Abstract
The accuracy of numerical results obtained by computers is hard to know mathematically. At present it is only natural to gain confidence in the results by computer runs. As a way to achieve reliability an interval method has been proposed for the eigenvalue problem. In the interval method eigenvalues are computed by solving the characteristic equation of a matrix. There are severe ristrictions on the starting intervals allowed for the method.
This paper describes an alternative way for finding the eigenvalues and eigenvectors of a given matrix with real elements. It employs Householder's transformation, the method of bisection and the QR algorithm, based on triplex arithmetic operations. The only limitation to matrices is that all the eigenvalues are to be simple. The matrices may be symmetric or nonsymmetric.
In triplex arithmetic the number is represented by describing simultaneously the approximate value and the error. In parallel to the result of operation, the rounding error produced in that operation is determined, which is summed up for each operation. When a series of operations is completed, the range of existence for the result can be determined automatically from the result of computation and the sum of errors.
Some numerical examples are given, 3×3 to 6×6 Lotkin matrices included.
