• 제목/요약/키워드: Legendre function

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Numerical Computation of Ultra-High-Degree Legendre Function

  • Kwon, Jay-Hyoun;Lee, Jong-Ki
    • 한국측량학회지
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    • 제25권1호
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    • pp.63-68
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    • 2007
  • The computations of an ultra-high degree associated Legendre functions and its first derivative up to degree and order of 10800 are reported. Not only the magnitude of orders for the ultra-high degree calculation is presented but the numerical stability and accuracy of the computed values are described in detail. The accuracy on the order of $10^{-25}\;and\;10^{-15}$ was obtained for the values of Legendre function and the first derivatives of Legendre functions, respectively. The computable highest degree and order of Legendre function in terms of latitudes and the linear relationship between the magnitude of the function with respect to degrees and orders is found. It is expected that the computed Legendre functions contribute in many geodetic and geophysical applications for simulations as well as theoretical verifications.

SOME IDENTITIES INVOLVING THE LEGENDRE'S CHI-FUNCTION

  • Choi, June-Sang
    • 대한수학회논문집
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    • 제22권2호
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    • pp.219-225
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    • 2007
  • Since the time of Euler, the dilogarithm and polylogarithm functions have been studied by many mathematicians who used various notations for the dilogarithm function $Li_2(z)$. These functions are related to many other mathematical functions and have a variety of application. The main objective of this paper is to present corrected versions of two equivalent factorization formulas involving the Legendre's Chi-function $\chi_2$ and an evaluation of a class of integrals which is useful to evaluate some integrals associated with the dilogarithm function.

A more efficient numerical evaluation of the green function in finite water depth

  • Xie, Zhitian;Liu, Yujie;Falzarano, Jeffrey
    • Ocean Systems Engineering
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    • 제7권4호
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    • pp.399-412
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    • 2017
  • The Gauss-Legendre integral method is applied to numerically evaluate the Green function and its derivatives in finite water depth. In this method, the singular point of the function in the traditional integral equation can be avoided. Moreover, based on the improved Gauss-Laguerre integral method proposed in the previous research, a new methodology is developed through the Gauss-Legendre integral. Using this new methodology, the Green function with the field and source points near the water surface can be obtained, which is less mentioned in the previous research. The accuracy and efficiency of this new method is investigated. The numerical results using a Gauss-Legendre integral method show good agreements with other numerical results of direct calculations and series form in the far field. Furthermore, the cases with the field and source points near the water surface are also considered. Considering the computational efficiency, the method using the Gauss-Legendre integral proposed in this paper could obtain the accurate numerical results of the Green function and its derivatives in finite water depth and can be adopted in the near field.

고차유한요소의 파랑변형해석에의 적용에 관한 소고 (A Study on Wave Transformation Analysis using Higher-Order Finite Element)

  • 정태화;이종인;김영택;류용욱
    • 한국해안·해양공학회논문집
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    • 제21권2호
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    • pp.108-116
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    • 2009
  • 유한요소법에 사용되어 효율적으로 파랑변형을 해석할 수 있는 (Legendre 보간 함수) 방법을 소개하였다. 고차의 보간함수를 사용하는 유한요소모형은 대부분이 Lagrangian 보간 함수를 사용한다. 이 경우, 적은 수의 요소를 사용하고도 정확한 결과를 얻을 수 있다는 장점이 있지만 결과를 얻기 위해 소요되는 시간이 증가한다는 단점이 있다. Mass lumping을 통하여 계산 시간을 절약할 수는 있지만 이 경우에는 해의 정확성이 떨어진다는 단점이 있어 정확도를 향상시키기 위하여 요소의 수를 증가시켜 다시 계산시간이 증대되는 문제가 생기게 된다. 본 연구에서 Lagrangian 보간 함수의 변형된 형태인 Legendre 보간함수에 수치적분을 사용하여 mass lumping을 수행한 것과 같이 대각 행렬을 만들어 시간 절약의 효과를 얻으면서도 정확도가 어느 정도 유지되는 방법을 소개하였다. Boussinesq 방정식을 이용한 다양한 수치 계산을 통하여 본 연구에서 제안하는 방법의 우수성을 검증하였다.

GENERATING FUNCTIONS FOR LEGENDRE-BASED POLY-BERNOULLI NUMBERS AND POLYNOMIALS

  • Khan, N.U.;Usman, Talha;Aman, Mohd
    • 호남수학학술지
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    • 제39권2호
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    • pp.217-231
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    • 2017
  • In this paper, we introduce a generating function for a Legendre-based poly-Bernoulli polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. By making use of the generating function method and some functional equations mentioned in the paper, we conduct a further investigation in order to obtain some implicit summation formulae for the Legendre-based poly-Bernoulli numbers and polynomials.

A PROOF OF THE LEGENDRE DUPLICATION FORMULA FOR THE GAMMA FUNCTION

  • Park, In-Hyok;Seo, Tae-Young
    • East Asian mathematical journal
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    • 제14권2호
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    • pp.321-327
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    • 1998
  • There have been various proofs of the Legendre duplication formula for the Gamma function. Another proof of the formula is given here and a brief history of the Gamma function is also provided.

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Legendre Tau Method for the 2-D Stokes Problem

  • Jun, SeRan;Kang, Sungkwon;Kwon, YongHoon
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제4권2호
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    • pp.111-133
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    • 2000
  • A Legendre spectral tau approximation scheme for solving the two-dimensional stationary incompressible Stokes equations is considered. Based on the vorticity-stream function formulation and variational forms, boundary value and normal derivative of vorticity are computed. A factorization technique for matrix stems based on the Schur decomposition is derived. Several numerical experiments are performed.

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Legendre 다항식을 이용한 주파수 응답 함수의 곡선접합과 모드 매개변수 규명 (Modal Parameter Identification from Frequency Response Functions Using Legendre Polynomials)

  • 박남규;전상윤;서정민;김형구;장영기;김규태
    • 한국소음진동공학회논문집
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    • 제16권7호
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    • pp.769-776
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    • 2006
  • A measured frequency response function can be represented as a ratio of two polynomials. A curve-fitting of frequency responses with Legendre polynomialis suggested in the paper. And the suggested curve-fitting algorithm is based on the least-square error method. Since the Legendre polynomials satisfy the orthogonality condition, the curve-fitting with the polynomials results to more reliable curve-fitting than ordinary polynomial method. Though the proposed curve-fitting with Legendre polynomials cannot cover all frequency range of interest, example shows that the suggested method is quite applicable in a limited frequency band.

르장드르 웨이블릿을 이용한 선형 시불변 시스템의 효율적 수치 해석 방법 (An Efficient Computational Method for Linear Time-invariant Systems via Legendre Wavelet)

  • 김범수
    • 제어로봇시스템학회논문지
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    • 제19권7호
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    • pp.577-582
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    • 2013
  • In this paper Legendre wavelets are used to approximate the solutions of linear time-invariant system. The Legendre wavelet and its integral operational matrix are presented and an efficient algorithm to solve the Sylvester matrix equation is proposed. The algorithm is based on the decomposition of the Sylvester matrix equation and the preorder traversal algorithm. Using the special structure of the Legendre wavelet's integral operational matrix, the full order Sylvester matrix equation can be solved in terms of the solutions of pure algebraic matrix equations, which reduce the computation time remarkably. Finally a numerical example is illustrated to demonstrate the validity of the proposed algorithm.

FUNCTION APPROXIMATION OVER TRIANGULAR DOMAIN USING CONSTRAINED Legendre POLYNOMIALS

  • Ahn, Young-Joon
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제9권2호
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    • pp.99-106
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    • 2005
  • We present a relation between the orthogonality of the constrained Legendre polynomials over the triangular domain and the BB ($B{\acute{e}zier}\;-Bernstein$) coefficients of the polynomials using the equivalence of orthogonal complements. Using it we also show that the best constrained degree reduction of polynomials in BB form equals the best approximation of weighted Euclidean norm of coefficients of given polynomial in BB form from the coefficients of polynomials of lower degree in BB form.

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