ARTICLE | doi:10.20944/preprints202304.0359.v1
Subject: Computer Science And Mathematics, Applied Mathematics Keywords: Differential algebraic equations; Lie group; Hessenberg; High index
Online: 14 April 2023 (09:35:54 CEST)
Hessenberg differential algebraic equations (Hessenberg-DAEs) with high index play a critical role in the modeling of mechanical systems and multibody dynamics. Motivated by the widely used Lie Group Differential Algebraic Equation (LGDAE) method which only handles index 2 systems, we propose a Modified Extended Lie Group Differential Algebraic Equation (MELGDAE) method for solving index 3 Hessenberg-DAEs, and provide theoretical analysis to deepen the foundation of the MELGDAE method.The performance of the MELGDAE method is compared with the standard methods RADAU and MEBDF on index 2 and 3 DAE systems, and it is demonstrated that the MELGDAE integrator exhibits competitive performance in terms of high accuracy and the preservation of algebraic constraints. In particular, all differential variables in index 3 Hessenberg DAEs achieve second-order convergence using the MELGDAE method, which suggests potential for extension to Hessenberg-DAEs with an index of 4 or higher.
ARTICLE | doi:10.20944/preprints202008.0272.v1
Subject: Computer Science And Mathematics, Computational Mathematics Keywords: Bohemian; Toeplitz matrix; Hessenberg matrix; tridiagonal matrix; pentadiagonal matrix
Online: 12 August 2020 (06:00:31 CEST)
In this paper, we deduce explicit formulas to evaluate the determinants of nonsymmetrical structure Toeplitz Bohemians by two determinants of specific Hessenberg Toeplitz matrices, which are linear combinations in terms of determinants of specific Hessenberg Toeplitz matrices. We get some new results very di¤erent from [Massimiliano Fasi, Gian Maria Negri Porzio, Determinants of normalized upper Hessenberg matrices, Electronic Journal of Linear Algebra, Volume 36, pp. 352-366, June 2020].
ARTICLE | doi:10.20944/preprints201703.0208.v1
Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: closed expression; Fibonacci number; Fibonacci polynomial; tridiagonal determinant; Hessenberg determinant
Online: 28 March 2017 (03:11:06 CEST)
In the paper, the authors nd a new closed expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant.
SHORT NOTE | doi:10.20944/preprints201610.0034.v1
Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: determinantal expression; recurrence relation; Euler polynomial; Euler number; Hessenberg determinant
Online: 11 October 2016 (10:40:02 CEST)
In the paper, by a very simple approach, the author establishes an expression in terms of a lower Hessenberg determinant for the Euler polynomials. By the determinantal expression, the author finds a recurrence relation for the Euler polynomials. By the way, the author derives the corresponding expression and recurrence relation for the Euler numbers.
ARTICLE | doi:10.20944/preprints201610.0035.v1
Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: derangement number; closed form; Hessenberg determinant; tridiagonal determinant; generating function; recurrence relation; derivative
Online: 11 October 2016 (10:53:07 CEST)
In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.