# Efficient implementations of the modified Gram–Schmidt orthogonalization with a non-standard inner product

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## Abstract

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 2*n* 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 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.

## Keywords

Modified Gram–Schmidt orthogonalization Non-standard inner product Efficient implementations Error bounds## Notes

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