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Further Details: Error Bounds for Linear Equality Constrained Least Squares Problems

In this subsection, we will summarize the available error bound. The reader may also refer to [2,13,18,50] for further details.

Let be the solution computed by the driver xGGLSE (see subsection 4.6). It is normwise stable in a mixed forward/backward sense [18,13]. Specifically, , where solves , and

q(m,n,p) is a modestly growing function of m, n, and p. We take q(m,n,p) = 1 in the code fragment above. Let denote the Moore-Penrose pseudo-inverse of X. Let ( = `CNDAB` above) and ( = `CNDBA` above) where and . When is small, the error is bounded by

When B = 0 and d = 0, we essentially recover error bounds for the linear least squares (LS) problem:

where . Note that the error in the standard least squares problem provided in section 4.5.1 is

since . If one assumes that q(m,n) = p(n) = 1, then the bounds are essentially the same.

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