SORBDB6: improve numerical stability

* Require unit-norm vector X for otherwise the following computations
  might underflow
* Avoid over- and underflows in the computation of the Euclidean norm of
  X
* Fix the Euclidean norm computation after the second Gram-Schmidt
  iteration
* Consider round-off errors when checking for zero vectors
* Update identifiers

SRC/sorbdb6.f

@@ -41,8 +41,9 @@
 *> with respect to the columns of
 *>      Q = [ Q1 ] .
 *>          [ Q2 ]
-*> The columns of Q must be orthonormal. The orthogonalized vector will
-*> be zero if and only if it lies entirely in the range of Q.
+*> The Euclidean norm of X must be one and the columns of Q must be
+*> orthonormal. The orthogonalized vector will be zero if and only if it
+*> lies entirely in the range of Q.
 *>
 *> The projection is computed with at most two iterations of the
 *> classical Gram-Schmidt algorithm, see
@@ -172,14 +173,17 @@ SUBROUTINE SORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
 *  =====================================================================
 *
 *     .. Parameters ..
-      REAL               ALPHASQ, REALZERO
-      PARAMETER          ( ALPHASQ = 0.01E0, REALZERO = 0.0E0 )
+      REAL               ALPHA, REALZERO
+      PARAMETER          ( ALPHA = 0.1E0, REALZERO = 0.0E0 )
       REAL               NEGONE, ONE, ZERO
       PARAMETER          ( NEGONE = -1.0E0, ONE = 1.0E0, ZERO = 0.0E0 )
 *     ..
 *     .. Local Scalars ..
       INTEGER            I, IX
-      REAL               NORMSQ1, NORMSQ2, SCL1, SCL2, SSQ1, SSQ2
+      REAL               EPS, NORM, NORM_NEW, SCL, SSQ
+*     ..
+*     .. External Functions ..
+      REAL               SLAMCH
 *     ..
 *     .. External Subroutines ..
       EXTERNAL           SGEMV, SLASSQ, XERBLA
@@ -214,26 +218,17 @@ SUBROUTINE SORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
          CALL XERBLA( 'SORBDB6', -INFO )
          RETURN
       END IF
+*
+      EPS = SLAMCH( 'Precision' )
 *
 *     First, project X onto the orthogonal complement of Q's column
 *     space
 *
-      SCL1 = REALZERO
-      SSQ1 = REALZERO
-      CALL SLASSQ( M1, X1, INCX1, SCL1, SSQ1 )
-      SCL2 = REALZERO
-      SSQ2 = REALZERO
-      CALL SLASSQ( M2, X2, INCX2, SCL2, SSQ2 )
-      NORMSQ1 = SCL1**2*SSQ1 + SCL2**2*SSQ2
-      IF ( NORMSQ1 .EQ. 0 ) THEN
-         DO IX = 1, 1 + (M1-1)*INCX1, INCX1
-           X1( IX ) = ZERO
-         END DO
-         DO IX = 1, 1 + (M2-1)*INCX2, INCX2
-           X2( IX ) = ZERO
-         END DO
-        RETURN
-      END IF
+*     Christoph Conrads: In debugging mode the norm should be computed
+*     and an assertion added comparing the norm with one. Alas, Fortran
+*     never made it into 1989 when assert() was introduced into the C
+*     programming language.
+      NORM = 1.0E0
 *
       IF( M1 .EQ. 0 ) THEN
          DO I = 1, N
@@ -251,23 +246,21 @@ SUBROUTINE SORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
       CALL SGEMV( 'N', M2, N, NEGONE, Q2, LDQ2, WORK, 1, ONE, X2,
      $            INCX2 )
 *
-      SCL1 = REALZERO
-      SSQ1 = REALZERO
-      CALL SLASSQ( M1, X1, INCX1, SCL1, SSQ1 )
-      SCL2 = REALZERO
-      SSQ2 = REALZERO
-      CALL SLASSQ( M2, X2, INCX2, SCL2, SSQ2 )
-      NORMSQ2 = SCL1**2*SSQ1 + SCL2**2*SSQ2
+      SCL = REALZERO
+      SSQ = REALZERO
+      CALL SLASSQ( M1, X1, INCX1, SCL, SSQ )
+      CALL SLASSQ( M2, X2, INCX2, SCL, SSQ )
+      NORM_NEW = SCL * SQRT(SSQ)
 *
 *     If projection is sufficiently large in norm, then stop.
 *     If projection is zero, then stop.
 *     Otherwise, project again.
 *
-      IF( NORMSQ2 .GE. ALPHASQ*NORMSQ1 ) THEN
+      IF( NORM_NEW .GE. ALPHA * NORM ) THEN
          RETURN
       END IF
 *
-      IF( NORMSQ2 .EQ. ZERO ) THEN
+      IF( NORMSQ2 .LE. N * EPS * NORM ) THEN
          DO IX = 1, 1 + (M1-1)*INCX1, INCX1
            X1( IX ) = ZERO
          END DO
@@ -277,7 +270,7 @@ SUBROUTINE SORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
          RETURN
       END IF
 *
-      NORMSQ1 = NORMSQ2
+      NORM = NORM_NEW
 *
       DO I = 1, N
          WORK(I) = ZERO
@@ -299,19 +292,17 @@ SUBROUTINE SORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
       CALL SGEMV( 'N', M2, N, NEGONE, Q2, LDQ2, WORK, 1, ONE, X2,
      $            INCX2 )
 *
-      SCL1 = REALZERO
-      SSQ1 = REALZERO
-      CALL SLASSQ( M1, X1, INCX1, SCL1, SSQ1 )
-      SCL2 = REALZERO
-      SSQ2 = REALZERO
-      CALL SLASSQ( M1, X1, INCX1, SCL1, SSQ1 )
-      NORMSQ2 = SCL1**2*SSQ1 + SCL2**2*SSQ2
+      SCL = REALZERO
+      SSQ = REALZERO
+      CALL SLASSQ( M1, X1, INCX1, SCL, SSQ )
+      CALL SLASSQ( M2, X2, INCX2, SCL, SSQ )
+      NORM_NEW = SCL * SQRT(SSQ)
 *
 *     If second projection is sufficiently large in norm, then do
 *     nothing more. Alternatively, if it shrunk significantly, then
 *     truncate it to zero.
 *
-      IF( NORMSQ2 .LT. ALPHASQ*NORMSQ1 ) THEN
+      IF( NORM_NEW .LT. ALPHA * NORM ) THEN
          DO IX = 1, 1 + (M1-1)*INCX1, INCX1
             X1(IX) = ZERO
          END DO