On the Aitken–Hotelling–Duncan Formula: Brief History, Direct Derivations, and Effective Conditioning

In the first part of this paper, we examine historical documents related to the well-known Sherman–Morrison–Woodbury (SMW) formula and suggest that Aitken–Hotelling–Duncan (AHD) is an appropriate name for the same formula. In the second part, we present two elementary direct derivations of the formula, including a slow method that utilizes a QR trick and a fast method that makes the common practice of depending on being given an inverse formula for verification unnecessary. In the third part, we develop significant improvements over the classic conditioning bounds of Yip. Numerical experiments show that our bounds can be several orders of magnitude sharper than Yip’s bounds. Yip’s conditioning bounds suggest that numerical instability of the AHD formula may be common. However, for an important formula that has found and continues to find diverse applications, its numerical stability in general is not expected to be pessimistic. Our improved bounds shed a new light on the generally perceived tricky stability issue related to the AHD formula and provide clearer theoretical support for its stable performance observed by different researchers in diverse applications.