The linear least squares problem is to find *x* that minimizes
.
We discuss error bounds for the most common case where *A* is *m*-by-*n*
with *m* > *n*, and *A* has full rank ;
this is called an *overdetermined least squares problem*
(the following code fragments deal with *m*=*n* as well).

Let be the solution computed by the driver routine
PxGELS (see section 3.2.2).
An approximate error
bound

may be computed in the following way:

EPSMCH = PSLAMCH( ICTXT, 'E' ) * Get the 2-norm of the right hand side B BNORM = PSLANGE( 'F', M, 1, B, IB, JB, DESCB, WORK ) * Solve the least squares problem; the solution X overwrites B CALL PSGELS( 'N', M, N, 1, A, IA, JA, DESCA, B, IB, JB, DESCB, $ WORK, LWORK, INFO ) IF( MIN( M, N ).GT.0 ) THEN * Get the 2-norm of the residual A*X-B RNORM = PSLANGE( 'F', M-N, 1, B, IB+N, JB, DESCB, WORK ) * Get the reciprocal condition number RCOND of A CALL PSTRCON( 'I', 'U', 'N', N, A, IA, JA, DESCA, RCOND, WORK, $ LWORK, IWORK, LIWORK, INFO ) RCOND = MAX( RCOND, EPSMCH ) IF( BNORM.GT.0.0 ) THEN SINT = RNORM / BNORM ELSE SINT = 0.0 END IF COST = MAX( SQRT( ( 1.0E0 - SINT )*( 1.0E0 + SINT ) ), EPSMCH ) TANT = SINT / COST ERRBD = EPSMCH*( 2.0E0 / ( RCOND*COST ) + TANT / RCOND**2 ) END IF

For example, if ,

then, to four decimal places,

,
,
,
, and the true error
is .

Note that in the preceding code fragment, the routine PSLANGE was
used to compute the two-norm of the right hand side matrix *B* and
the residual . This routine was chosen because the result
of the computation (BNORM or RNORM, respectively) is automatically
known on all process columns within the process grid. The routine
PSNRM2 could have also been
used to perform this calculation; however, the use of PSNRM2 in this
example would have required an additional communication broadcast,
because the resulting value of BNORM or RNORM, respectively, is known only
within the process column owning B(:,JB).

**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
ScaLAPACK using the least squares driver PxGELS
(see section 3.2.2).
We discuss the most common case, where *A* is
overdetermined
(i.e., has more rows than columns) and has full rank [71]:

The computed solution has a small normwise backward error. In other words, minimizes , whereEandfsatisfy

andp(n) is a modestly growing function ofn. We takep(n)=1 in the code fragments above. Let (approximated by 1/RCONDin the above code fragments), (=RNORMabove), and (SINT = RNORM / BNORMabove). Here, is the acute angle between the vectors andb. Then when is small, the error is bounded by

where =COSTand =TANTin 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 PxGESVD.
It may also be approximated by using PxTRCON following calls to
PxGELS. PxTRCON estimates or instead
of , but these can differ from by at most a factor of *n*.

Tue May 13 09:21:01 EDT 1997