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Further Details: Error Bounds for Linear Least Squares Problems

The conventional error analysis of linear least squares problems goes as follows. As above, let be the solution to minimizing computed by LAPACK using one of the least squares drivers xGELS, xGELSX, xGELSY, xGELSS or xGELSD (see subsection 2.3.2). We discuss the most common case, where A is overdetermined (i.e., has more rows than columns) and has full rank [16,25,55,67]:

> The computed solution has a small normwise backward error. In other words minimizes , where E and f satisfy

> and p(n) is a modestly growing function of n. We take p(n)=1 in the code fragments above. Let (approximated by 1/`RCOND` in the above code fragments), (= `RNORM` above), and (`SINT = RNORM / BNORM` above). Here, is the acute angle between the vectors and b. Then when is small, the error is bounded by

> where = `COST` and = `TANT` in the code fragments above.

We avoid overflow by making sure `RCOND` and `COST` are both at least `EPSMCH`, and by handling the case of a zero `B` matrix separately (`BNORM = 0`).

may be computed directly from the singular values of A returned by xGELSS or xGELSD (as in the code fragment) or by xGESVD or xGESDD. It may also be approximated by using xTRCON following calls to xGELS, xGELSX or xGELSY. xTRCON estimates or instead of , but these can differ from by at most a factor of n.

If A is rank-deficient, xGELSS, xGELSD, xGELSY and xGELSX can be used to regularize the problem by treating all singular values less than a user-specified threshold ( ) as exactly zero. The number of singular values treated as nonzero is returned in `RANK`. See [16,55,67] for error bounds in this case, as well as [28] for the underdetermined case. The ability to deal with rank-deficient matrices is the principal attraction of these four drivers, which are more expensive than the simple driver xGELS.

The solution of the overdetermined, full-rank problem may also be characterized as the solution of the linear system of equations

By solving this linear system using xyyRFS or xyySVX (see section 4.4) componentwise error bounds can also be obtained [7].

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