Generated Error in Rotational Tridiagonalization

The tridiagonalization of a symmetric matrix by a sequence of plane rotations constitutes the first of the two major steps in the computation of the proper values by one of the methods now in common use. This method was first programmed by J. W. Givens (1954) for the ORACLE, and his description includes a detailed analysis of the errors generated in the computation. The analysis of the errors in the tridiagonalization, however, is extremely laborious. I t is the purpose of the present note to outline an approach which, though yielding less sharp bounds, seems to be considerably simpler, and possibly capable of extension to other routines. The actual computation of the proper values themselves is carried out by an essentially independent routine, and the errors generated in this phase of the computation will not be considered here. Before the analysis can be presented it is necessary to review the nature of the computation. The first step is to rotate in the (2,3)-coordinate plane so tha t if A = A T = (ol~3) is the original matrix, and A' = (a'~j) the rotated matrix, ! ! then a13 = a~l = O. Tha t this is always possible will become clear presently. All subsequent steps can be reduced to a repetition of this following possible permutation. In fact, if the third column and the third row are permuted to the last place, a repetition annihilates a second element without affecting the zero already produced, and after n -2 such steps at most the first two elements will be non-null in the first row (column). Thereafter one operates on the principal minor which remains after the deletion of the first row and first column. Consider the matrix in partitioned form: