No abstract

No abstract

This paper is an account of the methods that have been used with the EDSAC for the solution of algebraic equations. Three repetitive or iterative methods are examined: Bernoulli's method, the root-squaring method, and the Newton-Raphson method. Experience with the EDSAC has shown that, as in hand computing, quadratically convergent methods are to be preferred to those less rapidly convergent. In particular, the Newton-Raphson method has proved the most useful. Several examples are given in the appendix.

No abstract

The paper draws attention to a method suggested by Lanczos for arriving at the reduced characteristic function of a matrix, and shows how it may be used with an electronic computer to evaluate automatically the characteristic roots and vectors of a real symmetric matrix, no matter how degenerate.(1) INTRODUCTION The importance of the characteristic-value problem in physics, chemistry and engineering has led to the use of high-speed electronic computers for the numerical evaluation of characteristic roots and vectors of matrices of high order. Wilkinson 5 has described some of the methods used on the Ace computer for this purpose. One of the purposes of the paper is to draw attention to a method suggested by Lanczos, 3 which has several features which make it suitable for use with automatic computers. First, since all the matrix operations can be carried out in terms of column vectors rather than row and column vectors, it is easily adaptable for machines employing a "partitioned" storage system such as a magnetic drum; secondly there are no "special cases," the method being equally applicable to matrices of high dispersion (i.e. roots widely separated) and to those having several roots close together. A disadvantage of the method is the necessity for floating-point operations.In the first part of the paper the theoretical basis of the method is discussed in relation to the classical algebraic method, which has also been suggested for numerical application. 2 In the second part numerical aspects of the method are discussed with particular reference to the separation of close characteristic values. A specially constructed example of a pathological matrix, suggested by Lanczos, has been analysed on the Manchester electronic computer, the machine evaluating completely automatically all the characteristic roots to 10 significant figures. (2) THE THEORETICAL BASIS OF THE METHOD Consider an arbitrary matrix A of size n x n. The classical analysis starts by constructing from an arbitrary vector b 0 the sequence of vectors b 0 , Ab 0 , A2b 0 , . . . A»>b 0 ,where A 2 b 0 = A(Ab 0 ), etc. There can be at most n linearly independent members in this sequence, so that there exists a linear relation of the form ao b 0 + ai (Ab 0 ) + . . . + a m (A^b 0 ) = 0, m < n) . (2) The value m is called the "grade" of the vector b 0 . Among all possible vectors there will be vectors of maximum grade M. Then, putting a M = 1, eqn. (2) can be written f M (A)b 0 = + a M _ x +...+ ai A+ a o )b o = 0 (3) The polynomial f M (x) has the following properties.* (a) /M(X) is the same for all vectors t>o of maximum grade, and is called the "reduced characteristic function"; the roots of /M(X) = 0 are characteristic roots of A. (b) /M(A) -0. i-e -a matrix satisfies its own reduced characteristic equation. (When M -n this is the Cayley-Hamilton theorem.) (c) For symmetric matrices the roots of /M(X) are all distinct. Lanczos 3 has given an alternative method of arriving at the reduced characteristic equation which has both theoretical and computation...