LAPACK Library Research Articles

Givens rotations for QR decomposition, SVD and PCA over database joins

This article introduces FiGaRo, an algorithm for computing the upper-triangular matrix in the QR decomposition of the matrix defined by the natural join over relational data. FiGaRo ’s main novelty is that it pushes the QR decomposition past the join. This leads to several desirable properties. For acyclic joins, it takes time linear in the database size and independent of the join size. Its execution is equivalent to the application of a sequence of Givens rotations proportional to the join size. Its number of rounding errors relative to the classical QR decomposition algorithms is on par with the database size relative to the join output size. The QR decomposition lies at the core of many linear algebra computations including the singular value decomposition (SVD) and the principal component analysis (PCA). We show how FiGaRo can be used to compute the orthogonal matrix in the QR decomposition, the SVD and the PCA of the join output without the need to materialize the join output. A suite of experiments validate that FiGaRo can outperform both in runtime performance and numerical accuracy the LAPACK library Intel MKL by a factor proportional to the gap between the sizes of the join output and input.

An Optimized Framework for Matrix Factorization on the New Sunway Many-core Platform

Matrix factorization functions are used in many areas and often play an important role in the overall performance of the applications. In the LAPACK library, matrix factorization functions are implemented with blocked factorization algorithm, shifting most of the workload to the high-performance Level-3 BLAS functions. But the non-blocked part, the panel factorization, becomes the performance bottleneck, especially for small- and medium-size matrices that are the common cases in many real applications. On the new Sunway many-core platform, the performance bottleneck of panel factorization can be alleviated by keeping the panel in the LDM for the panel factorization. Therefore, we propose a new framework for implementing matrix factorization functions on the new Sunway many-core platform, facilitating the in-LDM panel factorization. The framework provides a template class with wrapper functions, which integrates inter-CPE communication for the Level-1 and Level-2 BLAS functions with flexible interfaces and can accommodate different partitioning schemes. With the framework, writing panel factorization code with data residing in the LDM space can be done with much higher productivity. We implemented three functions ( dgetrf , dgeqrf , and dpotrf ) based on the framework and compared our work with a CPE_BLAS version, which uses the original LAPACK implementation linked with optimized BLAS library that runs on the CPE mesh. Using the most favorable partitioning, the panel factorization part achieves speedup of up to 26.3, 19.1, and 18.2 for the three matrix factorization functions. For the whole function, our implementation is based on a carefully tuned recursion framework, and we added specific optimization to some subroutines used in the factorization functions. Overall, we obtained average speedup of 9.76 on dgetrf , 10.12 on dgeqrf , and 4.16 on dpotrf , compared to the CPE_BLAS version. Based on the current template class, our work can be extended to support more categories of linear algebra functions.

ElATools: A tool for analyzing anisotropic elastic properties of the 2D and 3D materials

We introduce a computational method and a user-friendly code with a terminal-based graphical user interface (GUI), named ElATools, developed to analyze mechanical and anisotropic elastic properties. ElATools enables facile analysis of the second-order elastic stiffness tensor of two-dimensional (2D) and three-dimensional (3D) crystal systems. It computes and displays the main mechanical properties including the bulk modulus, Young's modulus, shear modulus, hardness, p-wave modulus, universal anisotropy index, Chung-Buessem anisotropy index, log-Euclidean anisotropy parameter, Cauchy pressures, Poisson's ratio, and Pugh's ratio, using three averaging schemes of Voigt, Reuss, and Hill. It includes an online and offline database from the Materials Project with more than 13,000 elastic stiffness constants for 3D materials. The program supports output files of the well-known computational codes IRelast, IRelast2D, ElaStic, and AELAS. Four types of plotting and visualization tools are integrated to conveniently interface with GNUPLOT, XMGRACE, view3dscene and plotly libraries, offering immediate post-processing of the results. ElATools provides reliable means to investigate the mechanical stability based on the calculation of six (three) eigenvalues of the elastic tensor in 3D (2D) materials. It can efficiently identify anomalous mechanical properties, such as negative linear compressibility, negative Poisson's ratio, and highly-anisotropic elastic modulus in 2D and 3D materials, which are central properties to design and develop high-performance nanoscale electromechanical devices. Moreover, ElATools can predict the behavior of the sound velocities and their anisotropic properties, such as acoustic phase/group velocities and power flow angles in materials, by solving the Christoffel equation. Six case studies on selected material systems, namely, ZnAu2(CN)4, CrB2, δ-phosphorene, Pd2O6Se2 monolayer, and GaAs, and a hypothetical set of systems with cubic symmetry are presented to demonstrate the descriptive and predictive capabilities of ElATools. Program summaryProgram Title:ElAToolsCPC Library link to program files:https://doi.org/10.17632/njh92x66c6.1Developer's repository link:https://github.com/shahramyalameha/ElAToolsLicensing provisions: GNU General Public Licence 3.0Programming language: Fortran 90External routines/libraries:LaPack, Blas, and Plotly Javascript libraries, GNUPLOT, XMGRACE, view3dscene.Nature of problem: Identifying anisotropic elastic properties of 2D and 3D materials, and calculating acoustic phase and group velocities in homogeneous solids.Solution method: Second-order elastic stiffness tensor analysis using transformation law and calculations of the elastic surfaces properties. Solving the Christoffel equation eigenvalue problem using diagonalization and calculations of the sound velocities.Additional comments: Documentation available at https://yalameha.gitlab.io/elastictools/index.html

K[formula omitted] — Open-source library for the shifted Krylov subspace method of the form [formula omitted

We develop Kω, an open-source linear algebra library for the shifted Krylov subspace methods. The methods solve a set of shifted linear equations (zkI−H)x(k)=b(k=0,1,2,…) for a given matrix H and a vector b, simultaneously. The leading order of the operational cost is the same as that for a single equation. The shift invariance of the Krylov subspace is the mathematical foundation of the shifted Krylov subspace methods. Applications in materials science are presented to demonstrate the advantages of the algorithm over the standard Krylov subspace methods such as the Lanczos method. We introduce benchmark calculations of (i) an excited (optical) spectrum and (ii) intermediate eigenvalues by the contour integral on the complex plane. In combination with the quantum lattice solver HΦ, Kω can realize parallel computation of excitation spectra and intermediate eigenvalues for various quantum lattice models. Program summaryProgram Title: Kω [kéi-óumig@]CPC Library link to program files:http://dx.doi.org/10.17632/mt928nz5r3.1Developer’s repository link:https://github.com/issp-center-dev/KomegaLicensing provisions: GNU Lesser General Public License Version 3.Programming language: Fortran 90External routines/libraries: BLAS library, LAPACK library (Used in the sample program), MPI library (Optional).Nature of problem: Efficient algorithms, called shifted Krylov subspace algorithms, designed to solve the shifted linear equations.Solution method: Shifted conjugate gradient method, Shifted conjugate orthogonal conjugate gradient method, shifted bi-conjugate gradient method.Additional comments: The present paper is accompanied by a frozen copy of Kω release 2.0.0 that is made publicly available on GitHub (repository https://github.com/issp-center-dev/Komega, commit hash fd5455328b102ec4fa13432496e41c404a0f5a9d).

The BLAS API of BLASFEO

Basic Linear Algebra Subroutines For Embedded Optimization (BLASFEO) is a dense linear algebra library providing high-performance implementations of BLAS- and LAPACK-like routines for use in embedded optimization and other applications targeting relatively small matrices. BLASFEO defines an application programming interface (API) which uses a packed matrix format as its native format. This format is analogous to the internal memory buffers of optimized BLAS, but it is exposed to the user and it removes the packing cost from the routine call. For matrices fitting in cache, BLASFEO outperforms optimized BLAS implementations, both open source and proprietary. This article investigates the addition of a standard BLAS API to the BLASFEO framework, and proposes an implementation switching between two or more algorithms optimized for different matrix sizes. Thanks to the modular assembly framework in BLASFEO, tailored linear algebra kernels with mixed column- and panel-major arguments are easily developed. This BLAS API has lower performance than the BLASFEO API, but it nonetheless outperforms optimized BLAS and especially LAPACK libraries for matrices fitting in cache. Therefore, it can boost a wide range of applications, where standard BLAS and LAPACK libraries are employed and the matrix size is moderate. In particular, this article investigates the benefits in scientific programming languages such as Octave, SciPy, and Julia.

Bidiagonal SVD Computation via an Associated Tridiagonal Eigenproblem

The Singular Value Decomposition (SVD) is widely used in numerical analysis and scientific computing applications, including dimensionality reduction, data compression and clustering, and computation of pseudo-inverses. In many cases, a crucial part of the SVD of a general matrix is to find the SVD of an associated bidiagonal matrix. This article discusses an algorithm to compute the SVD of a bidiagonal matrix through the eigenpairs of an associated symmetric tridiagonal matrix. The algorithm enables the computation of only a subset of singular values and corresponding vectors, with potential performance gains. The article focuses on a sequential version of the algorithm, and discusses special cases and implementation details. The implementation, called BDSVDX, has been included in the LAPACK library. We use a large set of bidiagonal matrices to assess the accuracy of the implementation, both in single and double precision, as well as to identify potential shortcomings. The results show that BDSVDX can be up to three orders of magnitude faster than existing algorithms, which are limited to the computation of a full SVD. We also show comparisons of an implementation that uses BDSVDX as a building block for the computation of the SVD of general matrices.

LAPACK library for mathematical modeling used in 3D object reconstruction

Over time, multiple solutions have been sought for building 3D models in the fastest and most accurate way possible. For this reason, this article proposes the use of LAPACK library for a quick solution of linear systems generated by least square method calculation. A mathematical and physical component used in the reconstruction of models consists in the use of Spherical Harmonics which represent some special functions often used in solving partial differential equations in different scientific areas. This problem finds its place in multiple domains such as medicine, engineering, programming. The target of the work described by this paper is to achieve optimization in the reconstruction of 3D models. We aim to perform real-time model reconstruction for large data sets by using Spherical Harmonics functions.

Features of formation of the temperature field in a well with a multilayer system during oil degassing

As is known, well thermometry is one of the main methods for diagnosing the condition of a well and formation while monitoring the development of oil and gas fields. Despite the widespread use of thermometry in the oil and gas industry, today the question of the formation of the temperature field in a well with a multilayer system under conditions of oil degassing remains poorly understood. Therefore, it is relevant to study thermohydrodynamic processes that form the temperature field in the well-reservoir system based on a mathematical model of the movement of a two-phase flow of oil and gas, taking into account oil degassing. Based on a numerical model describing the non-isothermal movement of a two-phase fluid in a well with a multi-layer system, the article discusses the problem of selecting soda oil from a well that operates two layers. The formation of the temperature field is due to the influence of the adiabatic and Joule-Thomson effects, as well as the heat of gas degassing. The system of differential equations is solved numerically, and discretization by the control volume method. The resulting system of nonlinear equations is linearized by the Newton-Raphson method, the roots of which are found by the Gauss method from the LAPACK library. The paper considers the features of temperature effects in a well with a two-layer system for various durations of pressure reduction at the wellhead, oil saturation pressure with gas, as well as the functional dependences of the relative phase permeabilities of phases on saturation. It is shown that a sharp decrease in pressure at the wellhead leads to significant cooling in the first hour of well operation. The results can be used in the planning of field geophysical studies and interpretation of the results of temperature studies of wells in the production of gas-cut oil.

The Parallel Tiled WZ Factorization Algorithm for Multicore Architectures

Abstract The aim of this paper is to investigate dense linear algebra algorithms on shared memory multicore architectures. The design and implementation of a parallel tiled WZ factorization algorithm which can fully exploit such architectures are presented. Three parallel implementations of the algorithm are studied. The first one relies only on exploiting multithreaded BLAS (basic linear algebra subprograms) operations. The second implementation, except for BLAS operations, employs the OpenMP standard to use the loop-level parallelism. The third implementation, except for BLAS operations, employs the OpenMP task directive with the depend clause. We report the computational performance and the speedup of the parallel tiled WZ factorization algorithm on shared memory multicore architectures for dense square diagonally dominant matrices. Then we compare our parallel implementations with the respective LU factorization from a vendor implemented LAPACK library. We also analyze the numerical accuracy. Two of our implementations can be achieved with near maximal theoretical speedup implied by Amdahl’s law.

TeraPCA: a fast and scalable software package to study genetic variation in tera-scale genotypes.

Principal Component Analysis is a key tool in the study of population structure in human genetics. As modern datasets become increasingly larger in size, traditional approaches based on loading the entire dataset in the system memory (Random Access Memory) become impractical and out-of-core implementations are the only viable alternative. We present TeraPCA, a C++ implementation of the Randomized Subspace Iteration method to perform Principal Component Analysis of large-scale datasets. TeraPCA can be applied both in-core and out-of-core and is able to successfully operate even on commodity hardware with a system memory of just a few gigabytes. Moreover, TeraPCA has minimal dependencies on external libraries and only requires a working installation of the BLAS and LAPACK libraries. When applied to a dataset containing a million individuals genotyped on a million markers, TeraPCA requires <5h (in multi-threaded mode) to accurately compute the 10 leading principal components. An extensive experimental analysis shows that TeraPCA is both fast and accurate and is competitive with current state-of-the-art software for the same task. Source code and documentation are both available at https://github.com/aritra90/TeraPCA. Supplementary data are available at Bioinformatics online.

High-accuracy numerical calculations of the bound states of a hydrogen atom in a constant magnetic field with arbitrary strength

We develop a simple and effective method for solving the Schrödinger equation of a hydrogen atom in a constant magnetic field with arbitrary strength. Energies are obtained not only for the ground and low-lying states but also for highly excited states with precision from 12 up to 20 decimal digits. The calculations are performed for an entire range of magnetic field intensity up to 9.4×108 Tesla, the strongest field ever observed. The strong point of the development of the method is the construction of an anharmonic oscillator model for a hydrogen atom in a constant magnetic field via the Kustaanheimo–Stiefel transformation. This model allows the use of purely algebraic calculations and the Feranchuk–Komarov (FK) operator method for effectively solving the Schrödinger equation. The advantages of the basis set in this work are also discussed to extend its application to other problems, such as multi-electron atoms in a constant magnetic field. We also provide a program written by FORTRAN for the solutions mentioned above. Program summaryProgram Title: CHAMFProgram Files doi:http://dx.doi.org/10.17632/s5shscgfcj.1Licensing provisions: BSD 3-clauseProgramming language: FORTRAN77Nature of problem: The Schrödinger equation for a hydrogen atom in a magnetic field is rewritten into that of a four-dimensional anharmonic oscillator by using the Kustaanheimo–Stiefel transformation. The later has a structure more convenient for applying the algebraic method based on annihilation and creation operators. Therefore, the FK operator method [1] can be applied for obtaining high-accuracy numerical solutions which means the obtained energies converge to a given precision (up to twenty decimal digits in this work). The method is developed for an entire range of magnetic field intensity up to 9.4×108 Tesla and for not only ground state but also highly excited states.Solution method: The FK operator method is modified and used for the high-accuracy numerical solutions. This method combines with the algebraic technique which is useful for calculating all the needed matrix elements. Thus, the Schrödinger equation becomes the symmetric generalized eigenvalue problem which can be solved by the subroutine dsygvx.f of the LAPACK library [2]. For the better accuracy with quadruple precision, we modify this subroutine and its dependence with using the real*16 number instead of the real*8. Moreover, a free parameter introduced by the FK operator method can be used for increasing the speed of convergence. The optimal choice of this free parameter is implemented in the program.Additional comments including restrictions and unusual features: Operating system: Linux. RAM: at least 1 GByte per core. The gfortran compiler is recommended for this program. The running time is from a centisecond to a few minutes depending on the required precision. However, for the super-strong magnetic fields (γ≥200), the running time may reach to two hours for the precision of twelve decimal digits, and RAM is needed to be at least 60 GB.[1] I. Feranchuk, A. Ivanov, Van-Hoang Le, A. Ulyanenkov, Non-perturbative Description of Quantum Systems, Springer, Switzerland, 2015. doi:10.1007/978-3-319-13006-4.[2] Netlib.org. LAPACK: Linear Algebra PACKage, Subroutine dsygvx.f. URL http://www.netlib.org/lapack/lapack-3.1.1/html/dsygvx.f.html

Accelerating the computation of FLAPW methods on heterogeneous architectures

Legacy codes in computational science and engineering have been very successful in providing essential functionality to researchers. However, they are not capable of exploiting the massive parallelism provided by emerging heterogeneous architectures. The lack of portable performance and scalability puts them at high risk: either they evolve or they are doomed to disappear. One example of legacy code which would heavily benefit from a modern design is FLEUR, a software for electronic structure calculations. In previous work, the computational bottleneck of FLEUR was partially re-engineered to have a modular design that relies on standard building blocks, namely BLAS and LAPACK. In this paper, we demonstrate how the initial redesign enables the portability to heterogeneous architectures. More specifically, we study different approaches to port the code to architectures consisting of multi-core CPUs equipped with one or more coprocessors such as Nvidia GPUs and Intel Xeon Phis. Our final code attains over 70\% of the architectures' peak performance, and outperforms Nvidia's and Intel's libraries. Finally, on JURECA, the supercomputer where FLEUR is often executed, the code takes advantage of the full power of the computing nodes, attaining $5\times$ speedup over the sole use of the CPUs.

Fast and accurate normalization of vectors and quaternions

We present fast and accurate ways to normalize two and three-dimensional vectors and quaternions and compute their length. Our approach is an adaptation of ideas used in the linear algebra library LAPACK, and we believe that the computational geometry and computer-aided design communities are not aware of the possibility of speeding up these fundamental operations in the robust way proposed here.

Базовые библиотеки линейной алгебры для высокопроизводительных расчетов

The article considers linear algebra libraries such as BLAS, LAPACK, ScaLAPACK, MKL, and ATLAS, which support high-performance computing (HPC) in modern architectures and are used both in well-known performance tests and in various applications. In the majority of applications, the most time-consuming computation stages are implemented by calling subroutines from such libraries; therefore, the optimal choice of a library is an important issue in setting up computations. The main aim of this review is to describe the invariant characteristics of libraries to achieve high performance of applications. High performance computations used in different fields of knowledge are briefly reviewed. Classification of linear algebra libraries in terms of their functionality and applied high-performance architectures is suggested. The basic low-level BLAS library implemented for all HPC architectures is demonstrated. It is pointed out that the BLAS library supports dividing of the entire computation process into several parallel flows in systems with a common memory field; for such systems, tools such as OpenMP or OpenACC are used. In the case of systems with distributed memory, the parallel version of this library, called PBLAS is used, which supports exchange of messages between nodes using the MPI standard. Higher-level libraries based on the BLAS, e.g., the LAPACK library, which contains a large set of different programs for linear algebra, are described. The ScaLAPACK library for the distributed memory model, which is based on the LAPACK and PBLAS libraries, as well as the Intel MKL library, which is its modern development, are presented. To support efficient operation of hybrid systems, the fundamentally new libraries MAGMA and PLASMA involving features for optimizing linear-algebraic computing of small dimension are analyzed. Libraries supporting solution of eigenvalue problems, such as the EISPACK, PeigS, and a number of other libraries, are investigated. It is pointed out that in the new ELPA library oriented to supercomputers, both OpenMP and MPI tools can be used. It is noted that operations on sparse matrices, especially multiplication of matrices, are very relevant for many applied fields of science; in this regard, the SparseBLAS library can be considered to be the basic standard for them. It is concluded that the optimal choice of a library depends essentially on both the particular application and on the used computing architecture.

The Singular Value Decomposition: Anatomy of Optimizing an Algorithm for Extreme Scale

The computation of the singular value decomposition, or SVD, has a long history with many improvements over the years, both in its implementations and algorithmically. Here, we survey the evolution of SVD algorithms for dense matrices, discussing the motivation and performance impacts of changes. There are two main branches of dense SVD methods: bidiagonalization and Jacobi. Bidiagonalization methods started with the implementation by Golub and Reinsch in Algol60, which was subsequently ported to Fortran in the EISPACK library, and was later more efficiently implemented in the LINPACK library, targeting contemporary vector machines. To address cache-based memory hierarchies, the SVD algorithm was reformulated to use Level 3 BLAS in the LAPACK library. To address new architectures, ScaLAPACK was introduced to take advantage of distributed computing, and MAGMA was developed for accelerators such as GPUs. Algorithmically, the divide and conquer and MRRR algorithms were developed to reduce the number of operations. Still, these methods remained memory bound, so two-stage algorithms were developed to reduce memory operations and increase the computational intensity, with efficient implementations in PLASMA, DPLASMA, and MAGMA. Jacobi methods started with the two-sided method of Kogbetliantz and the one-sided method of Hestenes. They have likewise had many developments, including parallel and block versions and preconditioning to improve convergence. In this paper, we investigate the impact of these changes by testing various historical and current implementations on a common, modern multicore machine and a distributed computing platform. We show that algorithmic and implementation improvements have increased the speed of the SVD by several orders of magnitude, while using up to 40 times less energy.

Linear algebra software for large-scale accelerated multicore computing

Many crucial scientific computing applications, ranging from national security to medical advances, rely on high-performance linear algebra algorithms and technologies, underscoring their importance and broad impact. Here we present the state-of-the-art design and implementation practices for the acceleration of the predominant linear algebra algorithms on large-scale accelerated multicore systems. Examples are given with fundamental dense linear algebra algorithms – from the LU, QR, Cholesky, and LDLT factorizations needed for solving linear systems of equations, to eigenvalue and singular value decomposition (SVD) problems. The implementations presented are readily available via the open-source PLASMA and MAGMA libraries, which represent the next generation modernization of the popular LAPACK library for accelerated multicore systems.To generate the extreme level of parallelism needed for the efficient use of these systems, algorithms of interest are redesigned and then split into well-chosen computational tasks. The task execution is scheduled over the computational components of a hybrid system of multicore CPUs with GPU accelerators and/or Xeon Phi coprocessors, using either static scheduling or light-weight runtime systems. The use of light-weight runtime systems keeps scheduling overheads low, similar to static scheduling, while enabling the expression of parallelism through sequential-like code. This simplifies the development effort and allows exploration of the unique strengths of the various hardware components. Finally, we emphasize the development of innovative linear algebra algorithms using three technologies – mixed precision arithmetic, batched operations, and asynchronous iterations – that are currently of high interest for accelerated multicore systems.

Parallel Programming Models for Dense Linear Algebra on Heterogeneous Systems

We present a review of the current best practices in parallel programming models for dense linear algebra DLA on heterogeneous architectures. We consider multicore CPUs, stand alone manycore coprocessors, GPUs, and combinations of these. Of interest is the evolution of the programming models for DLA libraries { in particular, the evolution from the popular LAPACK and ScaLAPACK libraries to their modernized counterparts PLASMA for multicore CPUs and MAGMA for heterogeneous architectures, as well as other programming models and libraries. Besides providing insights into the programming techniques of the libraries considered, we outline our view of the current strengths and weaknesses of their programming models { especially in regards to hardware trends and ease of programming high-performance numerical software that current applications need { in order to motivate work and future directions for the next generation of parallel programming models for high-performance linear algebra libraries on heterogeneous systems.

Singular Value Homogenization: a simple preconditioning technique for linearly constrained optimization and its potential applications in medical therapy

A wealth of problems occurring naturally in the applied sciences can be reformulated as optimization tasks whose argument is constrained to the solution set of a system of linear equations. Solving these efficiently typically requires computation of feasible descent directions and proper step sizes - the quality of which depends largely on conditioning of the linear equality constraint. In this paper we present a way of transforming such ill-conditioned problems into easily solvable, equivalent formulations by means of directly changing the singular values of the system’s associated matrix. This transformation allows us to solve problems for which corresponding routines in the LAPACK library as well as widely used projection methods converged either very slowly or not at all.

EMUstack: An open source route to insightful electromagnetic computation via the Bloch mode scattering matrix method

We describe EMUstack, an open-source implementation of the Scattering Matrix Method (SMM) for solving field problems in layered media. The fields inside nanostructured layers are described in terms of Bloch modes that are found using the Finite Element Method (FEM). Direct access to these modes allows the physical intuition of thin film optics to be extended to complex structures. The combination of the SMM and the FEM makes EMUstack ideally suited for studying lossy, high-index contrast structures, which challenge conventional SMMs. Program summaryProgram title: EMUstackCatalogue identifier: AEZI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZI_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 154301No. of bytes in distributed program, including test data, etc.: 5308635Distribution format: tar.gzProgramming language: Python, Fortran.Computer: Any computer with a Unix-like system with Python, a Fortran compiler and F2Py [1]. Also required are the following free libraries LAPACK and BLAS [2], UMFPACK [3]. Developed on 1.6 GHz Intel Core i7.Operating system: Any Unix-like system; developed on Ubuntu 14.04 (using Linux kernel 3.16).RAM: Problem dependent; specifically on the resolution of the FEM mesh and the number of modes included. The given example uses approximately 100 MB.Classification: 10.External routines: Required are the following free libraries LAPACK and BLAS [2], UMFPACK [3]. Optionally exploits additional commercial software packages: Intel MKL [4], Gmsh [5].Nature of problem: Time-harmonic electrodynamics in layered media.Solution method: Finite element method and the scattering matrix method.Running time: Problem dependent (typically about 3 s per wavelength including plane wave orders ≤3).

Numerical Verification of Transition’s Energies of Excitons in Quantum Well of ZnO with the Finite Difference Method

This paper shows that the experimental results of quantum well energy transitions can be found numerically. The cases of several ZnO-ZnMgO wells are considered and their excitonic transition energies were calculated using the finite difference method. In that way, the one-dimensional Schrodinger equation has been solved by using the BLAS and LAPACK libraries. The numerical results are in good agreement with the experimental ones.