Showing papers in "Numerische Mathematik in 1962"

Partitioning procedures for solving mixed-variables programming problems

Abstract: Paper presented to the 8th International Meeting of the Institute of Management Sciences, Brussels, August 23-26, 1961.

Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection

Abstract: The procedure bisect is designed to replace the procedures tridibi 1 and 2 given in [5]. All three procedures are based essentially on the following theorem.

A graph theoretic approach to matrix inversion by partitioning

Abstract: LetM be a square matrix whose entries are in some field. Our object is to find a permutation matrixP such thatPM P ?1 is completely reduced, i.e., is partitioned in block triangular form, so that all submatrices below its diagonal are 0 and all diagonal submatrices are square and irreducible. LetA be the binary (0, 1) matrix obtained fromM by preserving the 0's ofM and replacing the nonzero entries ofM by 1's. ThenA may be regarded as the adjacency matrix of a directed graphD. CallD strongly connected orstrong if any two points ofD are mutually reachable by directed paths. Astrong component ofD is a maximal strong subgraph. Thecondensation D * ofD is that digraph whose points are the strong components ofD and whose lines are induced by those ofD. By known methods, we constructD * from the digraph,D whose adjacency matrixA was obtained from the original matrixM. LetA * be the adjacency matrix ofD *. It is easy to show that there exists a permutation matrixQ such thatQA * Q ?1 is an upper triangular matrix. The determination of an appropriate permutation matrixP from this matrixQ is straightforward.

Revised report on the algorithmic languageAlgol 60

Abstract: The report gives a complete defining description of the international algorithmic languageAlgol 60. This is a language suitable for expressing a large class of numerical processes in a form sufficiently concise for direct automatic translation into the language of programmed automatic computers. The introduction contains an account of the preparatory work leading up to the final conference, where the language was defined. In addition the notions reference language, publication language, and hardware representations are explained. In the first chapter a survey of the basic constituents and features of the language is given, and the formal notation, by which the syntactic structure is defined, is explained. The second chapter lists all the basic symbols, and the syntactic units known as identifiers, numbers, and strings are defined. Further some important notions such as quantity and value are defined. The third chapter explains the rules for forming expressions and the meaning of these expressions. Three different types of expressions exist: arithmetic, Boolean (logical), and designational. The fourth chapter describes the operational units of the language, known as statements. The basic statements are: assignment statements (evaluation of a formula), go to statements (explicit break of the sequence of execution of statements), dummy statements, and procedure statements (call for execution of a closed process, defined by a procedure declaration). The formation of more complex structures, having statement character, is explained. These include: conditional statements, for statements, compound statements, and blocks. In the fifth chapter the units known as declarations, serving for defining permanent properties of the units entering into a process described in the language, are defined. The report ends with two detailed examples of the use of the language and an alphabetic index of definitions.

Comparison theorems for supremum norms

Abstract: 1 The problem of charac~erizing all supremum norms on a space of matrices or linear transformations is still unsolved. The theorems of this note are intended as a step towards solving this problem. Our most general result is Theorem 3. in which the assumption of finite dimensionality is not needed. Theorems 1 and 2 are special cases of Theorem 3. In view of the independent interest of Theorem 1, we have thought it desirable to include a separate proof of this case.