  The modified Gram-Schmidt (MGS) orthogonalization is one of the most
well-used algorithms for computing the thin QR factorization. MGS can be
straightforwardly extended to a non-standard inner product with respect to a
symmetric positive definite matrix $A$. For the thin QR factorization of an $m
\times n$ matrix with the non-standard inner product, a naive implementation of
MGS requires $2n$ matrix-vector multiplications (MV) with respect to $A$. In
this paper, we propose $n$-MV implementations: a high accuracy (HA) type and a
high performance (HP) type, of MGS. We also provide error bounds of the HA-type
implementation. Numerical experiments and analysis indicate that the proposed
implementations have competitive advantages over the naive implementation in
terms of both computational cost and accuracy.
