A proof of convergence for the tridiagonal QL algorithm in floating-point arithmetic
James G. Sanderson · UNM’s Digital Repository (University of New Mexico) · 1976
Numerous routines are available to find the eigenvalues of a real symmetric tridiagonal matrix. Since it is known to converge in exact arithmetic, the tridiagonal QL algorithm with origin shift is widely used. Here we analyze the algorithm in floating-point arithmetic. This analysis suggests two modifications to the EISPACK implementation TQLl that enable one to prove correctness and hence convergence of the routine. Also, it is known that the implicit and explicit versions of the QL algorithm produce the same results in exact arithmetic. A counter-example to the floating-point analog of this theorem is presented.