An Augmented Stability Result for the Lanczos Hermitian Matrix Tridiagonalization Process
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
The rational approximations of the unitary groups
Quantum Information Processing
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Charles Sheffield pointed out that the modified Gram-Schmidt (MGS) orthogonalization algorithm for the QR factorization of $B\!\in\!\R^{n\times k}$ is mathematically equivalent to the QR factorization applied to the matrix $B$ augmented with a $k\times k$ matrix of zero elements on top. This is true in theory for any method of QR factorization, but for Householder's method it is true in the presence of rounding errors as well. This knowledge has been the basis for several successful but difficult rounding error analyses of algorithms which in theory produce orthogonal vectors but significantly fail to do so because of rounding errors. Here we show that the same results can be found more directly and easily without recourse to the MGS connection. It is shown that for any sequence of $k$ unit 2-norm $n$-vectors there is a special $(n\!+\!k)$-square unitary matrix which we call a unitary augmentation of these vectors and that this matrix can be used in the analyses without appealing to the MGS connection. We describe the connection of this unitary matrix to Householder matrices. The new approach is applied to an earlier analysis to illustrate both the improvement in simplicity and advantages for future analyses. Some properties of this unitary matrix are derived. The main theorem on orthogonalization is then extended to cover the case of biorthogonalization.