A null-space approach for large-scale symmetric saddle point systems with a small and non zero (2, 2) block

Null-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.

Development of Data Sets for the Validation of Analytical Instrumentation

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.