Sparse Sets Of Linear Equations Research Articles

A Feasible Delay Margin Sensitivity Analysis Method

This letter proposes a method for calculating the sensitivity of delay margin with respect to system parameters. According to the property that the delay margin is achieved when the rightmost eigenvalues of the close-loop system locate exactly on the imaginary axis, a characteristic equation with inclusion of the delay margin is established. By differentiating the characteristic equation and further introducing normalization condition for the eigenvector, the sensitivity of the delay margin with respect to any given parameters can be obtained by solving a set of sparse linear equations. Due to the general differential and algebraic equation used in the derivation, the proposed delay margin sensitivity analysis method applies to various power system models and control problems.

Improved Self-Adaptive Glowworm Swarm Optimization Algorithm

The sparse sets of linear equations are produced in electrical surveying, how to raise the efficiency of the solution of equations is the key to Object-probed. Glowworm swarm optimization algorithm (GSO) algorithm put forward to solve linear equations.To overcome slow convergence and lower accuracy solution, independent movement and self-adaptive step was proposed to improve the GSO (IGSO). The experimental results prove that, IGSO has a better performance than GSO.

Improved Self-Adaptive Glowworm Swarm Optimization Algorithm

The sparse sets of linear equations are produced in electrical surveying, how to raise the efficiency of the solution of equations is the key to Object-probed. Glowworm swarm optimization algorithm (GSO) algorithm put forward to solve linear equations.To overcome slow convergence and lower accuracy solution, independent movement and self-adaptive step was proposed to improve the GSO (IGSO). The experimental results prove that, IGSO has a better performance than GSO.

Computation of Intermodulation Distortion in RF Circuits Using Single-Tone Moments Analysis

Obtaining the value of the third order intercept point using traditional simulation techniques typically requires a nonlinear steady state analysis with multitone inputs, which is very computationally expensive. In this paper, a new method is presented for the computation of the third order intercept point. Using the proposed approach, the necessary Volterra kernels are computed directly from the harmonic balance equations. The only computation cost is that of solving a set of sparse linear equations. Furthermore, only one input tone is required in this case, which greatly reduces the size of the equations and thus the computation cost.

Analysis and comparison of two general sparse solvers for distributed memory computers

This paper provides a comprehensive study and comparison of two state-of-the-art direct solvers for large sparse sets of linear equations on large-scale distributed-memory computers. One is a multifrontal solver called MUMPS, the other is a supernodal solver called superLU. We describe the main algorithmic features of the two solvers and compare their performance characteristics with respect to uniprocessor speed, interprocessor communication, and memory requirements. For both solvers, preorderings for numerical stability and sparsity play an important role in achieving high parallel efficiency. We analyse the results with various ordering algorithms. Our performance analysis is based on data obtained from runs on a 512-processor Cray T3E using a set of matrices from real applications. We also use regular 3D grid problems to study the scalability of the two solvers.

Hierarchical bases and the finite element method

In this work we present a brief introduction to hierarchical bases, and the important part they play in contemporary finite element calculations. In particular, we examine their role ina posteriorierror estimation, and in the formulation of iterative methods for solving the large sparse sets of linear equations arising from finite element discretization.

A W-matrix based fast decoupled load flow for contingency studies on vector computers

The authors deal with an application of the inverse factors method (W-matrix method) to a fast decoupled load flow procedure for steady-state contingency analysis. The W-matrix method, originally developed for the solution of sparse sets of linear equations on multiple instruction/multiple data (MIMD) computers, can also be made effective for vector computers. The recurrence problem is overcome by reordering the addition operations required in a forward and backward solution. Matrix partitioning is employed to find the best tradeoff between the number of fill-ins added to the W matrix and the increased efficiency of vector operations achieved through a reduced number of partitions. The effect of different bus ordering algorithms on the partition number is also considered. Operation reordering is employed to make the bus power computation phase very fast in comparison to traditional bus-wise calculation. Tests were performed on the IEEE 118 bus system, some different configurations of the Italian EHV system and a European equivalent network with up to about 700 buses using a 4-CPU CRAY Y-MP8/432 and a 4-CPU Alliant FX/80 computer.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A fast LU update for linear programming

This paper discusses sparse matrix kernels of simplex-based linear programming software. State-of-the-art implementations of the simplex method maintain an LU factorization of the basis matrix which is updated at each iteration. The LU factorization is used to solve two sparse sets of linear equations at each iteration. We present new implementation techniques for a modified Forrest-Tomlin LU update which reduce the time complexity of the update and the solution of the associated sparse linear systems. We present numerical results on Netlib and other real-life LP models.

Efficient factorization and solution algorithms on vector computers

Soon after their appearance, large vector computers were tested for the solution of power system problems such as security analysis and transient stability. The first results were discouraging, mainly because the consolidated sparsity exploiting algorithms performed poorly on vector computers, which are designed to work at their peak performance when operating with long full vectors. In this paper, the operations needed to perform factorization and solution of sparse sets of linear equations are rearranged in the form of long Fortran DO loops. The main result achieved by the proposed method is that faster factorization and solution algorithms are obtained by changing only the order of operations, without altering the operation count of the standard sparsity based algorithm. Tests on CRAY Y-MP8/432 and on IBM 3090/200S VF vector computers demonstrate the effectiveness of the proposed approach in comparison with the standard factorization and solution algorithms.

Use of the ${\text{P}}^4 $ and ${\text{P}}^5 $ Algorithms for In-Core Factorization of Sparse Matrices

Variants of the ${\text{P}}^4 $ algorithm of Hellerman and Rarick and the ${\text{P}}^5 $ algorithm of Erisman, Grimes, Lewis, and Poole, used for generating a bordered block triangular form for the in-core solution of sparse sets of linear equations, are considered. A particular concern is with maintaining numerical stability. Methods for ensuring stability and the extra cost that they entail are discussed. Different factorization schemes are also examined. The uses of matrix modification and iterative refinement are considered, and the best variant is compared with an established code for the solution of unsymmetric sparse sets of linear equations. The established code is usually found to be the most effective method.

APPLICATION OF GRAPH THEORY TO THE ORDERING OF LARGE SURVEY NETWORKS

AbstractMany researchers in surveying, geodesy and photogrammetry have a need for an efficient method of solving large, sparse sets of linear equations. Over the past decade or so, much research on this problem has been reported in the applied mathematics literature, but little of this material has found its way into the surveying literature. This paper surveys some modern algorithms, based on graph theory, for reordering systems of linear equations so as to reduce the bandwidth or the profile of the coefficient matrix, in order to improve the efficiency of solutions based on treating the coefficient matrix as having a banded, or a low profile, structure. Block elimination methods, based on partitioning the reordered coefficient matrix, and suited to the factorisation of a symmetric matrix such as those encountered in many surveying applications, are also briefly considered.

Vectorization of a Multiprocessor Multifrontal Code

We describe design changes that enhance the vectoriza tion of a multiprocessor version of a multifrontal code for the direct solution of large sparse sets of linear equations. These changes employ techniques used with success in full Gaussian elimination and are based on the use of matrix-vector and matrix-matrix kernels as implemented in the Level 2 and Level 3 BLAS. We illus trate the performance of the improved code by runs on the IBM 3090/VF, the ETA-10P, and the CRAY-2. Al though our experiments are principally on a single pro cessor of these machines, we briefly consider the influ ence of multiprocessing. Speedup factors of more than 11 are obtained, and the modified code performs at over 200 MFLOPS on standard structures problems on one processor of the CRAY-2.

Parallel implementation of sparse LU decomposition for chemical engineering applications

Parallel, shared-memory computer architectures can provide enormous computational resources for numerical simulations in many chemical engineering applications. The efficient solution of large, sparse sets of linear equations is often a critical requirement in these applications; thus the parallel implementation of a sparse linear solver is crucial. We examine methods for the parallel LU decomposition of sparse matrices arising from 3-D grid graphs (typically associated with finite difference discretizations in three dimensions). Parallel performance of the decomposition is significantly enhanced by the use of a new diagonal variant of nested dissection, in conjunction with a parallel dense solver at certain stages. The overall parallel efficiency exceeds that of the parallel dense solver. We also consider the extension of such methods to the less structured graphs that arise from sparse linear systems in equation-based process flowsheeting. We use performance results from the 3-D grid graphs to make inferences regarding generalizations of these methods of flowsheeting applications.

High accuracy ion optics computing

Computer simulation of focused ion beams for surface analysis of materials by SIMS, or for microfabrication by ion beam lithography plays an important role in the design of low energy ion beam transport and optical systems. Many computer packages currently available, are limited in their applications, being inaccurate or inappropriate for a number of practical purposes. This work describes an efficient and accurate computer programme which has been developed and tested for use on medium sized machines. The programme is written in Algol 68 and models the behaviour of a beam of charged particles through an electrostatic system. A variable grid finite difference method is used with a unique data structure, to calculate the eletric potential in an axially symmetric region, for arbitrary shaped boundaries. Emphasis has been placed upon finding an economic method of solving the resulting set of sparse linear equations in the calculation of the electric field and several of these are described. Applications include individual ion lenses, extraction optics for ions in surface analytical instruments and the design of columns for ion beam lithography. Computational results have been compared with analytical calculations and with some data obtained from individual einzel lenses by Harting and Read ( Electrostatic Lenses, Elsevier, Amsterdam ( 1976)).

Parallel implementation of multifrontal schemes

We consider the direct solution of large sparse sets of linear equations in an MIMD environment. We base our algorithm on a multifrontal approach and show how to distribute tasks among processors according to an elimination free that can be automatically generated from any pivot strategy. We study organizational aspects of the scheme for shared memory multiprocessor configurations and consider implications for multiprocessors with local memories and a communication network.

A CONSIDERATION ON NUMERICAL ERROR ACCORDING TO ELIMINATION METHODS FOR LINEAR ALGEBRAIC EQUATIONS

The application of the finite element method to structural analysis generally requires to solve a large and sparse set of linear equations, but it is well known that the reliability of the solution becomes low as the increase of the dimension of the linear system. The purpose of this investigation is to show how to prevent the development of the numerical error according to the elimination procedure. Through the theoretical investigation we show that the numerical error is governed by the conditioning number of the governing equation and also of matrices appearing during the elimination procedure. Successive experimental study concludes that the appropriate elimination ordering may decrease the error by an order of magnitude. Some other important informations on the elimination ordering are also given in this paper.

The Multifrontal Solution of Unsymmetric Sets of Linear Equations

We show that general sparse sets of linear equations whose pattern is symmetric (or nearly so) can be solved efficiently by a multifrontal technique. The main advantages are that the analysis time is small compared to the factorization time and that analysis can be performed in a predictable amount of storage. Additionally, there is scope for extra performance during factorization and solution on a vector or parallel machine. We show performance figures for examples run on the IBM 3081K and CRAY-1 computers.

An Efficient Parallel Algorithm for the Solution of Large Sparse Linear Matrix Equations

An algorithm for the parallel solution of large sparse sets of linear equations, given their factor matrices, is developed. It is aimed at efficient practical implementation on a processor of the multiple instruction multiple data stream (MIMD) type. The software required to implement the algorithm is described. In addition, the amount of memory necessary for data retention during execution is considered and related to that which is required on single processor systems. Hardware developed for the implementation of the algorithm is described. Bus contention for the system is outlined and shown to be insignificant. Possible bus contention problems for systems differing in the number of processors and speed of processing elements are also considered. A simulator modeling the execution of the algorithm on large systems has been implemented. The performance of the algorithm, in terms of execution speed enhancement relative to the theoretical maximum, is shown to be good.

Experience of sparse matrix codes on the CRAY-1

We consider direct methods based on Gaussian elimination for solving sparse sets of linear equations. Among conventional approaches, band and frontal methods are obviously vectorizable and general sparse methods equally do not vectorize easily since they involve indirect addressing in inner loops. We illustrate these effects with actual times from runs on the Cray at Harwell. To avoid indirect addressing we have been developing code that uses a “multi-frontal” technique. This moves the reals within storage in such a way that all operations are performed on full matrices although the pivotal strategy is minimum degree. We describe how the in-core and out-of-core versions perform on theCRAY-1.

A communication mechanism for multiprocessor simulation systems

This paper describes the organization of a multi processor system designed to analyze large systems such as power systems, global weather, and complex real-time systems. The configuration discussed handles up to 256 processors which can be inter connected using a new, high-speed communication method that takes advantage of the iterative com puting requirements of simulation problems. The communication scheme uses parallelism and address- mapping memories to achieve a very high interprocessor data-transfer rate. The solution of a large, sparse set of linear equations illustrates how the communi cation mechanism operates.