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Further Details: Error Bounds for General Linear Model Problems

In this subsection, we will summarize the available error bounds. The reader may also refer to [2,13,50,80] for further details.

Let and be the solutions by the driver routine xGGGLM (see subsection 4.6). Then is normwise backward stable and is stable in a mixed forward/backward sense [13]. Specifically, we have and , where and solve , and

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 errors and are bounded by

When B = I, the GLM problem is the standard LS problem. y is the residual vector y = d - Ax, and we have

and

where and . The error bound of is the same as in the LSE problem (see section 4.6.1.1), which is essentially the same as given in section 4.5.1. The bound on the error in is the same as that provided in [55, section 5.3.7].

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