A note on the pivoted symmetric LR iteration

We prove that the diagonally pivoted symmetric LR algorithm on a positive definite matrix is globally convergent. The “symmetric” or “Cholesky” LR iteration is a fairly old method of eigenreduction of a positive definite Hermitian matrix H. It reads H = H0 = R∗ 0R0 H1 = R0R 0 = R ∗ 1R1 .. (1) This process is linearly convergent [6], [5]. Recently, its singular value ’implicit’ equivalent R∗ k = QkRk+1, Qk unitary, Rk upper triangular, (2) was studied in [3]. An obvious modification of the algorithm (2) includes pivoting. Thus modified, (2) reads R∗ k = QkRk+1Pk, (3) where Pk is a permutation of standard column pivoting or, equivalently, of the diagonal pivoting within the ’explicit’ algorithm (1). This means that R = Rk has the property rii ≥ √ |ril| + · · ·+ |rll|, i = 1, . . . , n, l ≥ i. (4) Although the practical use of pivoting is limited to first few steps, it is of interest to know whether the pivoted algorithm itself is globally convergent. The answer is affirmative and this will be shown in our main theorem below. The importance of pivoting was stressed in [2], Cor. 5.4, 5.5, where it was shown that the pivoted implicit Cholesky iteration computes the singular values with high relative accuracy. We produce a simple example which shows that the non-pivoted algorithm really is worse in this respect. Set R = ( 1e− 12 2e− 12 1 1 ) Here the pivoted algorithm gives the correct small singular value 7.07106781186548e− 13 as expected, while the non-pivoted algorithm gives only 7.07099414738691e − 13. (Both algorithms were implemented in MATLAB by using its routine qr.1 ) ∗Lehrgebiet Mathematische Physik, Fernuniversität Hagen, Postf. 940, 58084 Hagen Germany, email: [email protected] The genuine MATLAB routine svd was also unsatisfactory, its outputs were min(svd(R)) = 7.07012634145304e− 13 and min(svd(R′)) = 7.07106781207136e− 13. 1