Diagonalization Method Research Articles

Optimal constant for generalized diagonal update method

Optimal constant for generalized diagonal update method

An Analytical Model of Walls with Openings Variation Using the Diagonal Strut Method

Abstract Improving the strength and lateral stiffness values of the reinforced concrete structure can be done by installing the brick wall. The capacity of the reinforced concrete structure cannot be predicted if the wall equipped with openings. Developing the modeling using the diagonal strut method was done in this analytical model for wall with openings in this research. The analytical modeling results will be verified with experimental and analytical models of brick walls with two openings and a door opening show comprehensible results with deviations of 19% and 7.9% for IFOW1, IFOP and IFOW2, respectively. In other words, the analytical equations developed are successfully applied.

New occurrence of corn and rice strains of Spodoptera frugiperda (Lepidoptera: Noctuidae) in Bali and Lesser Sunda: Genetic diversity, distribution, and damage

Abstract. Yudha IKW, Supartha IW, Susila IW, Sudiarta P, Wijaya IN, Wiradana PA. 2024. New occurrence of corn and rice strains of Spodoptera frugiperda (Lepidoptera: Noctuidae) in Bali and Lesser Sunda (Indonesia): Genetic diversity, distribution, and damage. Biodiversitas 25: 1890-1900. Spodoptera frugiperda is an invasive pest with rapid proliferation and is responsible for reducing corn productivity in several regions in Indonesia, including Bali and Lesser Sunda in Indonesia. Therefore, this study aimed to determine the geographic distribution, genetic diversity, and specific genes of S. frugiperda as well as assess its attacks in Bali and Lesser Sunda (NTB and NTT) regions. The study procedures were carried out using field surveys, followed by molecular identification to determine the genetic diversity of S. frugiperda with the COI locus. A total of 150 infected corn plants were sampled using the diagonal sampling method, with a unit size of 5 m × 5 m. The distribution data obtained were then analyzed using the QGIS application and SPSS 22 for statistical analysis. The geographic distribution results showed that S. frugiperda was evenly distributed in the provinces of Bali and Lesser Sunda, with the observation of 2 strains, namely corn and rice. Corn strain was more specifically found in the Karangasem, Flores, Lombok Highlands, and Timor areas, while rice strain occured in the Jembrana, Sumbawa, Sumba, Flores, and Lombok lowland areas. In addition, a total of 12 specific nucleotide bases from S. frugiperda gene were found in Bali, NTB, and NTT regions. The findings also showed that the haplotype network of Bali, NTB, and NTT populations was effectively isolated from other locations and was in the process of genetic divergence. The 3-year invasion of S. frugiperda in Indonesian territory was reported to have experienced a process of genetic population development. Lombok Island had the highest average plant damage at 54.63%, which was not significantly different from the value of Sumbawa, Bali, and Sumba at 47.79%, 42.65%, and 40.72%, respectively. On the other hand, the lowest plant damage level in Timor at 20.95%, followed by Flores (at 30.38%).

Second harmonic generation in Harmonic-Gaussian asymmetric double quantum well: effect of magnetic and intense laser fields

We have theoretically analyzed the effects of externally applied fields (magnetic and intense laser) as well as various structural parameters such as well width, barrier width, barrier height, and structural asymmetry on the second harmonic generation (SHG) coefficients of GaAs/Al0.3Ga0.7As Harmonic-Gaussian asymmetric double quantum wells (QW). The time-independent Schrödinger equation was solved using the diagonalization method and effective mass approximation to obtain the energy eigenvalues and eigenfunctions of the structure. An expression derived within the compact density matrix approach has been used to evaluate the second harmonic generation (SHG) coefficient. Our numerical results indicate that increasing the magnitude of the magnetic (intense laser) field causes the peak positions of SHG to shift towards higher energy regions/blue (lower energy regions/red). On the other hand, as the structural parameters A1 and A2 ( k and z0 ) increase, the peak positions of the SHG coefficients exhibit a shift towards higher energy regions/blue (lower energy regions/red). We expect that the results of our study will contribute to the development of new optoelectronic devices based on the Harmonic-Gaussian asymmetric double quantum well structure.

Correlation of Water Quality Parameters on Growth Performance of Seaweed (Kappaphycus alvarezii Doty, 1986) Cultivated with Diagonal Method

Kappaphycus alvarezii seaweed is an exported commodity that has economic value and is a leading commodity in aquaculture. The present study aims to analyze the correlation between water quality parameters and the growth of Kappaphycus alvarezii seaweed using the diagonal method. A completely randomized design using three treatments and four replications was applied for the experimental design. The research was carried out from May to June 2023 in the waters of Angkue Village, Bone Regency, Indonesia. Treatment in this diagonal model was based on the length of the diagonal rope, namely 2.5 m (treatment 1), 3.5 m (treatment 2), and 4.5 m (treatment 3) of the water depth and seaweed maintained for 42 days at a water depth of five meters. The results showed that the absolute growth of seaweed was 17,537.0 g during the study, with an average of 5845.67 ± 359.34 g. In treatment 1, the absolute growth was 1368.55 ± 5.29 g; in treatment 2, it was 1467.83 ± 4.58 g; and in treatment 3, it was 1547.88 ± 1.24 g, with the highest growth rate and the lowest growth rate in the first week. Water quality parameters all contribute positively to seaweed growth, but those that contribute most strongly to seaweed growth are brightness, phosphate, nitrate, and salinity.

A Heuristic Method for Solving Polynomial Matrix Equations

We propose a heuristic method to solve polynomial matrix equations of the type ∑k=1makXk=B, where ak are scalar coefficients and X and B are square matrices of order n. The method is based on the decomposition of the B matrix as a linear combination of the identity matrix and an idempotent, involutive, or nilpotent matrix. We prove that this decomposition is always possible when n=2. Moreover, in some cases we can compute solutions when we have an infinite number of them (singular solutions). This method has been coded in MATLAB and has been compared to other methods found in the existing literature, such as the diagonalization and the interpolation methods. It turns out that the proposed method is considerably faster than the latter methods. Furthermore, the proposed method can calculate solutions when diagonalization and interpolation methods fail or calculate singular solutions when these methods are not capable of doing so.

An exact in time Fourier pseudospectral method with multiple conservation laws for three-dimensional Maxwell’s equations

Maxwell’s equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an efficient fully discrete scheme for solving three-dimensional Maxwell’s equations. This is accomplished by combining Fourier pseudospectral methods in space and exact formulation in time. Fast computation is efficiently implemented in the scheme by using the matrix diagonalisation method and fast Fourier transform algorithm which are well known in scientific computations. An optimal error estimate which is not encumbered by the CFL condition is established and the resulting scheme is proved to be of spectral accuracy in space and exact in time. Furthermore, the scheme is shown to have multiple conservation laws including discrete energy, helicity, momentum, symplecticity, and divergence-free field conservations. All the theoretical results of the accuracy and conservations are numerically illustrated by two numerical tests.

Underdetermined blind source separation based on third‐order cumulant and tensor compression

AbstractA method for Underdetermined Blind Source Separation is proposed using third‐order cumulants and tensor compression. To effectively suppress symmetrical distributed noise, the third‐order cumulant is considered. Additionally, the complexity of high‐dimensional tensors can be reduced through high order singular value decomposition (HOSVD) for compression purposes. The method begins by calculating the third‐order cumulant tensor for whitening signals at different time delays, and then stacks several cumulants into a fourth‐order tensor. The HOSVD decomposition is applied to the fourth‐order tensor, compressing the high‐dimensional tensor into a low‐dimensional core tensor. Next, the core tensor is further decomposed using the canonical polyadic decomposition, and the resulting factor matrices are fused to obtain an estimation of the mixed matrix. Finally, leveraging the signal independence, a matrix diagonalisation method is employed to recover the source signals. Theoretical analysis and simulation results demonstrate that the proposed method effectively suppresses the influence of Gaussian noise, reduces computational complexity, and saves computational time. Moreover, compared with five representative approaches, the proposed method achieves superior separation results. Specifically, for the 3 × 4 mixed model with a signal‐to‐noise ratio of 20 dB, the average relative error of speech signal and radio signal are −11.02 and −6.8 dB respectively.

Simultaneous effects of the position dependent mass and magnetic field on quantum well with the improved Tietz potential

In this study, we explored the synergistic impact of position-dependent mass and magnetic field on the electronic and optical properties of quantum wells, featuring an enhanced Tietz potential confinement incorporating both standard Morse and improved Rosen–Morse potentials, as influenced by variations in the structural parameter. Calculations were conducted within the framework of effective mass and parabolic band approximations. The diagonalization method was employed, selecting a wave function basis of trigonometric orthonormal functions to determine the eigenvalues and eigenfunctions of the confined electron potential. Our findings reveal that alterations in the sizes and shapes of the quantum wells, magnetic field strength, geometric asymmetry, and confinement parameters lead to significant variations in electron energies, transition between electron states, and the absorption spectrum. These outcomes offer valuable insights for investigating the electronic and structural properties of materials and devising novel materials with tailored optical characteristics.

The negative theology of absolute infinity: Cantor, mathematics, and humility

Cantor argued that absolute infinity is beyond mathematical comprehension. His arguments imply that the domain of mathematics cannot be grasped by mathematical means. We argue that this inability constitutes a foundational problem. For Cantor, however, the domain of mathematics does not belong to mathematics, but to theology. We thus discuss the theological significance of Cantor’s treatment of absolute infinity and show that it can be interpreted in terms of negative theology. Proceeding from this interpretation, we refer to the recent debate on absolute generality and argue that the method of diagonalization constitutes a modern version of the vianegativa. On our reading, negative theology can evoke an attitude of humility with respect to the boundedness of the human condition. Along these lines, we think that the foundational problem of mathematics concerning its domain can be addressed through a methodological attitude of humility.

Construction of A-stable explicit last-stage diagonal implicit Runge–Kutta (ELDIRK) methods

ELDIRK methods are defined to have an Explicit Last stage in the general Butcher array of Diagonal Implicit Runge-Kutta methods, with the consequence, that no additional system of equations must be solved, compared to the embedded RK method. Two general formulations for second- and third-order ELDIRK methods have been obtained recently in Mahnken [21] with specific schemes, e.g. for the embedded implicit Euler method, the embedded trapezoidal-rule and the embedded Ellsiepen method. In the first part of this paper, we investigate some general stability characteristics of ELDIRK methods, and it will be shown that the above specific RK schemes are not A-stable. Therefore, in the second part, the above-mentioned general formulations are used for further stability investigations, with the aim to construct new second- and third-order ELDIRK methods which simultaneously are A-stable. Two numerical examples are concerned with the curing for a thermosetting material and phase-field RVE modeling for crystallinity and orientation. The numerical results confirm the theoretical results on convergence order and stability.

Performance evaluation of data embedding schemes for two-dimensional display field communication.

Display field communication (DFC) is an unobtrusive display-to-camera technology that transmits data within the frequency domain of images, ensuring that the embedded data are hidden and do not disrupt the viewing experience. The display embeds data into image frames, while the receiver captures the display and extracts it. Two-dimensional DFC (2D-DFC) focuses on embedding data in the width and height of an image. This study explores two methods to minimize the error rate in 2D-DFC without affecting the quality of the displayed image. The orthogonal method embeds data in the orthogonal direction of an image. On the other hand, the diagonal embedding method strategically embeds the data in the diagonal direction. Experiments show the diagonal method maintains a higher peak signal-to-noise ratio and surpasses the orthogonal embedding method in terms of bit error rate. 2D-DFC is expected to have practical applications in digital signage, advertising and informational displays at airports and train stations, as well as at large-scale displays for events, sports arenas, and performance venues.

Robust PRESB Preconditioning of a 3-Dimensional Space-Time Finite Element Method for Parabolic Problems

Abstract We present a recently developed preconditioning of square block matrices (PRESB) to be used within a parallel method to solve linear systems of equations arising from tensor-product discretizations of initial boundary-value problems for parabolic partial differential equations. We consider weak formulations in Bochner–Sobolev spaces and tensor-product finite element approximations for the heat and eddy current equations. The fast diagonalization method is employed to decouple the arising linear system of equations into auxiliary spatial complex-valued linear systems that can be solved concurrently. We prove that the real part of the system matrix is positive definite, which allows us to accelerate the flexible generalized minimal residual method (FGMRES) by the PRESB preconditioner. The action of PRESB on a vector includes two solutions with positive definite matrices. The spectrum of the preconditioned system lies between 1/2 and 1. Finally, we combine the PRESB-FGMRES method with multigrid-CG iterations and illustrate the numerical efficiency and the robustness for spatial discretizations up to 12 millions degrees of freedom.

Scrambling Dynamics and Out-of-Time-Ordered Correlators in Quantum Many-Body Systems

This tutorial article introduces the physics of quantum information scrambling in quantum many-body systems. The goals are to understand how to precisely quantify the spreading of quantum information and how causality emerges in complex quantum systems. We introduce a general framework to study the dynamics of quantum information, including detection and decoding. We show that the dynamics of quantum information is closely related to operator dynamics in the Heisenberg picture, and, under certain circumstances, can be precisely quantified by the so-called out-of-time-ordered correlator (OTOC). The general behavior of the OTOC is discussed based on several toy models, including the Sachdev-Ye-Kitaev model, random circuit models, and Brownian models, in which the OTOC is analytically tractable. We introduce numerical methods, including exact diagonalization and tensor network methods, to calculate the OTOC for generic quantum many-body systems. We also survey current experimental schemes for measuring the OTOC in various quantum simulators. Published by the American Physical Society 2024

Update of [formula omitted]: Newly added functions and methods in versions 2 and 3

HΦ [aitch-phi] is an open-source software package of numerically exact and stochastic calculations for a wide range of quantum many-body systems. In this paper, we present the newly added functions and the implemented methods in vers. 2 and 3. In ver. 2, we implement spectrum calculations by the shifted Krylov method, and low-energy excited state calculations by the locally optimal block preconditioned conjugate gradient (LOBPCG) method. In ver. 3, we implement the full diagonalization method using ScaLAPACK and GPGPU computing via MAGMA. We also implement a real-time evolution method and the canonical thermal pure quantum (cTPQ) state method for finite-temperature calculations. The Wannier90 format for specifying the Hamiltonians is also implemented. Using the Wannier90 format, it is possible to perform the calculations for the abinitio low-energy effective Hamiltonians of solids obtained by the open-source software RESPACK. We also update Standard mode—simplified input format in HΦ—to use these functions and methods. We explain the basics of the implemented methods and how to use them. Program summaryProgram Title:HΦ [aitch-phi]CPC Library link to program files:https://doi.org/10.17632/vnfthtyctm.2Developer's repository link:https://github.com/issp-center-dev/HPhiLicensing provisions: GPLv3Programming language: C, FortranJournal reference of previous version: Comput. Phys. Commun. 217 (2017) 180–192Does the new version supersede the previous version?: Yes. The latest version has compatibility with the old versions. Although most functions are available in newer versions, some redundant/unnecessary functions were abolished in newer versions.Reasons for the new version: Implementation of new functions and methods, development of utilities, and bug fixes.Summary of revisions: We added new functions to obtain excited spectrum, low-energy excited states, and time-dependent physical quantities. We also implemented the full diagonalization method by using MAGMA on GPGPUs and finite-temperature simulations by using the canonical thermal pure quantum state method. In addition, we developed utilities for connection with RESPACK and a submodule for a generator of input files.Nature of problem: Physical properties in quantum lattice models with finite system sizesSolution method: In HΦ, we implemented several numerical methods such as the Lanczos method, the full diagonalization method, the locally optimal block preconditioned conjugate gradient (LOBPCG) method, the real-time evolution method based on the Taylor expansion, the shifted Krylov method, and the microcanonical/canonical thermal pure quantum state method.

GPU Algorithms for Structured Sparse Matrix Multiplication with Diagonal Storage Schemes

Matrix–matrix multiplication is of singular importance in linear algebra operations with a multitude of applications in scientific and engineering computing. Data structures for storing matrix elements are designed to minimize overhead information as well as to optimize the operation count. In this study, we utilize the notion of the compact diagonal storage method (CDM), which builds upon the previously developed diagonal storage—an orientation-independent uniform scheme to store the nonzero elements of a range of matrices. This study exploits both these storage schemes and presents efficient GPU-accelerated parallel implementations of matrix multiplication when the input matrices are banded and/or structured sparse. We exploit the data layouts in the diagonal storage schemes to expose a substantial amount of fine-grained parallelism and effectively utilize the GPU shared memory to improve the locality of data access for numerical calculations. Results from an extensive set of numerical experiments with the aforementioned types of matrices demonstrate orders-of-magnitude speedups compared with the sequential performance.

High-accuracy solution of the Dirac equation for three-dimensional hydrogen atom using a variationally optimized diagonalization method

High-accuracy solution of the Dirac equation for three-dimensional (3D) hydrogen atom is theoretically presented using a variationally optimized diagonalization method. By imposing the asymptotical conditions, a complete set of orthogonal Laguerre basis functions is constructed explicitly for the solution of the Dirac equation, which eliminates the singularity difficulty at the point nucleus and gives a convergent integral. A fully explicit expression of the matrix elements of the Dirac Hamiltonian has been derived from the method of the generating function of Laguerre polynomials. The relative errors of the relativistic energies are ranged from 11 to 14 digits, showing that the calculated relativistic energies are in excellent agreement with the exact energies. These results show that my proposed orthogonal Laguerre basis set is extremely efficient and stable in calculating high-accuracy solution of the Dirac equation for 3D hydrogen atom.

Effects of anisotropy on the electronic structure and nonlinear optical rectification in a quantum dot with hydrogen molecular ion

In this study, the effects of anisotropy and quantum dot size on energy levels and nonlinear optical rectification of a D_{2}^{ + } molecular complex confined to a two-dimensional quantum dot are investigated. The energy eigenvalues and corresponding eigenfunctions of the resulting eigenvalue equation are obtained using the adiabatic approach and the two-dimensional diagonalization method. We conclude that the evolution of the energy spectrum depends predominantly on the anisotropy-induced change of geometric confinement size. In addition, it has been determined that the peak position of the nonlinear optical rectification shifts to lower or higher energies as the asymmetry of the system increases, depending on whether the QD is in the oblate or prolate case. This feature can be considered an advantage in tuning the nonlinear optical properties of systems based on two-dimensional semiconductor heterostructures.

ТЕОРЕМА ПРО НЕПОВНОТУ ДЛЯ ОБЧИСЛЮВАНИХ ЗАДАЧ

In the theory of recursive functions, a recursively enumerable (but not recursive) set K = {x |x ∈ Wx } is obtained by Cantor’s diagonal method. Based on Diophantine sets, K is expressed by some polynomial that has positive roots. On the contrary, the set K = {x |x ∉ Wx } is not recursively enumerable. None of the computable functions can enumerate all the elements of the set K . As a result of the productivity of the set K , a parameter exists for which the polynomial has no positive roots. However, it is impossible to prove their absence since this parameter does not belong to any recursively enumerable set. Keywords: diagonal method, Diophantine set, recursively enumerable sets.

New-generation electron-propagator methods for vertical electron detachment energies of molecular anions: benchmarks and applications to model green-fluorescent-protein chromophores.

Ab initio electron-propagator calculations continue to be useful companions to experimental investigations of electronic structure in molecular anions. A new generation of electron-propagator methods recently has surpassed its antecedents' predictive accuracy and computational efficiency. Interpretive clarity has been conserved, for no adjustable parameters have been introduced in the preparation of molecular orbitals or in the formulation of approximate self-energies. These methods have employed the diagonal self-energy approximation wherein each Dyson orbital equals a canonical Hartree-Fock orbital times the square root of a probability factor. Numerical tests indicate that explicitly renormalized, diagonal self-energies are needed when Dyson orbitals have large valence nitrogen, oxygen or fluorine components. They also demonstrate that even greater accuracy can be realized with generalizations that do not employ the diagonal self-energy approximation in the canonical Hartree-Fock basis. Whereas the diagonal methods have fifth-power arithmetic scaling factors, the non-diagonal generalizations introduce only non-iterative sixth-power contractions. Composite models conserve the accuracy of the most demanding combinations of self-energy approximations and flexible basis sets with drastically reduced computational effort. Composite-model results on anions that resemble the chromophore of the green fluorescent protein illustrate the interpretive capabilities of explicitly renormalized self-energies. Accurate predictions on the lowest vertical electron detachment energy of each anion confirm experimental data and the utility of the diagonal self-energy approximation.