A fully stable rational version of theQR algorithm for tridiagonal matrices

Summary A rational version of theQR algorithm for symmetric tridiagonal matrices is presented. Stability is ensured by calculating the elements of the transformed matrix by various formulas, depending on the angle of rotation. Virtual origin shifts are determined from perturbation estimates for the leading 2×2 and 3×3 submatrices; the size of these shifts can optionally serve as a convergence criterion. A number of test matrices, including one with several degeneracies, were diagonalized; an average of 1.3–1.5QR iterations per eigenvalue was needed for 12-figure precision, and of 1.7–2.0 for 22-figure precision.