Add notes to *GGLSE about how only exact rank-deficiency is checked

Author: GYT

SRC/cgglse.f

@@ -52,6 +52,16 @@
 *> matrices (B, A) given by
 *>
 *>    B = (0 R)*Q,   A = Z*T*Q.
+*>
+*> Callers of this subroutine should note that the singularity/rank-deficiency checks
+*> implemented in this subroutine are rudimentary. The CTRTRS subroutine called by this
+*> subroutine only signals a failure due to singularity if the problem is exactly singular.
+*>
+*> It is conceivable for one (or more) of the factors involved in the generalized RQ
+*> factorization of the pair (B, A) to be subnormally close to singularity without this
+*> subroutine signalling an error. The solutions computed for such almost-rank-deficient
+*> problems may be less accurate due to a loss of numerical precision.
+*> 
 *> \endverbatim
 *
 *  Arguments:
@@ -153,12 +163,12 @@
 *>          = 0:  successful exit.
 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
 *>          = 1:  the upper triangular factor R associated with B in the
-*>                generalized RQ factorization of the pair (B, A) is
+*>                generalized RQ factorization of the pair (B, A) is exactly
 *>                singular, so that rank(B) < P; the least squares
 *>                solution could not be computed.
 *>          = 2:  the (N-P) by (N-P) part of the upper trapezoidal factor
 *>                T associated with A in the generalized RQ factorization
-*>                of the pair (B, A) is singular, so that
+*>                of the pair (B, A) is exactly singular, so that
 *>                rank( (A) ) < N; the least squares solution could not
 *>                    ( (B) )
 *>                be computed.

SRC/dgglse.f

@@ -52,6 +52,16 @@
 *> matrices (B, A) given by
 *>
 *>    B = (0 R)*Q,   A = Z*T*Q.
+*>
+*> Callers of this subroutine should note that the singularity/rank-deficiency checks
+*> implemented in this subroutine are rudimentary. The DTRTRS subroutine called by this
+*> subroutine only signals a failure due to singularity if the problem is exactly singular.
+*>
+*> It is conceivable for one (or more) of the factors involved in the generalized RQ
+*> factorization of the pair (B, A) to be subnormally close to singularity without this
+*> subroutine signalling an error. The solutions computed for such almost-rank-deficient
+*> problems may be less accurate due to a loss of numerical precision.
+*> 
 *> \endverbatim
 *
 *  Arguments:
@@ -153,12 +163,12 @@
 *>          = 0:  successful exit.
 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
 *>          = 1:  the upper triangular factor R associated with B in the
-*>                generalized RQ factorization of the pair (B, A) is
+*>                generalized RQ factorization of the pair (B, A) is exactly
 *>                singular, so that rank(B) < P; the least squares
 *>                solution could not be computed.
 *>          = 2:  the (N-P) by (N-P) part of the upper trapezoidal factor
 *>                T associated with A in the generalized RQ factorization
-*>                of the pair (B, A) is singular, so that
+*>                of the pair (B, A) is exactly singular, so that
 *>                rank( (A) ) < N; the least squares solution could not
 *>                    ( (B) )
 *>                be computed.

SRC/sgglse.f

@@ -52,6 +52,16 @@
 *> matrices (B, A) given by
 *>
 *>    B = (0 R)*Q,   A = Z*T*Q.
+*>
+*> Callers of this subroutine should note that the singularity/rank-deficiency checks
+*> implemented in this subroutine are rudimentary. The STRTRS subroutine called by this
+*> subroutine only signals a failure due to singularity if the problem is exactly singular.
+*>
+*> It is conceivable for one (or more) of the factors involved in the generalized RQ
+*> factorization of the pair (B, A) to be subnormally close to singularity without this
+*> subroutine signalling an error. The solutions computed for such almost-rank-deficient
+*> problems may be less accurate due to a loss of numerical precision.
+*> 
 *> \endverbatim
 *
 *  Arguments:
@@ -153,12 +163,12 @@
 *>          = 0:  successful exit.
 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
 *>          = 1:  the upper triangular factor R associated with B in the
-*>                generalized RQ factorization of the pair (B, A) is
+*>                generalized RQ factorization of the pair (B, A) is exactly
 *>                singular, so that rank(B) < P; the least squares
 *>                solution could not be computed.
 *>          = 2:  the (N-P) by (N-P) part of the upper trapezoidal factor
 *>                T associated with A in the generalized RQ factorization
-*>                of the pair (B, A) is singular, so that
+*>                of the pair (B, A) is exactly singular, so that
 *>                rank( (A) ) < N; the least squares solution could not
 *>                    ( (B) )
 *>                be computed.

SRC/zgglse.f

@@ -52,6 +52,16 @@
 *> matrices (B, A) given by
 *>
 *>    B = (0 R)*Q,   A = Z*T*Q.
+*>
+*> Callers of this subroutine should note that the singularity/rank-deficiency checks
+*> implemented in this subroutine are rudimentary. The ZTRTRS subroutine called by this
+*> subroutine only signals a failure due to singularity if the problem is exactly singular.
+*>
+*> It is conceivable for one (or more) of the factors involved in the generalized RQ
+*> factorization of the pair (B, A) to be subnormally close to singularity without this
+*> subroutine signalling an error. The solutions computed for such almost-rank-deficient
+*> problems may be less accurate due to a loss of numerical precision.
+*> 
 *> \endverbatim
 *
 *  Arguments:
@@ -153,12 +163,12 @@
 *>          = 0:  successful exit.
 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
 *>          = 1:  the upper triangular factor R associated with B in the
-*>                generalized RQ factorization of the pair (B, A) is
+*>                generalized RQ factorization of the pair (B, A) is exactly
 *>                singular, so that rank(B) < P; the least squares
 *>                solution could not be computed.
 *>          = 2:  the (N-P) by (N-P) part of the upper trapezoidal factor
 *>                T associated with A in the generalized RQ factorization
-*>                of the pair (B, A) is singular, so that
+*>                of the pair (B, A) is exactly singular, so that
 *>                rank( (A) ) < N; the least squares solution could not
 *>                    ( (B) )
 *>                be computed.