A code developed recently by the authors, for counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, which lie in a given region of the complex plane, is ...A code developed recently by the authors, for counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, which lie in a given region of the complex plane, is modified to run in parallel on multi-core machines. A basic characteristic of this code (eventually pointing to its parallelization) is that it can proceed with: 1) partitioning the given region into an appropriate number of subregions;2) counting eigenvalues in each subregion;and 3) computing (already counted) eigenvalues in each subregion. Consequently, theoretically speaking, the whole code in itself parallelizes ideally. We carry out several numerical experiments with random complex tridiagonal matrices, and random complex polynomials as well, in order to study the behaviour of the parallel code, especially the degree of declination from theoretical expectations.展开更多
We present explicit inverses of two Brownian-type matrices, which are defined as Hadamard products of certain already known matrices. The matrices under consideration are defined by 3n - 1 parameters and their lower H...We present explicit inverses of two Brownian-type matrices, which are defined as Hadamard products of certain already known matrices. The matrices under consideration are defined by 3n - 1 parameters and their lower Hessenberg form inverses are expressed analytically in terms of these parameters. Such matrices are useful in the theory of digital signal processing and in testing matrix inversion algorithms.展开更多
The so-called “global polytropic model” is based on the assumption of hydrostatic equilibrium for the solar system, or for a planet’s system of statellites (like the Jovian system), described by the Lane-Emden diff...The so-called “global polytropic model” is based on the assumption of hydrostatic equilibrium for the solar system, or for a planet’s system of statellites (like the Jovian system), described by the Lane-Emden differential equation. A polytropic sphere of polytropic index?n?and radius?R1?represents the central component?S1?(Sun or planet) of a polytropic configuration with further components the polytropic spherical shells?S2,?S3,?..., defined by the pairs of radi (R1,?R2), (R2,?R3),?..., respectively.?R1,?R2,?R3,?..., are the roots of the real part Re(θ) of the complex Lane-Emden function?θ. Each polytropic shell is assumed to be an appropriate place for a planet, or a planet’s satellite, to be “born” and “live”. This scenario has been studied numerically for the cases of the solar and the Jovian systems. In the present paper, the Lane-Emden differential equation is solved numerically in the complex plane by using the Fortran code DCRKF54 (modified Runge-Kutta-Fehlberg code of fourth and fifth order for solving initial value problems in the complex plane along complex paths). We include in our numerical study some trans-Neptunian objects.展开更多
文摘A code developed recently by the authors, for counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, which lie in a given region of the complex plane, is modified to run in parallel on multi-core machines. A basic characteristic of this code (eventually pointing to its parallelization) is that it can proceed with: 1) partitioning the given region into an appropriate number of subregions;2) counting eigenvalues in each subregion;and 3) computing (already counted) eigenvalues in each subregion. Consequently, theoretically speaking, the whole code in itself parallelizes ideally. We carry out several numerical experiments with random complex tridiagonal matrices, and random complex polynomials as well, in order to study the behaviour of the parallel code, especially the degree of declination from theoretical expectations.
文摘We present explicit inverses of two Brownian-type matrices, which are defined as Hadamard products of certain already known matrices. The matrices under consideration are defined by 3n - 1 parameters and their lower Hessenberg form inverses are expressed analytically in terms of these parameters. Such matrices are useful in the theory of digital signal processing and in testing matrix inversion algorithms.
文摘The so-called “global polytropic model” is based on the assumption of hydrostatic equilibrium for the solar system, or for a planet’s system of statellites (like the Jovian system), described by the Lane-Emden differential equation. A polytropic sphere of polytropic index?n?and radius?R1?represents the central component?S1?(Sun or planet) of a polytropic configuration with further components the polytropic spherical shells?S2,?S3,?..., defined by the pairs of radi (R1,?R2), (R2,?R3),?..., respectively.?R1,?R2,?R3,?..., are the roots of the real part Re(θ) of the complex Lane-Emden function?θ. Each polytropic shell is assumed to be an appropriate place for a planet, or a planet’s satellite, to be “born” and “live”. This scenario has been studied numerically for the cases of the solar and the Jovian systems. In the present paper, the Lane-Emden differential equation is solved numerically in the complex plane by using the Fortran code DCRKF54 (modified Runge-Kutta-Fehlberg code of fourth and fifth order for solving initial value problems in the complex plane along complex paths). We include in our numerical study some trans-Neptunian objects.
