Prestructuring sparse matrices with dense rows and columns via null space methods

AbstractNull-space methods have long been used to solve large sparse n × n symmetric saddle point systems of equations in which the (2, 2) block is zero. This paper focuses on the case where the (1, 1) block is ill conditioned or rank deficient and the k × k (2, 2) block is non zero and small (k ≪ n). Additionally, the (2, 1) block may be rank deficient. Such systems arise in a range of practical applications. A novel null-space approach is proposed that transforms the system matrix into a nicer symmetric saddle point matrix of order n that has a non zero (2, 2) block of order at most 2k and, importantly, the (1, 1) block is symmetric positive definite. Success of any null-space approach depends on constructing a suitable null-space basis. We propose methods for wide matrices having far fewer rows than columns with the aim of balancing stability of the transformed saddle point matrix with preserving sparsity in the (1, 1) block. Linear least squares problems that contain a small number of dense rows are an important motivation and are used to illustrate our ideas and to explore their potential for solving large-scale systems.

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Data reconciliation and variable classification by null space methods

Measurement ◽

10.1016/j.measurement.2009.12.032 ◽

2010 ◽

Vol 43 (5) ◽

pp. 702-707 ◽

Cited By ~ 5

Author(s):

Christos L. Mitsas

Keyword(s):

Null Space ◽

Data Reconciliation ◽

Null Space Methods

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Development of Data Sets for the Validation of Analytical Instrumentation

Journal of AOAC International ◽

10.1093/jaoac/77.3.777 ◽

1994 ◽

Vol 77 (3) ◽

pp. 777-781 ◽

Cited By ~ 1

Author(s):

Stephen L R Ellison ◽

Maurice G Cox ◽

Alastatr B Forbes ◽

Bernard P Butler ◽

Simon A Hannaby ◽

...

Keyword(s):

Multivariate Regression ◽

Reference Data ◽

Dynamic Range ◽

Null Space ◽

Data Sets ◽

Statistical Parameters ◽

Incorrect Choice ◽

Wide Range ◽

Null Space Methods ◽

Numerical Precision

Abstract Analytical chemistry makes use of a wide range of basic statistical operations, including means; standard deviations; significance tests based on assumed distributions; and linear, polynomial, and multivariate regression. The effects of limited numerical precision, poor choice of algorithm, and extreme dynamic range on these common statistical operations are discussed. The effects of incorrect choice of algorithm on calculations of basic statistical parameters and calibration lines are illustrated by examples. Some approaches to validation of such software are considered. The preparation of reference data sets for testing statistical software is discussed. The use of ‘null space’ methods for producing reference data sets is described, and an example is given. These data sets have well-characterized properties and can be used to test the accuracy of basic statistical procedures. Specific properties that are controlled include the numerical precision required to represent the sets exactly and the analytically correct answers. A further property of some of the data sets under development is the predictability of the deviation from the expected results resulting from poor choice of algorithm.

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EFFICIENCY ASSESSMENT OF THE DIRECT INTEGRATION METHOD BY CENTRAL DIFFERENCES (DIMCD)

DYNA INGENIERIA E INDUSTRIA ◽

10.6036/10004 ◽

2021 ◽

Vol 96 (5) ◽

pp. 512-519

Author(s):

GORKA URKULLU MARTIN ◽

IGOR FERNANDEZ DE BUSTOS ◽

ANDER OLABARRIETA ◽

RUBEN ANSOLA

Keyword(s):

Computational Efficiency ◽

Integration Method ◽

Null Space ◽

Sparse Matrices ◽

Point Of View ◽

Direct Integration ◽

Efficiency Assessment ◽

Direct Integration Method ◽

Dense Matrices

The direct integration method by central differences (DIMCD) is an explicit method of order two for integrating the equations governing the dynamic analysis of multibody systems. So far, development has focused only on verifying the quality of the results. In this paper, it is shown that in addition to providing optimal results, it is also competitive from the point of view of computational efficiency, at least for systems with up to six bodies. For this purpose, an appropriate implementation of the method in a compiled language is presented. In turn, it is shown that the methodology is suitable for modeling in sparse matrices, although the proposed implementation is based on dense matrices. The resulting code is applied to different benchmark examples. Results from various commercial software are also included. Keywords: Computational efficiency, multibody dynamics, central differences, null space, dense matrices, quaternions

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