Errors of solutions of linear algebraic systems

Abstract

Error estimates are derived for the following methods: the sweepout method for tridiagonal systems, the method of square roots, the bordering method, and the method of reflection matrices. The book of S. K. Godunov is devoted to the last method; he altered the method so that it takes any matrix into a bidiagonal matrix; a considerable part of that book is devoted to the error of this alteration. In the present paper the method of reflection matrices is studied in the form in which it is expounded in the familiar book of D. K. Faddeev and V. N. Faddeeva. Recurrent formulas are obtained for the sweepout method which make it possible to successively estimate errors of the components of the solution vector. In the method of square roots the error of the solution vector is estimated by the quantity

Here ɛ_i and ɛ are small quantities; the first characterizes the accuracy of small arithmetics effects, and the second the round-off error in the reverse step. Further, A is the matrix of the system, m, is its order, f is the vector of free terms, and C and β are constants with 0 ⩽β<1. We shall not present here the rather involved estimates for the bordering method. The error of the solution vector obtained by the method of reflection matrices is estimated by the quantity (p_A is the conditioning number of the matrix A)

All estimates are obtained up to terms of higher order of smallness than ɛ and ɛ₁. The estimates themselves are related to the classification of errors of computing processes proposed by the author in recent years.