Pentadiagonal Matrix Research Articles

On Hessenberg and pentadiagonal determinants related with Fibonacci and Fibonacci-like numbers

In this paper, we establish several new connections between the generalizations of Fibonacci and Lucas sequences and Hessenberg determinants. We also give an interesting conjecture related to the determinant of an infinite pentadiagonal matrix with the classical Fibonacci and Gaussian Fibonacci numbers.

On determinant of certain pentadiagonal matrix

In this paper, using the LU factorization, the relation between the determinant of a certain pentadiagonal matrix and the determinant of a corresponding tridiagonal matrix will be derived. Moreover, it will be shown that determinant of this special pentadiagonal matrix can be calculated by applying the fourth order homogeneous linear difference equation.

Numerical Solution of Range-Dependent Acoustic Propagation

The direct global matrix approach can be applied to modeling of range-dependent sound propagation in order to achieve numerically stable and accurate solutions. By solving the global system directly, this method features high efficiency as well as accuracy by avoiding error accumulation. It is an important issue to solve linear systems numerically in the direct global matrix approach, especially for the large-scale problems. An efficient and memory-saving algorithm is developed for solving the global system, in which the global coefficient matrix is treated as a block pentadiagonal matrix. As a result, this numerical model has the ability to solve large-scale problems on regular computers. Numerical examples are also presented to demonstrate the accuracy and efficiency of this method.

Inverse eigenvalue problem for pentadiagonal matrices

In this paper, we propose an algorithm for constructing a pentadiagonal matrix with given prescribed three spectra. Sufficient conditions for solvability of the problem are given. We generate an algorithmic procedure to construct the solution matrices and we given a numerical example illustrating the construction algorithm.

Quasi-disjoint pentadiagonal matrix systems for the parallelization of compact finite-difference schemes and filters

This paper proposes a novel systematic approach for the parallelization of pentadiagonal compact finite-difference schemes and filters based on domain decomposition. The proposed approach allows a pentadiagonal banded matrix system to be split into quasi-disjoint subsystems by using a linear–algebraic transformation technique. As a result the inversion of pentadiagonal matrices can be implemented within each subdomain in an independent manner subject to a conventional halo-exchange process. The proposed matrix transformation leads to new subdomain boundary (SB) compact schemes and filters that require three halo terms to exchange with neighboring subdomains. The internode communication overhead in the present approach is equivalent to that of standard explicit schemes and filters based on seven-point discretization stencils. The new SB compact schemes and filters demand additional arithmetic operations compared to the original serial ones. However, it is shown that the additional cost becomes sufficiently low by choosing optimal sizes of their discretization stencils. Compared to earlier published results, the proposed SB compact schemes and filters successfully reduce parallelization artifacts arising from subdomain boundaries to a level sufficiently negligible for sophisticated aeroacoustic simulations without degrading parallel efficiency. The overall performance and parallel efficiency of the proposed approach are demonstrated by stringent benchmark tests.

A New Algorithm for Computing the Determinant and the Inverse of a Pentadiagonal Toeplitz Matrix

An effective numerical algorithm for computing the determinant of a pentadiagonal Toeplitz matrix has been proposed by Xiao-Guang Lv and others [1]. The complexity of the algorithm is (9n + 3). In this paper, a new algorithm with the cost of (4n + 6) is presented to compute the determinant of a pentadiagonal Toeplitz matrix. The inverse of a pentadiagonal Toeplitz matrix is also considered.

A note for an explicit formula for the determinant of pentadiagonal and heptadiagonal symmetric Toeplitz matrices

More recently, the author gave an explicit formula for the determinant of a pentadiagonal symmetric Toeplitz matrix. The objective of this note is to make this formula more explicit. The heptadiagonal symmetric matrices are also considered, we make a precise conjecture about their explicit determinants.

Simulation of water balance in a clayey, subsurface drained agricultural field with three-dimensional FLUSH model

Summary Water flow is a key component in the evaluation of soil erosion and nutrient loads from agricultural fields. Field cultivation is the main non-point pollution source threatening water quality of surface waters in Nordic and many other countries. Few models exist that can describe key hydrological processes in clayey soils, i.e. overland flow, preferential flow in macropores and soil shrinkage and swelling. A new three-dimensional (3-D) distributed numerical model called FLUSH is introduced in this study to simulate these processes. FLUSH describes overland flow with the diffuse wave simplification of the Saint Venant equations and subsurface flow with a dual-permeability approach using the Richards equation in both macropore and matrix pore systems. A method based on the pentadiagonal matrix algorithm solves flow in both macropore and matrix systems directly in a column of cells in the computational grid. Flow between the columns is solved with iteration accelerated with OpenMP parallelisation. The model validity is tested with data from a 3-D analytical model and a clayey subsurface drained agricultural field in southern Finland. According to the simulation results, over 99% of the drainflow originated from the macropore system and drainflow started in some cases within the same hour when precipitation started indicating preferential flow in the profile. The moisture content of the clay soil had a profound effect on runoff distribution between surface runoff and drainflow. In summer, when the soil was dry and cracked, drainflow dominated the total runoff, while in autumn, when the shrinkage crack network had swollen shut, surface runoff fraction clearly increased. Observed differences in surface runoff fraction before and after tillage indicated that the operation decreased hydraulic conductivity of the profile.

REVERSIBLE CELLULAR AUTOMATA WITH PENTA-CYCLIC RULE AND ECCs

The reversibility problem for linear cellular automata with null boundary defined by a rule matrix in the form of a pentadiagonal matrix was studied over the binary field ℤ2 by Martín del Rey et al. [Appl. Math. Comput.217, 8360 (2011)]. Recently, the reversibility problem of 1D Cellular automata with periodic boundary has been extended to ternary fields and further to finite primitive fields ℤp by Cinkir et al. [J. Stat. Phys.143, 807 (2011)]. In this work, we restudy some of the work done in Cinkir et al. [J. Stat. Phys.143, 807 (2011)] by using a different approach which is based on the theory of error correcting codes. While we reestablish some of the theorems already presented in Cinkir et al. [J. Stat. Phys.143, 807 (2011)], we further extend the results to more general cases. Also, a conjecture that is left open in Cinkir et al. [J. Stat. Phys.143, 807 (2011)] is also solved here. We conclude by presenting an application to Error Correcting Codes (ECCs) where reversibility of cellular automata is crucial.

A fast numerical algorithm for solving nearly penta-diagonal linear systems

In this paper, we present a fast numerical algorithm for solving nearly penta-diagonal linear systems and show that the computational cost is less than those of three algorithms in El-Mikkawy and Rahmo, [Symbolic algorithm for inverting cyclic penta-diagonal matrices recursively–Derivation and implementation, Comput. Math. Appl. 59 (2010), pp. 1386–1396], Lv and Le [A note on solving nearly penta-diagonal linear systems, Appl. Math. Comput. 204 (2008), pp. 707–712] and Neossi Nguetchue and Abelman [A computational algorithm for solving nearly penta-diagonal linear systems, Appl. Math. Comput. 203 (2008), pp. 629–634.]. In addition, an efficient way of evaluating the determinant of a nearly penta-diagonal matrix is also discussed. The algorithm is suited for implementation using computer algebra systems (CAS) such as MATLAB, MACSYMA and MAPLE. Some numerical examples are given in order to illustrate the efficiency of our algorithm.

A Numerical Model for the Heat Transfer Process in an Oil Field: A Homogeneous Equilibrium Model

The authors present the numerical solution of the one-equation averaging model in thermodynamic equilibrium to describe the processes of heat transfer in an oilfield. The transient and 2-dimensional numerical model was obtained using the finite difference method in an implicit form that results in a pentadiagonal matrix in which a solution is obtained by the method of Gaussian elimination with iterative refinement. To demonstrate the performance a set of unit tests representing dynamic simulations of heat transfer processes in a reservoir is presented and the numerical results show that the mechanisms of heat transfer by convection and conduction are important for temperature distribution prediction.

Eulerian Oil Spills Model Using Finite-Volume Method with Moving Boundary and Wet-Dry Fronts

The world production of crude oil is about 3 billion tons per year. The overall objective of the model in present study is supporting the decision makers in planning and conducting preventive and emergency interventions. The conservative equation for the slick dynamics was derived from layer-averaged Navier-Stokes (LNS) equations, averaged over the slick thickness. Eulerian approach is applied across the model, based on nonlinear shallow water Reynolds-averaged Navier-Stokes (RANS) equations. Depth-integrated standard k-εturbulence schemes have been included in the model. Wetting and drying fronts of intertidal zone and moving boundary are treated within the numerical model. A highly accurate algorithm based on a fourth-degree accurate shape function has been used through an alternating-direction implicit (ADI) scheme which separates the operators into locally one-dimensional (LOD) components. The solution has been achieved by the application of KPENTA algorithm for the set of the flow equations which constitutes a pentadiagonal matrix. Hydrodynamic model was validated for a channel with a sudden expansion in width. For validation of oil spill model, predicted results are compared with experimental data from a physical modeling of oil spill in a laboratory wave basin under controlled conditions.

An elementary algorithm for computing the determinant of pentadiagonal Toeplitz matrices

Over the last 25 years, various fast algorithms for computing the determinant of a pentadiagonal Toeplitz matrices were developed. In this paper, we give a new kind of elementary algorithm requiring 56⋅⌊n−4k⌋+30k+O(logn) operations, where k≥4 is an integer that needs to be chosen freely at the beginning of the algorithm. For example, we can compute det(Tn) in n+O(logn) and 82n+O(logn) operations if we choose k as 56 and ⌊2815(n−4)⌋, respectively. For various applications, it will be enough to test if the determinant of a pentadiagonal Toeplitz matrix is zero or not. As in another result of this paper, we used modular arithmetic to give a fast algorithm determining when determinants of such matrices are non-zero. This second algorithm works only for Toeplitz matrices with rational entries.

An explicit formula for the determinant of a skew-symmetric pentadiagonal Toeplitz matrix

We give the explicit determinant of the general skew-symmetric pentadiagonal Toeplitz matrix in terms of the Tchebechev polynomials.

An eigenvalue localization theorem for pentadiagonal symmetric Toeplitz matrices

We express the eigenvalues of a pentadiagonal symmetric Toeplitz matrix as the zeros of explicitly given rational functions.

Reversibility of 1D Cellular Automata with Periodic Boundary over Finite Fields $\pmb{ {\mathbb{Z}}}_{p}$

The reversibility problem for linear cellular automata with null boundary defined by a rule matrix in the form of a pentadiagonal matrix was studied recently over the binary field ℤ2 (del Rey and Rodriguez Sanchez in Appl. Math. Comput., 2011, doi:10.1016/j.amc.2011.03.033). In this paper, we study one-dimensional linear cellular automata with periodic boundary conditions over any finite field ℤp. For any given p≥2, we show that the reversibility problem can be reduced to solving a recurrence relation depending on the number of cells and the coefficients of the local rules defining the one-dimensional linear cellular automata. More specifically, for any given values (from any fixed field ℤp) of the coefficients of the local rules, we outline a computer algorithm determining the recurrence relation which can be solved by testing reversibility of the cellular automaton for some finite number of cells. As an example, we give the full criteria for the reversibility of the one-dimensional linear cellular automata over the fields ℤ2 and ℤ3.

Reversibility of linear cellular automata

In this paper the reversibility problem for linear cellular automata defined by a characteristic matrix of the form of a pentadiagonal matrix is tackled. Specifically, a criterion for the reversibility in terms of the number of cells is stated and, in these cases, the inverse cellular automata are explicitly computed.

High-order compact filters with variable cut-off wavenumber and stable boundary treatment

This paper presents a new system of compact discrete filters based on a seven-point stencil and pentadiagonal matrix formulation which can be readjusted with ease for many different high-order finite difference schemes. The filter coefficients are determined by imposing a required cut-off wavenumber which is a free parameter for the system. The filters incorporate a high-order boundary treatment which enables the closure of the pentadiagonal matrix system and ensures its stable implementation. The overall accuracy of the proposed filter system is proved to be sixth order following grid convergence tests. It is found that a locally non-uniform allocation of cut-off wavenumbers at the boundary nodes (boundary weighting) can enhance the numerical stability of nonlinear solutions especially where a complex geometry is concerned. An optimal value of the boundary weighting factor can be obtained through eigenvalue analysis. The eigenvalue analysis procedure shown in this paper will set an exemplar for the others to customise and optimise the filters for their own use. The new filters are presented in a differential form which has computational benefits in implementation. The accuracy and performance of the proposed filters are demonstrated through a variety of benchmark test cases.

A fast algorithm for solving Toeplitz penta-diagonal systems

In this paper a fast algorithm for solving a large system with a symmetric Toeplitz penta-diagonal coefficient matrix is presented. This efficient method is based on the idea of a system perturbation followed by corrections and is competitive with standard methods. The error analysis is also given.

BECOOL: Ballooning eigensolver with COOL finite elements

An incompressible variational ideal ballooning mode equation is discretized with the COOL finite element discretization scheme using basis functions composed of variable order Legendre polynomials. This reduces the second order ordinary differential equation to a special block pentadiagonal matrix equation that is solved using an inverse vector iteration method. A benchmark test of BECOOL (Ballooning Eigensolver using COOL finite elements) with second order Legendre polynomials recovers precisely the eigenvalues computed by the VVBAL shooting code. Timing runs reveal the need to determine an optimal lower order case. Eigenvalue convergence runs show that cubic Legendre polynomials construct the optimal ballooning mode equation for intensive computations.