C60, 65T50, 91G60

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Section: Numerical Integration Quadrature Rules and Improvementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Improved FFT approximations of probability functions based on modified quadrature rules

Hürlimann¹

2013

imf

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“…For further information on formula (1.2) and other modified quadrature formulas, see [18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Modified Midpoint Method for Solving System of Linear Fredholm Integral Equations of the Second Kind

Majeed¹

2014

AJAM

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Abstract:In this paper numerical solution to system of linear Fredholm integral equations by modified midpoint method is considered. This method transforms the system of linear Fredholm integral equations into a system of linear algebraic equations that can be solved easily with any of the usual methods. Finally, some illustrative examples are presented to test this method and the results reveal that this method is very effective and convenient by comparison with exact solution and with other numerical methods such as midpoint method, trapezoidal method, Simpson's method and modified trapezoidal method. All results are computed by using a programs written in Matlab R2012b.

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“…if those values are easy to calculate. Quadrature formulas of this form are sometimes called ''corrected'' (see [1,2]). As special cases, corrected Boole's, the corrected Gauss 3-point, the corrected Lobatto 4-point and finally, the classical Lobatto 5-point formula are obtained.…”

Section: Introductionmentioning

confidence: 99%