Pentadiagonal Matrix Research Articles

Sobre matrizes pentadiagonais não estritamente diagonais dominantes

Based on Crout's method, we will present, in this work, new non singularity criteria and sufficient conditions for existence of the LU factorization, for non strictly diagonally dominant pentadiagonal matrices. Crout's method is a recursive process of n stages that obtains the factorization A = LU of a pentadiagonal matrix of order n. In this recursive process of obtaining both the lower triangular matrix L and the upper triangular matrix U, the parameters alpha_i, 1 <= i <= n, must be non-zero to ensure that det(A) neq 0 and A = LU. Crout's recursive method is replaced by the analysis of sufficient conditions that can be verified simultaneously with low computational cost.

Continuum solutions to the two–center Coulomb problem in prolate spheroidal coordinates

We present a computational and theoretical framework for solving the Schrödinger equation (SE) for the two–center Coulomb problem in prolate spheroidal coordinates when the energy of the SE is positive. A general and robust computer code has been produced that calculates the separation constants, spheroidal harmonic expansion coefficients, regular quasi–radial two–center Coulomb wave functions, and two–center Coulomb phase shifts. These quantities can be calculated over a range of internuclear separations, angular momentum projections, and continuum electron momenta. A representative set of results are presented and compared with previous calculations, excellent agreement is found in many cases while significant disagreements are found in others. Program summaryProgram title:spheroidal-cwCPC Library link to program files:https://doi.org/10.17632/nf5gw7vjnh.1Code Ocean capsule:https://codeocean.com/capsule/7638744Licensing provisions: MITProgramming language: Fortran 90Nature of problem: A robust and accurate computer code for calculating Coulomb wave functions in prolate spheroidal coordinate is necessary for the calculation of cross sections for photonionization, and electron and positron scattering on charged diatomic targets.Solution method: The angular solution is expanded in a series of spherical harmonics leading to the diagonalization of a pentadiagonal matrix. The eigenvectors contain the expansion coefficients and the eigenvalues are the separation constants which, unlike the SE in spherical coordinates, are also functions of the energy. The radial two–center Coulomb wave functions are started using a power series solution and then propagated using a linear multistep method. An asymptotic expansion is used to calculate the two–center Coulomb phase shift, and to normalize the radial wave function.Additional comments including restrictions and unusual features: Two–center Coulomb radial wave functions and phase shifts can be accurately calculated for internuclear separations from 0.001 to 60.0 a0, angular momentum projection |m| from 0 to 44, momentum k from 0.01 to 100.0 a.u., and a combined nuclear charge Z+ from 1 to 12.

Two Inverse Eigenproblems for Certain Symmetric and Nonsymmetric Pentadiagonal Matrices

In this paper, we give sufficient conditions for the construction of certain symmetric and nonsymmetric pentadiagonal matrices from particular spectral information. The construction of the symmetric pentadiagonal matrix considers the extreme eigenvalues of its leading principal submatrices and a prescribed entry, and the construction of the nonsymmetric pentadiagonal matrix also considers an eigenvector and two prescribed entries.

The Harmonic Lagrange Top and the Confluent Heun Equation

The harmonic Lagrange top is the Lagrange top plus a quadratic (harmonic) potential term. We describe the top in the space fixed frame using a global description with a Poisson structure on \(T^{*}S^{3}\). This global description naturally leads to a rational parametrisation of the set of critical values of the energy-momentum map. We show that there are 4 different topological types for generic parameter values. The quantum mechanics of the harmonic Lagrange top is described by the most general confluent Heun equation (also known as the generalised spheroidal wave equation). We derive formulas for an infinite pentadiagonal symmetric matrix representing the Hamiltonian from which the spectrum is computed.

Mathematical formulation of large eddy simulation using hybrid finite difference methods and poisson equations

This study uses the hybrid finite-difference method (HFDM), which combines the finite-difference and spectral approaches, to describe the Magnetohydrodynamic (MHD) turbulence decay. By using a finite-difference approach in conjunction with a cyclic Penta-diagonal matrix, the numerical algorithm of the hybrid method solves the Navier-Stokes equations and the magnetic field equation with the fourth order's precision in space and second order in time. The spectral approach is used to solve the pressure Poisson equation. The time-dependent turbulence features of this flow were in excellent agreement with the appropriate analytical solution, which is valid for short timeframes, for the classical issue of the 3-D Taylor and Green vortex flow without taking the magnetic field into account. We also show how the effective numerical approach may be utilised to model the decline of magnetohydrodynamic turbulence at various magnetic Reynolds numbers.

Construction of H-Symmetric pentadiagonal matrices by three spectra

The set of eigenvalues of a square matrix P is denoted by , and the set of eigenvalues of the submatrix obtained from P by deleting its first i rows and columns of P is denoted by . It is known that a symmetric pentadiagonal matrix may be constructed from and . The pairs and must interlace; the construction is not unique. In this paper we solve the inverse eigenvalue problem for H-Symmetric pentadiagonal matrices, not necessarily symmetric. Using , not necessarily interlacing, and may contain complex eigenvalues, we construct the solution by modified Lanczos algorithm.

Exact Inversion of Pentadiagonal Matrix for Semi-Analytic Solution of 2D Poisson Equation

This work essentially consists in inverting in an exact, explicit, and original way the pentadiagonal Toeplitz matrix or tridiagonal block matrix resulting from the discretization of the two-dimensional Laplace operator. This method is an algorithm facilitating the resolution of a large number of problems governed by PDEs involving the Laplacian in two dimensions. It guarantees high precision and high efficiency in solving various differential equations.

Direct Computation of the PEC Body of Revolution Modal Green Function for the Electric Field Integral Equation

We propose a five-term recurrence relation for the direct computation of the electric field integral equation&#x00A0;(EFIE) modal Green function (MGF) of bodies of revolution. It is shown that, by considering it as an infinite penta-diagonal matrix, the proposed five-term recurrence relation can be solved in a stable manner in <inline-formula><tex-math notation="LaTeX">$O(M)$</tex-math></inline-formula> steps for <inline-formula><tex-math notation="LaTeX">$M$</tex-math></inline-formula> modes with high and controllable accuracy. By evaluating the performance of the proposed five-term recurrence relation for several scatterers of different geometries, we show that the proposed approach enables an accurate computation with a simple algorithm.

Mathematical formulation F large eddy simulation using hybrid finite difference methods and poisson equations

This study uses the hybrid finite-difference method (HFDM), which combines the finite-difference and spectral approaches, to describe the Magnetohydrodynamic (MHD) turbulence decay. By using a finite-difference approach in conjunction with a cyclic Penta-diagonal matrix, the numerical algorithm of the hybrid method solves the Navier-Stokes equations and the magnetic field equation with the fourth order's precision in space and second order in time. The spectral approach is used to solve the pressure Poisson equation. The time-dependent turbulence features of this flow were in excellent agreement with the appropriate analytical solution, which is valid for short timeframes, for the classical issue of the 3-D Taylor and Green vortex flow without taking the magnetic field into account. We also show how the effective numerical approach may be utilised to model the decline of magnetohydrodynamic turbulence at various magnetic Reynolds numbers.

The Explicit Formulae of the Determinants of Specific Pentadiagonal Toeplitz Hessenberg Matrices

The pentadiagonal Toeplitz matrix is a special kind of sparse matrix widely used in linear algebra, combinatorics, computational mathematics, and has been attracted much attention. We use the determinants of two specific Hessenberg matrices to represent the recurrence relations to prove two explicit formulae to evaluate the determinants of specific pentadiagonal Toeplitz matrices proposed in a recent paper [3]. Further, four new results are established.

Analysis of the CCFD method for MC-based image denoising problems

Image denoising using mean curvature leads to the problem of solving a nonlinear fourth-order integro-differential equation. The nonlinear fourth-order term comes from the mean curvature regularization functional. In this paper, we treat this high-order nonlinearity by reducing the nonlinear fourth-order integro-differential equation to a system of first-order equations. Then a cell-centered finite difference scheme is applied to this system. With a lexicographical ordering of the unknowns, the discretization of the mean curvature functional leads to a block pentadiagonal matrix. Our contributions are fourfold: (i) we give a new method for treating the high-order nonlinearity term; (ii) we express the discretization of this term in terms of simple matrices; (iii) we give an analysis for this new method and establish that the error is of first order; and (iv) we verify this theoretical result by illustrating the convergence rates in numerical experiments.

Trigonometric quintic B-spline collocation method for singularly perturbed turning point boundary value problems

A trigonometric quintic B-spline method is proposed for the solution of a class of turning point singularly perturbed boundary value problems (SP-BVPs) whose solution exhibits either twin boundary layers near both endpoints of the interval of consideration or an interior layer near the turning point. To resolve the boundary/interior layer(s) trigonometric quintic B-spline basis functions are used with a piecewise-uniform mesh generated with the help of a transition parameter that separates the layer and regular regions. The proposed method reduces the problem into a system of algebraic equations which can be written in matrix form with the penta-diagonal coefficient matrix. The well-known fast penta-diagonal system solver algorithm is used to solve the system. The method is shown almost fourth-order convergent irrespective of the size of the diffusion parameter ϵ. The theoretical error bounds are verified by taking some relevant test examples computationally.

Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

SummaryLet P(t) be an n × n (complex) exponentially parameterized pentadiagonal matrix. In this article, using a theorem of Akian, Bapat, and Gaubert, we present explicit formulas for asymptotics of the moduli of the eigenvalues of P(t) as t → ∞. Our approach is based on exploiting the relation with tropical algebra and the weighted digraphs of matrices. We prove that this asymptotics tends to a unique limit or two limits. Also, for n − 2 largest magnitude eigenvalues of P(t) we compute the asymptotics as n → ∞, in addition to t. When P(t) is also symmetric, these formulas allow us to compute the asymptotics of the 2‐norm condition number. The number of arithmetic operations involved, does not depend on n. We illustrate our results by some numerical tests.

Wave functions for pentadiagonal matrices in the weak coupling limit

We consider a pentadiagonal matrix which will be described in this paper. We demonstrate practical methods for obtaining weak coupling expressions for the lowest eigenvector in terms of the parameters in the matrix, v and w. It is found that the expressions simplify if the wave function coefficients are put in the denominator.

The Numerical Solution of Heat Equation by the Fourth-Order Iterative Alternating Decomposition Explicit Method with MPI

The numerical solution of the heat equation in one space dimension is obtained using the Fourth-Order Iterative Alternating Decomposition Explicit Method (4-IADE) on a parallel platform with Message Passing Interface (MPI). Here, a higher fourth-order Crank-Nicolson type scheme is used in the approximation which gives rise to a Penta diagonal matrix in the solution of the system at each time level. The method employs a splitting strategy which is applied alternately at each half time step. The method is shown to be computationally stable and appropriate parameters chosen to accelerate convergence. The accuracy of the method is comparable to that of existing well known methods. Results obtained by this method for several different problems were compared with the exact solution and agreed closely with those obtained by other finite-difference methods with correlation between speedup and efficiency.

On a structure-preserving matrix factorization for the determinants of cyclic pentadiagonal Toeplitz matrices

In recent years, a number of numerical algorithms of O(n) for computing the determinants of cyclic pentadiagonal matrices have been developed. In this paper, a cost-efficient numerical algorithm for the determinant of an n-by-n cyclic pentadiagonal Toeplitz matrix is proposed whose computational cost is estimated at $$O(\log n)$$ . The algorithm is based on a structure-preserving matrix factorization and a three-term recurrence relation. We provide some numerical results with simulations in Matlab implementation in order to demonstrate the accuracy and effectiveness of the proposed algorithm, and its competitiveness with other existing algorithms.

Binomial Coefficients and Powers of One Type of Large Pentadiagonal Matrices

In this paper, we derive a general expression for the entries of the $r$th power $\left( r\in \mathbb{N} \right) $ of one type of the $n\times n$ complex pentadiagonal matrix for all $n \geq 4( r-1)$, in terms of binomial coefficients.

Optimized compact finite difference scheme for frequency-domain acoustic wave equation

Frequency-domain numerical simulation is the most important foundation of frequency-domain full-waveform inversion and reverse time migration. The accuracy of numerical simulation seriously affects the results of the seismic inversion and image. In this article, we develop an optimized compact finite difference scheme for acoustic wave equation in frequency domain to improve numerical simulation accuracy. For the sake of avoiding the extra memory and computational costs caused by solving the inverse of a pentadiagonal band matrix, we calculate the optimized compact finite difference discrete operator for the Laplace operator in the numerical simulation. Although the optimized compact finite difference scheme has only second-order formal accuracy, it has a spectral-like resolution feature. This method can significantly reduce the numerical dispersion and the numerical anisotropy. We find that the results of the optimized compact finite difference scheme agree well with the analytic solution according to accuracy analysis. Two numerical simulations are done to verify the theoretical analysis of the optimized compact finite difference scheme.

CuPentBatch—A batched pentadiagonal solver for NVIDIA GPUs

We introduce cuPentBatch—our own pentadiagonal solver for NVIDIA GPUs. The development of cuPentBatch has been motivated by applications involving numerical solutions of parabolic partial differential equations, which we describe. Our solver is written with batch processing in mind (as necessitated by parameter studies of various physical models). In particular, our solver is directed at those problems where only the right-hand side of the matrix changes as the batch solutions are generated. As such, we demonstrate that cuPentBatch outperforms the NVIDIA standard pentadiagonal batch solver gpsvInterleavedBatch for the class of physically-relevant computational problems encountered herein. Program summaryProgram title: cuPentBatchProgram Files doi:http://dx.doi.org/10.17632/ccrp2pyyyj.1Licensing Provisions: Apache License 2.0Programming Languages: C, C++, CUDANature of Problem: cuPentBatch has been developed to solve batches of pentadiagonal systems in parallel. Pentadiagonal systems appear commonly in the numerical solution of 4th order derivatives and in 4th order accurate 2nd order derivatives. The need for batch solving these is apparent when preforming parameter studies or when solving 2D/3D systems with inversion methods such as ADI.Solution Method: cuPentBatch implements an LU factorisation for a banded pentadiagonal matrix on a GPU, solving batches of them in parallel across threads. This leads to greatly improved performance over serial and OpenMP implementations, also improves on current state of the art gpsvInterleavedBatch which is part of the cuSPRASE library in CUDA.

Stuart number effect on 3-d mhd convection in a cubic area

In this paper mathematical modeling of magnetohydrodynamics natural convection in three dimensional area at different Stuart numbers has been considered. The magnetic field is considered vertically and results have been shown at different planes of 3-D enclosure. The modeling of natural convection is based on the solution of a filtered unsteady three - dimensional Navier- Stokes equation and the equation for temperature. The problem is solved numerically: the equations of motion and temperature – by a finite-difference method in combination with penta-diagonal matrix using the Adams-Bashfort scheme, the equation for pressure – by spectral Fourier method with combination of matrix factorization. Change the dynamic of natural convection is gained over the time depending on the different values of Stuart numbers. As result of modelling, isothermal surfaces, velocity and temerature contoures, also profiles for different Sturart numbers are obtained.