Some Algorithms Providing Rigorous Bounds for the Eigenvalue of a Matrix

Three algorithms providing rigourous bounds for the eigenvalues of a real matrix are presented. The first is an implementation of the bisection algorithm for a symmetric tridiagonal matrix using IEEE floating-point arithmetic. The two others use interval arithmetic with directed rounding and are deduced from the Jacobi method for a symmetric matrix and the Jacobi-like method of Eberlein for an unsymmetric matrix. 1 Bisection Algorithm for a Symmetric Tridiagonal Matrix Let A be a symmetric tridiagonal matrix of order n: A = An =  a1 b2 b2 a2 . . . . . . . . . bn bn an  Set b1 = 0, and suppose bk 6= 0, k = 2, . . . , n. The bisection method is based on the fact that the sequence dk(x) of principal minors of A − xI is a Sturm sequence: dk(x) = det(Ak − xIk), k = 1, . . . , n, d0 = 1 In floating point arithmetic, as pointed out in [Barth, Martin, Wilkinson 1971], the direct use of this sequence is quite impossible: even for small n underflow and overflow are unavoidable. So they consider (the hypothesis bk 6= 0, k = 2, . . . , n can then be removed): pk = dk dk−1 , k = 1, . . . , n This new sequence satisfies the recurrence relations: (Sx) pk :=  ak − x if bk = 0 or pk−1 = −∞ −∞ if pk−1 = 0 ak − x− b 2 k pk−1 otherwise for k = 1, . . . , n Journal of Universal Computer Science, vol. 1, no. 7 (1995), 548-559 submitted: 15/12/94, accepted: 26/6/95, appeared: 28/7/95Springer Pub. Co.