Webb Miller

Software for roundoff analysis, II

Many roundoff analyses of nomteratlve methods from numerical linear algebra can be performed, at least in part, using off-the-shelf software. Such software is presented here and its use is illustrated with examples. The package presented differs from Its predecessor in four important respects. First, a mmicompfler allows easy specIficatmn of the algorithm being tested. Second, the package can test the simultaneous effect of rounding error upon several values. Third, it deals with branching in numerical methods, e.g. with pivoting in Gaussian elimination. Fourth, m addition to comparing rounding error with perturbations of the problem, it can directly compare rounding errors in two competing algorithms

Algorithm 532: software for roundoff analysis [Z]

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Roundoff analyses and sparse data

SummaryTechniques are emerging which automatically determine the effects of rounding error upon numerical methods of a certain type. Meaningful testing presupposes a reasonable basis for comparison. Typically, the errors effects upon a given method are compared either (i) with the effects of perturbing the computational problem or (ii) with the effects of rounding error upon a competing method. We show that the result of a type (ii) comparison often remains valid when the two methods are adapted for “sparse” data, though the comparison might be based upon a model of error propagation which requires that special care be taken. This observation sometimes provides a rationale for preferring comparisons (ii), since the results of type (i) comparisons may well not carry over to sparse data.

CONCEPTS FROM LINEAR ALGEBRA AND ANALYSIS

This chapter reviews concepts from linear algebra and analysis. It discusses Wilkinson numbers and their computations, norm functions, linear transformations, and differentiation. The chapter describes at linear transformations and their effect on Wilkinson numbers. It also discusses the problem of computing Wilkinson numbers. The chapter introduces the notion of reverse operator norm, a type of Wilkinson number.

5 – CASE STUDIES

Publisher Summary This chapter reviews some case studies under varied themes. The themes for which case studies are provided are the Cholesky factorization, Cholesky factors after rank-one modifications, Gaussian elimination, Gaussian elimination with iterative improvement, Gauss-Jordan elimination, Householder transformations for least-squares problems, rational qr methods, downdating the qr factorization, the characteristic polynomial, the representations of symmetric matrices, and finally the variants of the Gram–Schmidt method. Automatic roundoff analysis using the software can reproduce a part of Wilkinsons error analysis of Gaussian elimination. Instability is easily diagnosed when nothing is done to control the growth of entries in the successive reduced matrices.

4 – SOFTWARE FOR ROUNDOFF ANALYSIS

Publisher Summary This chapter discusses the method of utilization of the software for roundoff analysis. It discusses measuring rounding errors in terms of the extent to which the computational problem must be perturbed to achieve the same effect, the weak composition model of rounding error, reverse condition numbers, the direct comparison of two algorithms; it also provides a users guide to the software. In backward error analysis, the rounding error is reflected back into the data (hence the name backward error analysis) and bounds are given for the difference between the original data and a set of data for which the computed values are the exact solution. The output values computed under the weak composition model will form a subset of the standard set of possible computed values. Thereafter, the reverse notion measures the degree of data perturbation needed to account for rounding of the true solution. Reverse condition numbers also give information on the stability of a composite program without reference to the weak composition model. The software package consists of a number of programs in a portable subset of Fortran (PFORT). Subsequently, the minicompiler operates on programs in a simple programming language designed for coding matrix algorithms; the minicompiler translates the program into a sequence of assignment statements. Finally, the action of the software forces certain restrictions on the size of the compiled program.

3 – DIRECTED GRAPHS

Publisher Summary This chapter discusses directed graphs. Graphs provide the proper framework for the development of many elementary results about rounding errors, the results that do not depend on idiosyncrasies of machine arithmetic or on difficult or tedious analytical inequalities. A program can be abstracted to represent a graph in which the nodes correspond to program inputs, arithmetic operations, and program outputs. A straight-line program using operations +, - , x, / gives rise to a directed acyclic graph with a node for each input value, output value, and operation; there are arcs from each arithmetic node (that is, one corresponding to an operation) to the nodes for its operands and from each output node to the operation that computes its value.

Performing armchair roundoff analyses of statistical algorithms

Numerical stability is but one of many desirable properties which should be considered when designing statistical software. However, rigorous roundoff analysis is rarely done because it seems not worth the price; the influence of rounding error is usually of secondary importance, and the analysis is thought to be beyond the reach of all but a few specialists. This note discusses the role of roundoff analysis in the design of a statistical program and shows that new techniques sometimes make assessment of the effect of rounding errors no more difficult than the verification of other program properties.