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## Über dieses Buch

A. Ghizzetti: a) Lezioni sui procedimenti di quasilinearizzazione e applicazioni. b) Nozioni fondamentali sulle equazioni alle differenze e sulle frazioni continue.- P. Wynn: Four lectures on the numerical application of continued fractions.- W. Gautschi: Strength and weakness of three-term recurrence relation.- F.L. Bauer: Use of continued fractions and algorithms related to them.

## Inhaltsverzeichnis

### Lezioni Sui Procedimenti di Quasilinearizzazione

Queste lezioni dovevano essere tenute da R.Bellman. Poichè non gli è stato possibile partecipare personalmente al corso, mi ha pregato di sostituirlo, tenendo conto di un manoscritto da Lui redatto in collaborazione con R.Kalaba. Gli argomenti contenuti in tale manoscritto sono all'incirca quelli esposti in un precedente lavoro di R.Kalaba [6].

Ho scelto fra tali argomenti quelli che mi son sembrati più significativi e ne ho rifatta l'esposizione, seguendo un ordine inverso a quello adottato dai predetti Autori. Precisamente, dopo un primo § contenente nozioni generali sulla quasilinearizzazione, espongo nel § 2 un'applicazione ad un tipo abbastanza generale di problemi sulle equazioni differenziali, ordinarie o a derivate parziali. I casi particolari (premessi da Bellman e Kalaba) sono esposti successivamente nei § 3, 4, 5, 6 ; ciò mi ha consentito di evitare la.ripetizione di dimostrazioni del medesimo tipo. Ho introdotto anche alcune modificazioni di forma per mettere bene in evidenza le ipotesi essenziali su cui sono fondate le applicazioni considerate.

Aldo Ghizzetti

### Four Lectures on the Numerical Application of Continued Fractions

Those of you who have ever attempted to read the classical expositions of the theory of continued fractions will realise how difficult it is to extract those parts of the theory which are primarily of practical interest. As you know we shall be concerned during the course of these lectures with the numerical application of continued fractions. We shall be interested that is to say not primarily in continued fraction in themselves but in their exploitation. It is this consideration which determines the scope of these lectures.

Obviously it is not possible during this short course of lectures to deal with the subject as completely as is desirable and in a manner which answers fully to the finest pedagogic precepts. In these lectures the material is presented in such a manner as to facilitate the continuity of the exposition : the formal properties of continued fractions are described at the most convenient point and the more rigorous discussion of convergence behaviour takes place when a number of formal expansions have already been made available to illustrate it.

P. Wynn

### Strength and Weakness of Three-Term Recurrence Relations

The expression “three-term recurrence relation,” as used here, is meant to be synonymous with “second-order linear difference equation.” We are thus considering relations of the form

(0.1)

$$y_{n + 1} + a_n y_n + b_n y_{n-1} = 0$$

We assume that a

n

and b

n

are given real or complex-valued functions, defined on the set of positive integers, and regard y

n

as the unknown function, with its domain being the set of non-negative integers. Relation (0.1) is required, then, to hold for n = 1,2,3,…

(1′)

Walter Gautschi

### Use of Continued Fractions and Algorithms Related to Them

A. In these lectures, we are dealing with the explicite or implicite use of continued fractions for the numerical solution of a number of problems. This means, that we are dealing with algorithms related to continued fractions. Since continued fractions have a profound connection with a certain class of (infinite) matrices, and since in our algorithms necessarily the approximants of continued fractions, which correspond to finite segments of these matrices play a role, there is good reason to base our lectures on the matrix theory of continued fractions. In Part I, we give an introduction to concepts and notations, and derive the principal algorithms, which we subsume under the name rhombus algorithms.

F. L. Bauer
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