Publication Date

11-23-1967

Abstract

INTRODUCTION

This paper will amount to an inquiry into the mathematical structure and properties of a matrix of approximate integration formulas due to Werner Romberg 1 which have found considerable favor in numerical analysis during the past decade. Like other devices for approximate integration the Romberg method has its origin in the exhaustion techniques developed by the Greeks for determining the areas of figures enclosed by lines in the plane. It is particularly closely associated with the scheme employed by Archimedes2 (circa 250 B.C.) for approximating the value of [3.14]. This involves a doubling process which is essentially the basis for the Romberg method. Archimedes began with a circle of radius one within which was inscribed some regular polygon whose area A00 he could readily compute. After doubling the number of sides he was able to compute a new area A01, then a third area A02 after a second doubling and so on to A03, .... ,A0n, ect. It is geometrically clear that such a sequence should converge to [3.14] from below. As an example, if Aon corresponds to a 24-sided polygon it will he found that

Aon= 3.(32,(,,Zq) Ao,

the latter correct for [3.14] to 3 figures.

Much later, in about 1654 in fact, Christian Huygens rather ingeneously deduced an improved version of the Archimedes sequence. This involves combing two successive Archimedes terms with the weiphts,-1/3 and 4/3, to produce a new sequence

Ain = 4 Ao,(n+1) -Aon

3

which Huygens showed would converge to [3.14] much more rapidly. Thus

for term A1n, corresponding to a 48-sided polygon it will be found that

A in = 3. 141590

a value correct for [3.14] to 6 figures!

Huygens' numerical innovation lay practically unnoticed for

almost three centuries (most likely because of the emphasis on

methods of infinitesimal calculus during this period) when in

1936 K. Komqerellˆ3 undertook to extend it by constructing additional

sequences, the first of which, A2n, combines two successive weighted

Huygens' terms, the second of which, A2n, combines two successive weighted terms of A2n, etc. The general prescription Kommerellused for the kth sequence starting Kˆth, sequence, starting with k=1 for the case of Huygens, was

ARN= 4ˆR (R-1), (n+1)- A (R-1), n

4ˆR-1

This scheme, of course, produces as many sequences as desired each one of which, as it turns out, converges to [3.14] faster than any of its predecessors. 4 As evidence of the speed of convergence, the first term in the highest sequence corresponding to a 48-sided polygon gives [3.14] accurate to 9 figures.

It remained for Romberg in 1955 to obtain the notion of

Applying the gist of Kommerell’s results to the problem of finding the area under the graphs of function in the problem of approximate integration of such functions. In so doing he provided a rationale for the choice of the weighting factors and developed the matrix of integration formulas that are to be the subject of this paper.

It was while studying a discussion of the Romberg method in

a textbook by E. Stiefelˆ5 that the writer’s interest was aroused.

In particular there occurred in Stiefel’s development some unusual "bonus" properties in which formulas constructed to be exact for polynomials of some degree n continually turned out to hold as well for degree (n+1). These results, which were merely pointed out by Stiefel with little comment, motivated the writer to find their explanations. Doing so led to a fairly complete study of the Romberg method as a whole which culminated in the writing of this paper.

The primary purpose of the paper will be simply to elaborate on the structure of the Romberg method. This will be accomplished by developing the matrix of formulas in considerable detail, while emphasizing completeness in the presentation and analysis of all basic ideas essential to understand the structure. In the process, explanations will be provided for the aforementioned "bonus" properties as well as others. A secondary purpose, which in many ways overlaps the primary one, will he to evaluate the effectiveness of the Romberg matrix as a numerical integration tool. To this end some theorems on convergence obtained from the liter­ature will be presented and discussed (without proof), and a generalization will be used in the procedure for formula construction which, together with some computations, will produce graphical demonstrations of the capabilities of the Romberg method.

Degree Name

Mathematics

Level of Degree

Masters

Department Name

Mathematics & Statistics

First Committee Member (Chair)

James V. Lewis

Second Committee Member

George Milton Wing

Third Committee Member

Julius Rubin Blum

Document Type

Thesis

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