Symmetric Linear Equations Research Articles

Lopsided PMQHSS and double lopsided PMQHSS iteration methods for solving complex symmetric linear equations

Lopsided PMQHSS and double lopsided PMQHSS iteration methods for solving complex symmetric linear equations

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Structure Preprocessing Method for the System of Unclosed Linear Algebraic Equations

The complexity of open linear algebraic equations makes it difficult to obtain analytical solutions, and preprocessing techniques can be applied to coefficient matrices, which has become an effective method to accelerate the convergence of iterative methods. Therefore, it is important to preprocess the structure of open linear algebraic equations to reduce their complexity. Open linear algebraic equations can be divided into symmetric linear equations and asymmetric linear equations. The former is based on 2 × 2. The latter is preprocessed by the improved QMRGCGS method, and the applications of the two methods are analyzed, respectively. The experimental results show that when the step is 500, the pretreatment time of quasi‐minimal residual generalized conjugate gradient square 2 method is 34.23 s, that of conjugate gradient square 2 method is 35.14 s, and that of conjugate gradient square method is 45.20 s, providing a new reference method and idea for solving and preprocessing non‐closed linear algebraic equations.

Research on the Convergence of Iterative Method Using Mixed Precision Calculation Solving Complex Symmetric Linear Equation

This article investigates the complex symmetric linear equations that appear in high-frequency electromagnetic-field analysis. As an efficient linear solver, we propose a mixed-precision iterative method with double- and double-double (DD)-precision floating-point numbers and an efficient implementation of DD-precision arithmetic with fused multiply-add instructions. Using the proposed method, we successfully reduce both iteration count and calculation time compared with the conventional method. Moreover, we demonstrate the relationship between multiple-precision arithmetic and the acceleration factor of the incomplete Cholesky factorization.

Solving complex Sylvester matrix equation by accelerated double-step scale splitting (ADSS) method

Accelerated double-step scale splitting (ADSS) approach is an efficient and fast method for solving a class of large complex symmetric linear equations which has been recently presented by parameterized DSS method. In this paper, we will apply ADSS scheme for solving complex Sylvester matrix equation. It will be proved analytically that the ADSS iteration method is faster than the DSS iteration method. Moreover, we minimize the upper bound of the spectral radius of iteration matrix. Finally, some test problems will be given and results will be reported to support the theoretical claims.

A study on parallelization and optimization of iterative method using mixed precision calculation for complex symmetric linear equations

A study on parallelization and optimization of iterative method using mixed precision calculation for complex symmetric linear equations

Performance Evaluation of Multiple-precision Conjugate Orthogonal Conjugate Gradient Method in Complex Symmetric Linear Equations

Performance Evaluation of Multiple-precision Conjugate Orthogonal Conjugate Gradient Method in Complex Symmetric Linear Equations

On the Practical Implementation of ICCG Method Solving Real Symmetric Linear Equation Arising in Edge-based Finite Element Analysis

On the Practical Implementation of ICCG Method Solving Real Symmetric Linear Equation Arising in Edge-based Finite Element Analysis

DC resistivity modelling in anisotropic media with Gaussian quadrature grids

It has been shown that the spectral method (Trefethen 2000) and the spectral element method (Komatitsch and Tromp 1999) have more attractive features than the traditional finite difference (FD) and the finite element (FE) numerical methods used in resistivity modelling. The main advantages lie in the capability to simulate complex physical models and the exponential power convergence. They have been successfully applied to fluid flow dynamic modelling (Boyd 1989), seismic wave simulations (Komatitsch and Tromp 1999) and electromagnetic computations (Martinec 1999). The spectral method uses some global series of orthogonal functions to represent the unknown solution at the irregular collocation points, subject to boundary conditions. The resulting linear system matrix is full. The spectral element method combines the spectral method and the finite element method, and it possesses the main advantages of each. This includes the capability to handle various model shapes, the sparse matrix format of the FEM and the exponential power convergence of the spectral method. In a recent paper (Zhou et al., 2008) we presented the theory for a new resistivity modelling method based on Gaussian Quadrature Grids (GQG). It readily enables calculation of the electric potential in 2.5-D / 3-D heterogeneous, anisotropic models having arbitrary surface topography. The method co-operatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it transforms the 2.5-D / 3-D resistivity modelling problem into a sparse and symmetric linear equation system, which can be solved by an iterative or matrix inversion method.

2.5-D/3-D resistivity modelling in anisotropic media using Gaussian quadrature grids

SUMMARY We present a new numerical scheme for 2.5-D/3-D direct current resistivity modelling in heterogeneous, anisotropic media. This method, named the ‘Gaussian quadrature grid’ (GQG) method, cooperatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it has the main advantages of the spectral element method. The formulation shows that the GQG method is a modification of the spectral element method but does not employ the constant elements or require the mesh generator to match the Earth's surface. This makes it much easier to deal with geological models having a 2-D/3-D complex topography than using traditional numerical methods. The GQG technique can achieve a similar convergence rate to the spectral element method. We show it transforms the 2.5-D/3-D resistivity modelling problem into a sparse and symmetric linear equation system that can be solved by an iterative or matrix inversion method. Comparison with analytic solutions for homogeneous isotropic and anisotropic models shows that the error depends on the Gaussian quadrature order (abscissa number) and the subdomain size. The higher the order or the smaller the subdomain size that is employed, the more accurate are the results obtained. Several other synthetic examples, both homogeneous and inhomogeneous, incorporating sloping, undulating and severe topography, are presented and found to yield results comparable to finite element solutions involving a dense mesh.

Deflated and Restarted Symmetric Lanczos Methods for Eigenvalues and Linear Equations with Multiple Right-Hand Sides

A deflated restarted Lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Meanwhile, the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. Some reorthogonalization is necessary to control roundoff error, and several approaches are discussed. The eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right-hand sides. Experiments are given with large matrices from quantum chromodynamics that have many right-hand sides.

Eigenvalues of second-order symmetric equations on time scales with periodic and antiperiodic boundary conditions

This paper studies periodic and antiperiodic boundary value problems for second-order symmetric linear equations on time scales. By properties of eigenvalues of the Dirichlet boundary value problem and some oscillation results, existence of eigenvalues of these two different boundary value problems is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only unify the existing ones of periodic and antiperiodic boundary value problems for second-order symmetric differential and difference equations but also contain more complicated time scales.

DC resistivity modelling with a Gaussian quadrature grid

We present a new numerical scheme for 2D/3D direct current resistivity modelling. This method co-operatively combines the solution of the variational principle of the partial differential equation, Gaussian global quadrature abscissae and local cardinal functions so that it has the main advantages of the finite element method and the spectral method. The formulation shows that the method is close to the spectral element method, but it does not require the element mesh or the element integrations, and it makes it much easier to deal with geological models having a 2D/3D complex topography than the traditional numerical methods. It can achieve a similar convergence rate to the spectral element method. We show it transforms the 2D/3D resistivity modelling problem into a sparse and symmetric linear equation system, which can be solved by an iterative or matrix inversion method.

MA57---a code for the solution of sparse symmetric definite and indefinite systems

We introduce a new code for the direct solution of sparse symmetric linear equations that solves indefinite systems with 2 × 2 pivoting for stability. This code, called MA57, is in HSL 2002 and supersedes the well used HSL code MA27. We describe some of the implementation details and emphasize the novel features of MA57. These include restart facilities, matrix modification, partial solution for matrix factors, solution of multiple right-hand sides, and iterative refinement and error analysis. The code is written in Fortran 77, but there are additional facilities within a Fortran 90 implementation that include the ability to identify and change pivots. Several of these facilities have been developed particularly to support optimization applications, and we illustrate the performance of the code on problems arising therefrom.

A modified solution for large sparse symmetric linear systems in electromagnetic field analysis

A modified preconditioned conjugate gradient method for solution of large sparse symmetric linear equations is presented. Two controlling parameters are used to reduce the computation time of the incomplete decomposition and each iterative step of the conjugate gradient method. Theoretical analysis and numerical examples show that the method is problem-independent to a certain extent, and could decrease the computation time by 30%/spl sim/50% than the conventional method.

A segregated finite element solution for non-isothermal flow

A segregated finite element algorithm for the solution of the SUPG formulation of the incompressible steady-state Navier–Stokes equations for non-isothermal flow is presented in this paper. The method features equal order interpolation for all the flow variables. The SIMPLER and SIMPLEST algorithms are employed and the sets of non-symmetric linear equations are solved by means of the preconditioned conjugate gradient squared solver, whilst the preconditioned conjugate gradient solver is used to solve the sets of symmetric linear equations. Three cases are used as a basis for the study, i.e. the flow in a square cavity, the flow between parallel plates and natural convection in a square cavity. The effect that the choice of the values of the relaxation parameters has on the number of iterations which is needed to obtain a converged solution is investigated. In each case there is an optimum combination which is amongst others a function of the grid and the flow parameters. A non-optimum choice may lead either to an unnecessary number of iterations or the solution becoming unstable. From the results it is concluded that the method performs well when the optimum combination of the values for the relaxation parameters is used. However, much can still be done to improve the robustness of the method.

A three‐dimensional symmetric linear equation solver

AbstractFactorization procedures for the efficient solution of large sparse linear finite difference systems have been introduced recently. In these procedures the large sparse symmetric coefficient matrix of a certain structure is factorized exactly, yielding a direct solution method. Furthermore, approximate factorization procedures yeild implicit preconditioning iterative methods for the finite difference solution. The numerical implementation of these algorithms is presented and Fortran subroutines for the efficient solution of the resulting sparse symmetric linear system of algebraic equations are given.

Bounds for the round-off errors in the richardson second order method

The application of the Richardson second order iterative method to positive definite, symmetric linear equations is investigated. Absolute and statistical bounds for the round-off error are derived. The statistical theory agrees well with numerical experiments, until the accumulated round-off error becomes of the order of magnitude of the error in the computed solution. After this point the statistical dependence between the local round-off errors makes the observed variances larger than the theoretical variances.