Stabilized hyperbolic Householder transformations

The authors introduce a modification of the hyperbolic Householder scheme that is demonstrably stable theoretically (according to an established stability criterion) and exhibits superior numerical behavior in simulations. The modified transform scheme effects downdating of Cholesky factors by applying conventional orthonormal, rather than hyperbolic, Householder transformations to the data. The latter have preferable numerical properties. However, the construction of these orthonormal operators itself requires hyperbolic computations. Thus the method is, in some sense, half hyperbolic and half orthonormal. There is no computational penalty incurred with these stabilized hyperbolic Householder transforms; they enjoy an operation count identical to that of their conventional counterparts. This approach offers roughly a twofold saving over schemes based on the multiple application of single vector downdating schemes, and it is just as stable