A new proof of the global convergence for QL method with Wilkinson Shift
Jiang Erxiong · Journal of Natural Science of Heilongjiang University · 2004
In many practice problems such as to know the resonnance frequency of a structure and to know the critical value for the stability of a dynamical system,often need to compute the eigenvalues of a symmetric matrix. The chief method to compute the eigenvalues of a symmetric matrix is ?rst to transformate the matrix to a symmetric tridiagonal matrix similarly and orthogonally, then to use the QR(QL) method with shift to the symmetric tridiagonal matrix. For any unreducible symmetric tridiagonal, it is always convergent when use QR(QL) Algorithm with Wilkinson shift. This is a basic theorem of QR(QL) Algorithm theory. The ?rst proof of this theorem was given by J.H. Wilkinson at 1968. The proof is very complicated [7]. 1978, W.Ho?man and B.N.Parlett gave an other proof. It is a very nice proof [1]. But it is not so simple too. A new proof of this theorem is given, simply and elementarily. The new proof satis?es to write in textbooks.