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

Author: GYT

SRC/cggglm.f

@@ -61,6 +61,16 @@
 *>                  x
 *>
 *> where inv(B) denotes the inverse of B.
+*>
+*> 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 QR
+*> factorization of the pair (A, B) 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:
@@ -159,12 +169,12 @@
 *>          = 0:  successful exit.
 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
 *>          = 1:  the upper triangular factor R associated with A in the
-*>                generalized QR factorization of the pair (A, B) is
+*>                generalized QR factorization of the pair (A, B) is exactly
 *>                singular, so that rank(A) < M; the least squares
 *>                solution could not be computed.
 *>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
 *>                factor T associated with B in the generalized QR
-*>                factorization of the pair (A, B) is singular, so that
+*>                factorization of the pair (A, B) is exactly singular, so that
 *>                rank( A B ) < N; the least squares solution could not
 *>                be computed.
 *> \endverbatim

SRC/dggglm.f

@@ -61,6 +61,16 @@
 *>                  x
 *>
 *> where inv(B) denotes the inverse of B.
+*>
+*> 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 QR
+*> factorization of the pair (A, B) 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:
@@ -159,12 +169,12 @@
 *>          = 0:  successful exit.
 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
 *>          = 1:  the upper triangular factor R associated with A in the
-*>                generalized QR factorization of the pair (A, B) is
+*>                generalized QR factorization of the pair (A, B) is exactly
 *>                singular, so that rank(A) < M; the least squares
 *>                solution could not be computed.
 *>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
 *>                factor T associated with B in the generalized QR
-*>                factorization of the pair (A, B) is singular, so that
+*>                factorization of the pair (A, B) is exactly singular, so that
 *>                rank( A B ) < N; the least squares solution could not
 *>                be computed.
 *> \endverbatim

SRC/sggglm.f

@@ -61,6 +61,16 @@
 *>                  x
 *>
 *> where inv(B) denotes the inverse of B.
+*>
+*> 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 QR
+*> factorization of the pair (A, B) 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:
@@ -159,12 +169,12 @@
 *>          = 0:  successful exit.
 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
 *>          = 1:  the upper triangular factor R associated with A in the
-*>                generalized QR factorization of the pair (A, B) is
+*>                generalized QR factorization of the pair (A, B) is exactly
 *>                singular, so that rank(A) < M; the least squares
 *>                solution could not be computed.
 *>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
 *>                factor T associated with B in the generalized QR
-*>                factorization of the pair (A, B) is singular, so that
+*>                factorization of the pair (A, B) is exactly singular, so that
 *>                rank( A B ) < N; the least squares solution could not
 *>                be computed.
 *> \endverbatim

SRC/zggglm.f

@@ -61,6 +61,16 @@
 *>                  x
 *>
 *> where inv(B) denotes the inverse of B.
+*>
+*> 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 QR
+*> factorization of the pair (A, B) 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:
@@ -159,12 +169,12 @@
 *>          = 0:  successful exit.
 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
 *>          = 1:  the upper triangular factor R associated with A in the
-*>                generalized QR factorization of the pair (A, B) is
+*>                generalized QR factorization of the pair (A, B) is exactly
 *>                singular, so that rank(A) < M; the least squares
 *>                solution could not be computed.
 *>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
 *>                factor T associated with B in the generalized QR
-*>                factorization of the pair (A, B) is singular, so that
+*>                factorization of the pair (A, B) is exactly singular, so that
 *>                rank( A B ) < N; the least squares solution could not
 *>                be computed.
 *> \endverbatim