• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.2 no.2
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• pp.57-65
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• 1998

An estimation of remainder for Simpson's quadrature formula for differentiable mappings whose derivatives belong to $L_p$-spaces and applications in theory of special means (logarithmic mean, identric mean etc...) are given.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.65-79
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• 2007

An optimal 3-point quadrature formula of closed type is derived. It is shown that the optimal quadrature formula has a better error bound than the well-known Simpson's rule. A corrected formula is also considered. Various error inequalities for these formulas are established. Applications in numerical integration are given.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.615-622
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• 2005

In this paper, we discuss the average case errors of some numerical quadratures, namely Trapezoidal and Simpson's, in the numerical integration problem. Our integrands are r-fold Wiener functions from the interval [0,1] and only at finite number of points the function values are evaluated. We study average case errors of these quadratures theoretically and then compare it with our practical (a posteriori) researches. Monte-Carlo simulation is used to perform these empirical researches. Finally we empirically compute the error bounds of studied quadratures for the higher degrees of Wiener functions.

• Communications of the Korean Mathematical Society
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• v.16 no.4
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• pp.691-701
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• 2001

We computed zero mean Gaussian of average error bounds pf Simpsons quadrature with convariances in [2]. In this paper, we compute zero mean Gaussian of average error bounds between Simpsons quadrature and composite Simpsons quadra-ture on four consecutive subintervals. The reason why we compute these on subintervals is because these results enable us to compute a posteriori error bounds on the whole interval in the later paper.

• Progress in Superconductivity and Cryogenics
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• v.16 no.4
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• pp.53-56
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• 2014

A numerical method of magnetic field calculation for the air-core solenoid is presented in this paper. In application of the Biot-Savart law, the magnetic field induced from the source current can be obtained by a double integration ormula. The numerical method named composite Simpson's rule for the integration is applied to the program and the adaptive quadrature method is used to adjust the step size in the calculation according to the precision we need. When the target point is in the solenoid and the intergrand's denominator may be zeroin the process of calculation, the method sill can provide an appropriate result. We have developed a program which calculates the magnetic field with at least 1ppm precision and named it as rzBI() to implement this method. The method has been used in the design of an MRI magnet, and the result show it is very flexible and convenient.

• Communications of the Korean Mathematical Society
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• v.15 no.4
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• pp.715-721
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• 2000

It is known that one can generate functions distributed according to ${\gamma}$-fold Wiener measure. So we could estimate the average case errors in a similar way as in Monte-Carlo method. Hence we study the basic properties of the generator of random functions. n addition, because the ${\gamma}$-fold Wiener process is truly infinitely dimensional and a computer can only handle finitely dimensional spaces, we study in this paper, the properties of generator for an m-dimensional approximation of the ${\gamma}$-fold Wiener process.