Make vector orthogonalization more reliable (Reference-LAPACK PR 930)

lapack-netlib/SRC/clarfgp.f

@@ -97,7 +97,7 @@
 *> \author Univ. of Colorado Denver
 *> \author NAG Ltd.
 *
-*> \ingroup complexOTHERauxiliary
+*> \ingroup larfgp
 *
 *  =====================================================================
       SUBROUTINE CLARFGP( N, ALPHA, X, INCX, TAU )
@@ -122,7 +122,7 @@
 *     ..
 *     .. Local Scalars ..
       INTEGER            J, KNT
-      REAL               ALPHI, ALPHR, BETA, BIGNUM, SMLNUM, XNORM
+      REAL               ALPHI, ALPHR, BETA, BIGNUM, EPS, SMLNUM, XNORM
       COMPLEX            SAVEALPHA
 *     ..
 *     .. External Functions ..
@@ -143,11 +143,12 @@
          RETURN
       END IF
 *
+      EPS = SLAMCH( 'Precision' )
       XNORM = SCNRM2( N-1, X, INCX )
       ALPHR = REAL( ALPHA )
       ALPHI = AIMAG( ALPHA )
 *
-      IF( XNORM.EQ.ZERO ) THEN
+      IF( XNORM.LE.EPS*ABS(ALPHA) ) THEN
 *
 *        H  =  [1-alpha/abs(alpha) 0; 0 I], sign chosen so ALPHA >= 0.
 *

lapack-netlib/SRC/cunbdb5.f

@@ -148,7 +148,7 @@
 *> \author Univ. of Colorado Denver
 *> \author NAG Ltd.
 *
-*> \ingroup complexOTHERcomputational
+*> \ingroup unbdb5
 *
 *  =====================================================================
       SUBROUTINE CUNBDB5( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
@@ -169,18 +169,21 @@
 *  =====================================================================
 *
 *     .. Parameters ..
+      REAL               REALZERO
+      PARAMETER          ( REALZERO = 0.0E0 )
       COMPLEX            ONE, ZERO
       PARAMETER          ( ONE = (1.0E0,0.0E0), ZERO = (0.0E0,0.0E0) )
 *     ..
 *     .. Local Scalars ..
       INTEGER            CHILDINFO, I, J
+      REAL               EPS, NORM, SCL, SSQ
 *     ..
 *     .. External Subroutines ..
-      EXTERNAL           CUNBDB6, XERBLA
+      EXTERNAL           CLASSQ, CUNBDB6, CSCAL, XERBLA
 *     ..
 *     .. External Functions ..
-      REAL               SCNRM2
-      EXTERNAL           SCNRM2
+      REAL               SLAMCH, SCNRM2
+      EXTERNAL           SLAMCH, SCNRM2
 *     ..
 *     .. Intrinsic Function ..
       INTRINSIC          MAX
@@ -213,16 +216,33 @@
          RETURN
       END IF
 *
-*     Project X onto the orthogonal complement of Q
+      EPS = SLAMCH( 'Precision' )
 *
-      CALL CUNBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2,
-     $              WORK, LWORK, CHILDINFO )
+*     Project X onto the orthogonal complement of Q if X is nonzero
 *
-*     If the projection is nonzero, then return
+      SCL = REALZERO
+      SSQ = REALZERO
+      CALL CLASSQ( M1, X1, INCX1, SCL, SSQ )
+      CALL CLASSQ( M2, X2, INCX2, SCL, SSQ )
+      NORM = SCL * SQRT( SSQ )
 *
-      IF( SCNRM2(M1,X1,INCX1) .NE. ZERO
-     $    .OR. SCNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
-         RETURN
+      IF( NORM .GT. N * EPS ) THEN
+*        Scale vector to unit norm to avoid problems in the caller code.
+*        Computing the reciprocal is undesirable but
+*         * xLASCL cannot be used because of the vector increments and
+*         * the round-off error has a negligible impact on
+*           orthogonalization.
+         CALL CSCAL( M1, ONE / NORM, X1, INCX1 )
+         CALL CSCAL( M2, ONE / NORM, X2, INCX2 )
+         CALL CUNBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
+     $              LDQ2, WORK, LWORK, CHILDINFO )
+*
+*        If the projection is nonzero, then return
+*
+         IF( SCNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. SCNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
+            RETURN
+         END IF
       END IF
 *
 *     Project each standard basis vector e_1,...,e_M1 in turn, stopping
@@ -238,8 +258,8 @@
          END DO
          CALL CUNBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
      $                 LDQ2, WORK, LWORK, CHILDINFO )
-         IF( SCNRM2(M1,X1,INCX1) .NE. ZERO
-     $       .OR. SCNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
+         IF( SCNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. SCNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
             RETURN
          END IF
       END DO
@@ -257,8 +277,8 @@
          X2(I) = ONE
          CALL CUNBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
      $                 LDQ2, WORK, LWORK, CHILDINFO )
-         IF( SCNRM2(M1,X1,INCX1) .NE. ZERO
-     $       .OR. SCNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
+         IF( SCNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. SCNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
             RETURN
          END IF
       END DO

lapack-netlib/SRC/cunbdb6.f

@@ -41,9 +41,8 @@
 *> with respect to the columns of
 *>      Q = [ Q1 ] .
 *>          [ Q2 ]
-*> 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 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
@@ -174,7 +173,7 @@
 *
 *     .. Parameters ..
       REAL               ALPHA, REALONE, REALZERO
-      PARAMETER          ( ALPHA = 0.1E0, REALONE = 1.0E0,
+      PARAMETER          ( ALPHA = 0.83E0, REALONE = 1.0E0,
      $                     REALZERO = 0.0E0 )
       COMPLEX            NEGONE, ONE, ZERO
       PARAMETER          ( NEGONE = (-1.0E0,0.0E0), ONE = (1.0E0,0.0E0),
@@ -223,14 +222,16 @@
 *
       EPS = SLAMCH( 'Precision' )
 *
+*     Compute the Euclidean norm of X
+*
+      SCL = REALZERO
+      SSQ = REALZERO
+      CALL CLASSQ( M1, X1, INCX1, SCL, SSQ )
+      CALL CLASSQ( M2, X2, INCX2, SCL, SSQ )
+      NORM = SCL * SQRT( SSQ )
+*
 *     First, project X onto the orthogonal complement of Q's column
 *     space
-*
-*     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 = REALONE
 *
       IF( M1 .EQ. 0 ) THEN
          DO I = 1, N

lapack-netlib/SRC/dlarfgp.f

@@ -97,7 +97,7 @@
 *> \author Univ. of Colorado Denver
 *> \author NAG Ltd.
 *
-*> \ingroup doubleOTHERauxiliary
+*> \ingroup larfgp
 *
 *  =====================================================================
       SUBROUTINE DLARFGP( N, ALPHA, X, INCX, TAU )
@@ -122,7 +122,7 @@
 *     ..
 *     .. Local Scalars ..
       INTEGER            J, KNT
-      DOUBLE PRECISION   BETA, BIGNUM, SAVEALPHA, SMLNUM, XNORM
+      DOUBLE PRECISION   BETA, BIGNUM, EPS, SAVEALPHA, SMLNUM, XNORM
 *     ..
 *     .. External Functions ..
       DOUBLE PRECISION   DLAMCH, DLAPY2, DNRM2
@@ -141,11 +141,12 @@
          RETURN
       END IF
 *
+      EPS = DLAMCH( 'Precision' )
       XNORM = DNRM2( N-1, X, INCX )
 *
-      IF( XNORM.EQ.ZERO ) THEN
+      IF( XNORM.LE.EPS*ABS(ALPHA) ) THEN
 *
-*        H  =  [+/-1, 0; I], sign chosen so ALPHA >= 0
+*        H  =  [+/-1, 0; I], sign chosen so ALPHA >= 0.
 *
          IF( ALPHA.GE.ZERO ) THEN
 *           When TAU.eq.ZERO, the vector is special-cased to be

lapack-netlib/SRC/dorbdb5.f

@@ -148,7 +148,7 @@
 *> \author Univ. of Colorado Denver
 *> \author NAG Ltd.
 *
-*> \ingroup doubleOTHERcomputational
+*> \ingroup unbdb5
 *
 *  =====================================================================
       SUBROUTINE DORBDB5( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
@@ -169,18 +169,21 @@
 *  =====================================================================
 *
 *     .. Parameters ..
+      DOUBLE PRECISION   REALZERO
+      PARAMETER          ( REALZERO = 0.0D0 )
       DOUBLE PRECISION   ONE, ZERO
       PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0 )
 *     ..
 *     .. Local Scalars ..
       INTEGER            CHILDINFO, I, J
+      DOUBLE PRECISION   EPS, NORM, SCL, SSQ
 *     ..
 *     .. External Subroutines ..
-      EXTERNAL           DORBDB6, XERBLA
+      EXTERNAL           DLASSQ, DORBDB6, DSCAL, XERBLA
 *     ..
 *     .. External Functions ..
-      DOUBLE PRECISION   DNRM2
-      EXTERNAL           DNRM2
+      DOUBLE PRECISION   DLAMCH, DNRM2
+      EXTERNAL           DLAMCH, DNRM2
 *     ..
 *     .. Intrinsic Function ..
       INTRINSIC          MAX
@@ -213,16 +216,33 @@
          RETURN
       END IF
 *
-*     Project X onto the orthogonal complement of Q
+      EPS = DLAMCH( 'Precision' )
 *
-      CALL DORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2,
-     $              WORK, LWORK, CHILDINFO )
+*     Project X onto the orthogonal complement of Q if X is nonzero
 *
-*     If the projection is nonzero, then return
+      SCL = REALZERO
+      SSQ = REALZERO
+      CALL DLASSQ( M1, X1, INCX1, SCL, SSQ )
+      CALL DLASSQ( M2, X2, INCX2, SCL, SSQ )
+      NORM = SCL * SQRT( SSQ )
 *
-      IF( DNRM2(M1,X1,INCX1) .NE. ZERO
-     $    .OR. DNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
-         RETURN
+      IF( NORM .GT. N * EPS ) THEN
+*        Scale vector to unit norm to avoid problems in the caller code.
+*        Computing the reciprocal is undesirable but
+*         * xLASCL cannot be used because of the vector increments and
+*         * the round-off error has a negligible impact on
+*           orthogonalization.
+         CALL DSCAL( M1, ONE / NORM, X1, INCX1 )
+         CALL DSCAL( M2, ONE / NORM, X2, INCX2 )
+         CALL DORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
+     $              LDQ2, WORK, LWORK, CHILDINFO )
+*
+*        If the projection is nonzero, then return
+*
+         IF( DNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. DNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
+            RETURN
+         END IF
       END IF
 *
 *     Project each standard basis vector e_1,...,e_M1 in turn, stopping
@@ -238,8 +258,8 @@
          END DO
          CALL DORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
      $                 LDQ2, WORK, LWORK, CHILDINFO )
-         IF( DNRM2(M1,X1,INCX1) .NE. ZERO
-     $       .OR. DNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
+         IF( DNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. DNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
             RETURN
          END IF
       END DO
@@ -257,8 +277,8 @@
          X2(I) = ONE
          CALL DORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
      $                 LDQ2, WORK, LWORK, CHILDINFO )
-         IF( DNRM2(M1,X1,INCX1) .NE. ZERO
-     $       .OR. DNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
+         IF( DNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. DNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
             RETURN
          END IF
       END DO

lapack-netlib/SRC/dorbdb6.f

@@ -41,9 +41,8 @@
 *> with respect to the columns of
 *>      Q = [ Q1 ] .
 *>          [ Q2 ]
-*> 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 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
@@ -174,7 +173,7 @@
 *
 *     .. Parameters ..
       DOUBLE PRECISION   ALPHA, REALONE, REALZERO
-      PARAMETER          ( ALPHA = 0.1D0, REALONE = 1.0D0,
+      PARAMETER          ( ALPHA = 0.83D0, REALONE = 1.0D0,
      $                     REALZERO = 0.0D0 )
       DOUBLE PRECISION   NEGONE, ONE, ZERO
       PARAMETER          ( NEGONE = -1.0D0, ONE = 1.0D0, ZERO = 0.0D0 )
@@ -222,14 +221,16 @@
 *
       EPS = DLAMCH( 'Precision' )
 *
+*     Compute the Euclidean norm of X
+*
+      SCL = REALZERO
+      SSQ = REALZERO
+      CALL DLASSQ( M1, X1, INCX1, SCL, SSQ )
+      CALL DLASSQ( M2, X2, INCX2, SCL, SSQ )
+      NORM = SCL * SQRT( SSQ )
+*
 *     First, project X onto the orthogonal complement of Q's column
 *     space
-*
-*     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 = REALONE
 *
       IF( M1 .EQ. 0 ) THEN
          DO I = 1, N

lapack-netlib/SRC/slarfgp.f

@@ -97,7 +97,7 @@
 *> \author Univ. of Colorado Denver
 *> \author NAG Ltd.
 *
-*> \ingroup realOTHERauxiliary
+*> \ingroup larfgp
 *
 *  =====================================================================
       SUBROUTINE SLARFGP( N, ALPHA, X, INCX, TAU )
@@ -122,7 +122,7 @@
 *     ..
 *     .. Local Scalars ..
       INTEGER            J, KNT
-      REAL               BETA, BIGNUM, SAVEALPHA, SMLNUM, XNORM
+      REAL               BETA, BIGNUM, EPS, SAVEALPHA, SMLNUM, XNORM
 *     ..
 *     .. External Functions ..
       REAL               SLAMCH, SLAPY2, SNRM2
@@ -141,9 +141,10 @@
          RETURN
       END IF
 *
+      EPS = SLAMCH( 'Precision' )
       XNORM = SNRM2( N-1, X, INCX )
 *
-      IF( XNORM.EQ.ZERO ) THEN
+      IF( XNORM.LE.EPS*ABS(ALPHA) ) THEN
 *
 *        H  =  [+/-1, 0; I], sign chosen so ALPHA >= 0.
 *

lapack-netlib/SRC/sorbdb5.f

@@ -148,7 +148,7 @@
 *> \author Univ. of Colorado Denver
 *> \author NAG Ltd.
 *
-*> \ingroup realOTHERcomputational
+*> \ingroup unbdb5
 *
 *  =====================================================================
       SUBROUTINE SORBDB5( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
@@ -169,18 +169,21 @@
 *  =====================================================================
 *
 *     .. Parameters ..
+      REAL               REALZERO
+      PARAMETER          ( REALZERO = 0.0E0 )
       REAL               ONE, ZERO
       PARAMETER          ( ONE = 1.0E0, ZERO = 0.0E0 )
 *     ..
 *     .. Local Scalars ..
       INTEGER            CHILDINFO, I, J
+      REAL               EPS, NORM, SCL, SSQ
 *     ..
 *     .. External Subroutines ..
-      EXTERNAL           SORBDB6, XERBLA
+      EXTERNAL           SLASSQ, SORBDB6, SSCAL, XERBLA
 *     ..
 *     .. External Functions ..
-      REAL               SNRM2
-      EXTERNAL           SNRM2
+      REAL               SLAMCH, SNRM2
+      EXTERNAL           SLAMCH, SNRM2
 *     ..
 *     .. Intrinsic Function ..
       INTRINSIC          MAX
@@ -213,16 +216,33 @@
          RETURN
       END IF
 *
-*     Project X onto the orthogonal complement of Q
+      EPS = SLAMCH( 'Precision' )
 *
-      CALL SORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2,
-     $              WORK, LWORK, CHILDINFO )
+*     Project X onto the orthogonal complement of Q if X is nonzero
 *
-*     If the projection is nonzero, then return
+      SCL = REALZERO
+      SSQ = REALZERO
+      CALL SLASSQ( M1, X1, INCX1, SCL, SSQ )
+      CALL SLASSQ( M2, X2, INCX2, SCL, SSQ )
+      NORM = SCL * SQRT( SSQ )
 *
-      IF( SNRM2(M1,X1,INCX1) .NE. ZERO
-     $    .OR. SNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
-         RETURN
+      IF( NORM .GT. N * EPS ) THEN
+*        Scale vector to unit norm to avoid problems in the caller code.
+*        Computing the reciprocal is undesirable but
+*         * xLASCL cannot be used because of the vector increments and
+*         * the round-off error has a negligible impact on
+*           orthogonalization.
+         CALL SSCAL( M1, ONE / NORM, X1, INCX1 )
+         CALL SSCAL( M2, ONE / NORM, X2, INCX2 )
+         CALL SORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
+     $              LDQ2, WORK, LWORK, CHILDINFO )
+*
+*        If the projection is nonzero, then return
+*
+         IF( SNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. SNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
+            RETURN
+         END IF
       END IF
 *
 *     Project each standard basis vector e_1,...,e_M1 in turn, stopping
@@ -238,8 +258,8 @@
          END DO
          CALL SORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
      $                 LDQ2, WORK, LWORK, CHILDINFO )
-         IF( SNRM2(M1,X1,INCX1) .NE. ZERO
-     $       .OR. SNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
+         IF( SNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. SNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
             RETURN
          END IF
       END DO
@@ -257,8 +277,8 @@
          X2(I) = ONE
          CALL SORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
      $                 LDQ2, WORK, LWORK, CHILDINFO )
-         IF( SNRM2(M1,X1,INCX1) .NE. ZERO
-     $       .OR. SNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
+         IF( SNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. SNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
             RETURN
          END IF
       END DO

lapack-netlib/SRC/sorbdb6.f

@@ -41,9 +41,8 @@
 *> with respect to the columns of
 *>      Q = [ Q1 ] .
 *>          [ Q2 ]
-*> 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 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
@@ -174,7 +173,7 @@
 *
 *     .. Parameters ..
       REAL               ALPHA, REALONE, REALZERO
-      PARAMETER          ( ALPHA = 0.1E0, REALONE = 1.0E0,
+      PARAMETER          ( ALPHA = 0.83E0, REALONE = 1.0E0,
      $                     REALZERO = 0.0E0 )
       REAL               NEGONE, ONE, ZERO
       PARAMETER          ( NEGONE = -1.0E0, ONE = 1.0E0, ZERO = 0.0E0 )
@@ -222,14 +221,16 @@
 *
       EPS = SLAMCH( 'Precision' )
 *
+*     Compute the Euclidean norm of X
+*
+      SCL = REALZERO
+      SSQ = REALZERO
+      CALL SLASSQ( M1, X1, INCX1, SCL, SSQ )
+      CALL SLASSQ( M2, X2, INCX2, SCL, SSQ )
+      NORM = SCL * SQRT( SSQ )
+*
 *     First, project X onto the orthogonal complement of Q's column
 *     space
-*
-*     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 = REALONE
 *
       IF( M1 .EQ. 0 ) THEN
          DO I = 1, N

lapack-netlib/SRC/zlarfgp.f

@@ -97,7 +97,7 @@
 *> \author Univ. of Colorado Denver
 *> \author NAG Ltd.
 *
-*> \ingroup complex16OTHERauxiliary
+*> \ingroup larfgp
 *
 *  =====================================================================
       SUBROUTINE ZLARFGP( N, ALPHA, X, INCX, TAU )
@@ -122,7 +122,7 @@
 *     ..
 *     .. Local Scalars ..
       INTEGER            J, KNT
-      DOUBLE PRECISION   ALPHI, ALPHR, BETA, BIGNUM, SMLNUM, XNORM
+      DOUBLE PRECISION   ALPHI, ALPHR, BETA, BIGNUM, EPS, SMLNUM, XNORM
       COMPLEX*16         SAVEALPHA
 *     ..
 *     .. External Functions ..
@@ -143,11 +143,12 @@
          RETURN
       END IF
 *
+      EPS = DLAMCH( 'Precision' )
       XNORM = DZNRM2( N-1, X, INCX )
       ALPHR = DBLE( ALPHA )
       ALPHI = DIMAG( ALPHA )
 *
-      IF( XNORM.EQ.ZERO ) THEN
+      IF( XNORM.LE.EPS*ABS(ALPHA) ) THEN
 *
 *        H  =  [1-alpha/abs(alpha) 0; 0 I], sign chosen so ALPHA >= 0.
 *

lapack-netlib/SRC/zunbdb5.f

@@ -148,7 +148,7 @@
 *> \author Univ. of Colorado Denver
 *> \author NAG Ltd.
 *
-*> \ingroup complex16OTHERcomputational
+*> \ingroup unbdb5
 *
 *  =====================================================================
       SUBROUTINE ZUNBDB5( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
@@ -169,18 +169,21 @@
 *  =====================================================================
 *
 *     .. Parameters ..
+      DOUBLE PRECISION   REALZERO
+      PARAMETER          ( REALZERO = 0.0D0 )
       COMPLEX*16         ONE, ZERO
       PARAMETER          ( ONE = (1.0D0,0.0D0), ZERO = (0.0D0,0.0D0) )
 *     ..
 *     .. Local Scalars ..
       INTEGER            CHILDINFO, I, J
+      DOUBLE PRECISION   EPS, NORM, SCL, SSQ
 *     ..
 *     .. External Subroutines ..
-      EXTERNAL           ZUNBDB6, XERBLA
+      EXTERNAL           ZLASSQ, ZUNBDB6, ZSCAL, XERBLA
 *     ..
 *     .. External Functions ..
-      DOUBLE PRECISION   DZNRM2
-      EXTERNAL           DZNRM2
+      DOUBLE PRECISION   DLAMCH, DZNRM2
+      EXTERNAL           DLAMCH, DZNRM2
 *     ..
 *     .. Intrinsic Function ..
       INTRINSIC          MAX
@@ -213,16 +216,33 @@
          RETURN
       END IF
 *
-*     Project X onto the orthogonal complement of Q
+      EPS = DLAMCH( 'Precision' )
 *
-      CALL ZUNBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2,
-     $              WORK, LWORK, CHILDINFO )
+*     Project X onto the orthogonal complement of Q if X is nonzero
 *
-*     If the projection is nonzero, then return
+      SCL = REALZERO
+      SSQ = REALZERO
+      CALL ZLASSQ( M1, X1, INCX1, SCL, SSQ )
+      CALL ZLASSQ( M2, X2, INCX2, SCL, SSQ )
+      NORM = SCL * SQRT( SSQ )
 *
-      IF( DZNRM2(M1,X1,INCX1) .NE. ZERO
-     $    .OR. DZNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
-         RETURN
+      IF( NORM .GT. N * EPS ) THEN
+*        Scale vector to unit norm to avoid problems in the caller code.
+*        Computing the reciprocal is undesirable but
+*         * xLASCL cannot be used because of the vector increments and
+*         * the round-off error has a negligible impact on
+*           orthogonalization.
+         CALL ZSCAL( M1, ONE / NORM, X1, INCX1 )
+         CALL ZSCAL( M2, ONE / NORM, X2, INCX2 )
+         CALL ZUNBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
+     $              LDQ2, WORK, LWORK, CHILDINFO )
+*
+*        If the projection is nonzero, then return
+*
+         IF( DZNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. DZNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
+            RETURN
+         END IF
       END IF
 *
 *     Project each standard basis vector e_1,...,e_M1 in turn, stopping
@@ -238,8 +258,8 @@
          END DO
          CALL ZUNBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
      $                 LDQ2, WORK, LWORK, CHILDINFO )
-         IF( DZNRM2(M1,X1,INCX1) .NE. ZERO
-     $       .OR. DZNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
+         IF( DZNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. DZNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
             RETURN
          END IF
       END DO
@@ -257,8 +277,8 @@
          X2(I) = ONE
          CALL ZUNBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
      $                 LDQ2, WORK, LWORK, CHILDINFO )
-         IF( DZNRM2(M1,X1,INCX1) .NE. ZERO
-     $       .OR. DZNRM2(M2,X2,INCX2) .NE. ZERO ) THEN
+         IF( DZNRM2(M1,X1,INCX1) .NE. REALZERO
+     $       .OR. DZNRM2(M2,X2,INCX2) .NE. REALZERO ) THEN
             RETURN
          END IF
       END DO

lapack-netlib/SRC/zunbdb6.f

@@ -41,9 +41,8 @@
 *> with respect to the columns of
 *>      Q = [ Q1 ] .
 *>          [ Q2 ]
-*> 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 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
@@ -174,7 +173,7 @@
 *
 *     .. Parameters ..
       DOUBLE PRECISION   ALPHA, REALONE, REALZERO
-      PARAMETER          ( ALPHA = 0.1D0, REALONE = 1.0D0,
+      PARAMETER          ( ALPHA = 0.83D0, REALONE = 1.0D0,
      $                     REALZERO = 0.0D0 )
       COMPLEX*16         NEGONE, ONE, ZERO
       PARAMETER          ( NEGONE = (-1.0D0,0.0D0), ONE = (1.0D0,0.0D0),
@@ -223,14 +222,16 @@
 *
       EPS = DLAMCH( 'Precision' )
 *
+*     Compute the Euclidean norm of X
+*
+      SCL = REALZERO
+      SSQ = REALZERO
+      CALL ZLASSQ( M1, X1, INCX1, SCL, SSQ )
+      CALL ZLASSQ( M2, X2, INCX2, SCL, SSQ )
+      NORM = SCL * SQRT( SSQ )
+*
 *     First, project X onto the orthogonal complement of Q's column
 *     space
-*
-*     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 = REALONE
 *
       IF( M1 .EQ. 0 ) THEN
          DO I = 1, N