The BR Eigenvalue Algorithm

T h e B R algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. I t is a bulge-chasingalgorithm like the Q R algorithm, but , unlike the QR algorithm, it is well adapted to computing the eigenvaluesof the narrow-band, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the B R algorithm and gives numerical evidence tha t it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the QR algorithm by a factor of 30-60 in computing time and a factor of over 100 in matrix storage space.