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where A is an m-by-n matrix, b is a given m element vector
and x is the n element solution vector.
In the most usual case,  and
 and  .
In this case the
solution to problem (3.1) is unique.
The problem is also
referred to as finding a least squares solution to an
overdetermined  system of linear equations.
.
In this case the
solution to problem (3.1) is unique.
The problem is also
referred to as finding a least squares solution to an
overdetermined  system of linear equations.
When m < n and  , there are an infinite number
 of solutions x
that exactly satisfy b-Ax=0. In this case it is often useful to find
the unique solution x that minimizes
, there are an infinite number
 of solutions x
that exactly satisfy b-Ax=0. In this case it is often useful to find
the unique solution x that minimizes  ,
and the problem
is referred to as finding a minimum norm solution  to an
underdetermined  system of linear equations.
,
and the problem
is referred to as finding a minimum norm solution  to an
underdetermined  system of linear equations.
The driver routine PxGELS 
solves problem (3.1) on the assumption that
 -- in other words, A has full rank --
finding a least squares solution of an overdetermined   system
when m > n, and a minimum norm solution of an underdetermined  system
when m < n.
PxGELS     uses a QR or LQ
factorization   of A and also allows A to be 
replaced by
 -- in other words, A has full rank --
finding a least squares solution of an overdetermined   system
when m > n, and a minimum norm solution of an underdetermined  system
when m < n.
PxGELS     uses a QR or LQ
factorization   of A and also allows A to be 
replaced by  in the statement of the problem (or by
 in the statement of the problem (or by  if A is 
complex).
 if A is 
complex).
In the general case when we may have
 -- in other words, 
A may be rank-deficient  --
we seek the minimum norm least squares solution  x
that minimizes both
 -- in other words, 
A may be rank-deficient  --
we seek the minimum norm least squares solution  x
that minimizes both  and
 and  .
.
The LLS driver routines are listed in table 3.3.
All routines allow several right-hand-side vectors b and corresponding
solutions x to be handled in a single call, storing these vectors as columns
of matrices B and X, respectively. 
Note, however, that equation 3.1 is solved for
each right-hand-side vector independently; this is not the same as
finding a matrix X that minimizes  .
.
  
Table 3.3: Driver routines for linear least squares problems
 
  
  
  
 