Remove manipulation of lambda due to missing guard digit

* Rename dlamda -> dlambda
* Remove those calls to lamc3 that are a workaround for
  old Cray machines
* Document the purpose of the remaining lamc3 calls
* Update documentation where dlambda has become an input
  rather than input & output

Thanks to @langou and Prof Demmel for investigating
the purpose of the lamc3 calls and suggesting
a solution.

SRC/claed8.f

@@ -18,7 +18,7 @@
 *  Definition:
 *  ===========
 *
-*       SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
+*       SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMBDA,
 *                          Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR,
 *                          GIVCOL, GIVNUM, INFO )
 *
@@ -29,7 +29,7 @@
 *       .. Array Arguments ..
 *       INTEGER            GIVCOL( 2, * ), INDX( * ), INDXP( * ),
 *      $                   INDXQ( * ), PERM( * )
-*       REAL               D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ),
+*       REAL               D( * ), DLAMBDA( * ), GIVNUM( 2, * ), W( * ),
 *      $                   Z( * )
 *       COMPLEX            Q( LDQ, * ), Q2( LDQ2, * )
 *       ..
@@ -122,9 +122,9 @@
 *>         destroyed during the updating process.
 *> \endverbatim
 *>
-*> \param[out] DLAMDA
+*> \param[out] DLAMBDA
 *> \verbatim
-*>          DLAMDA is REAL array, dimension (N)
+*>          DLAMBDA is REAL array, dimension (N)
 *>         Contains a copy of the first K eigenvalues which will be used
 *>         by SLAED3 to form the secular equation.
 *> \endverbatim
@@ -222,7 +222,7 @@
 *> \ingroup complexOTHERcomputational
 *
 *  =====================================================================
-      SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
+      SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMBDA,
      $                   Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR,
      $                   GIVCOL, GIVNUM, INFO )
 *
@@ -237,7 +237,7 @@ SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
 *     .. Array Arguments ..
       INTEGER            GIVCOL( 2, * ), INDX( * ), INDXP( * ),
      $                   INDXQ( * ), PERM( * )
-      REAL               D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ),
+      REAL               D( * ), DLAMBDA( * ), GIVNUM( 2, * ), W( * ),
      $                   Z( * )
       COMPLEX            Q( LDQ, * ), Q2( LDQ2, * )
 *     ..
@@ -322,14 +322,14 @@ SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
          INDXQ( I ) = INDXQ( I ) + CUTPNT
    20 CONTINUE
       DO 30 I = 1, N
-         DLAMDA( I ) = D( INDXQ( I ) )
+         DLAMBDA( I ) = D( INDXQ( I ) )
          W( I ) = Z( INDXQ( I ) )
    30 CONTINUE
       I = 1
       J = CUTPNT + 1
-      CALL SLAMRG( N1, N2, DLAMDA, 1, 1, INDX )
+      CALL SLAMRG( N1, N2, DLAMBDA, 1, 1, INDX )
       DO 40 I = 1, N
-         D( I ) = DLAMDA( INDX( I ) )
+         D( I ) = DLAMBDA( INDX( I ) )
          Z( I ) = W( INDX( I ) )
    40 CONTINUE
 *
@@ -438,7 +438,7 @@ SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
          ELSE
             K = K + 1
             W( K ) = Z( JLAM )
-            DLAMDA( K ) = D( JLAM )
+            DLAMBDA( K ) = D( JLAM )
             INDXP( K ) = JLAM
             JLAM = J
          END IF
@@ -450,19 +450,19 @@ SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
 *
       K = K + 1
       W( K ) = Z( JLAM )
-      DLAMDA( K ) = D( JLAM )
+      DLAMBDA( K ) = D( JLAM )
       INDXP( K ) = JLAM
 *
   100 CONTINUE
 *
-*     Sort the eigenvalues and corresponding eigenvectors into DLAMDA
+*     Sort the eigenvalues and corresponding eigenvectors into DLAMBDA
 *     and Q2 respectively.  The eigenvalues/vectors which were not
-*     deflated go into the first K slots of DLAMDA and Q2 respectively,
+*     deflated go into the first K slots of DLAMBDA and Q2 respectively,
 *     while those which were deflated go into the last N - K slots.
 *
       DO 110 J = 1, N
          JP = INDXP( J )
-         DLAMDA( J ) = D( JP )
+         DLAMBDA( J ) = D( JP )
          PERM( J ) = INDXQ( INDX( JP ) )
          CALL CCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
   110 CONTINUE
@@ -471,7 +471,7 @@ SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
 *     into the last N - K slots of D and Q respectively.
 *
       IF( K.LT.N ) THEN
-         CALL SCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
+         CALL SCOPY( N-K, DLAMBDA( K+1 ), 1, D( K+1 ), 1 )
          CALL CLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2, Q( 1, K+1 ),
      $                LDQ )
       END IF

SRC/clals0.f

@@ -392,6 +392,11 @@ SUBROUTINE CLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX,
      $                ( POLES( I, 2 ).EQ.ZERO ) ) THEN
                      RWORK( I ) = ZERO
                   ELSE
+*
+*                    Use calls to the subroutine SLAMC3 to enforce the
+*                    parentheses (x+y)+z. The goal is to prevent
+*                    optimizing compilers from doing x+(y+z).
+*
                      RWORK( I ) = POLES( I, 2 )*Z( I ) /
      $                            ( SLAMC3( POLES( I, 2 ), DSIGJ )-
      $                            DIFLJ ) / ( POLES( I, 2 )+DJ )
@@ -470,6 +475,11 @@ SUBROUTINE CLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX,
                   IF( Z( J ).EQ.ZERO ) THEN
                      RWORK( I ) = ZERO
                   ELSE
+*
+*                    Use calls to the subroutine SLAMC3 to enforce the
+*                    parentheses (x+y)+z. The goal is to prevent optimizing
+*                    compilers from doing x+(y+z).
+*
                      RWORK( I ) = Z( J ) / ( SLAMC3( DSIGJ, -POLES( I+1,
      $                            2 ) )-DIFR( I, 1 ) ) /
      $                            ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 )

SRC/dlaed2.f

@@ -18,7 +18,7 @@
 *  Definition:
 *  ===========
 *
-*       SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
+*       SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMBDA, W,
 *                          Q2, INDX, INDXC, INDXP, COLTYP, INFO )
 *
 *       .. Scalar Arguments ..
@@ -28,7 +28,7 @@
 *       .. Array Arguments ..
 *       INTEGER            COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
 *      $                   INDXQ( * )
-*       DOUBLE PRECISION   D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
+*       DOUBLE PRECISION   D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
 *      $                   W( * ), Z( * )
 *       ..
 *
@@ -123,9 +123,9 @@
 *>         process.
 *> \endverbatim
 *>
-*> \param[out] DLAMDA
+*> \param[out] DLAMBDA
 *> \verbatim
-*>          DLAMDA is DOUBLE PRECISION array, dimension (N)
+*>          DLAMBDA is DOUBLE PRECISION array, dimension (N)
 *>         A copy of the first K eigenvalues which will be used by
 *>         DLAED3 to form the secular equation.
 *> \endverbatim
@@ -148,7 +148,7 @@
 *> \param[out] INDX
 *> \verbatim
 *>          INDX is INTEGER array, dimension (N)
-*>         The permutation used to sort the contents of DLAMDA into
+*>         The permutation used to sort the contents of DLAMBDA into
 *>         ascending order.
 *> \endverbatim
 *>
@@ -207,7 +207,7 @@
 *>  Modified by Francoise Tisseur, University of Tennessee
 *>
 *  =====================================================================
-      SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
+      SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMBDA, W,
      $                   Q2, INDX, INDXC, INDXP, COLTYP, INFO )
 *
 *  -- LAPACK computational routine --
@@ -221,7 +221,7 @@ SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
 *     .. Array Arguments ..
       INTEGER            COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
      $                   INDXQ( * )
-      DOUBLE PRECISION   D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
+      DOUBLE PRECISION   D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
      $                   W( * ), Z( * )
 *     ..
 *
@@ -300,9 +300,9 @@ SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
 *     re-integrate the deflated parts from the last pass
 *
       DO 20 I = 1, N
-         DLAMDA( I ) = D( INDXQ( I ) )
+         DLAMBDA( I ) = D( INDXQ( I ) )
    20 CONTINUE
-      CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDXC )
+      CALL DLAMRG( N1, N2, DLAMBDA, 1, 1, INDXC )
       DO 30 I = 1, N
          INDX( I ) = INDXQ( INDXC( I ) )
    30 CONTINUE
@@ -324,11 +324,11 @@ SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
          DO 40 J = 1, N
             I = INDX( J )
             CALL DCOPY( N, Q( 1, I ), 1, Q2( IQ2 ), 1 )
-            DLAMDA( J ) = D( I )
+            DLAMBDA( J ) = D( I )
             IQ2 = IQ2 + N
    40    CONTINUE
          CALL DLACPY( 'A', N, N, Q2, N, Q, LDQ )
-         CALL DCOPY( N, DLAMDA, 1, D, 1 )
+         CALL DCOPY( N, DLAMBDA, 1, D, 1 )
          GO TO 190
       END IF
 *
@@ -421,7 +421,7 @@ SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
             PJ = NJ
          ELSE
             K = K + 1
-            DLAMDA( K ) = D( PJ )
+            DLAMBDA( K ) = D( PJ )
             W( K ) = Z( PJ )
             INDXP( K ) = PJ
             PJ = NJ
@@ -433,7 +433,7 @@ SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
 *     Record the last eigenvalue.
 *
       K = K + 1
-      DLAMDA( K ) = D( PJ )
+      DLAMBDA( K ) = D( PJ )
       W( K ) = Z( PJ )
       INDXP( K ) = PJ
 *
@@ -470,9 +470,9 @@ SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
          PSM( CT ) = PSM( CT ) + 1
   130 CONTINUE
 *
-*     Sort the eigenvalues and corresponding eigenvectors into DLAMDA
+*     Sort the eigenvalues and corresponding eigenvectors into DLAMBDA
 *     and Q2 respectively.  The eigenvalues/vectors which were not
-*     deflated go into the first K slots of DLAMDA and Q2 respectively,
+*     deflated go into the first K slots of DLAMBDA and Q2 respectively,
 *     while those which were deflated go into the last N - K slots.
 *
       I = 1

SRC/dlaed3.f

@@ -18,7 +18,7 @@
 *  Definition:
 *  ===========
 *
-*       SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
+*       SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMBDA, Q2, INDX,
 *                          CTOT, W, S, INFO )
 *
 *       .. Scalar Arguments ..
@@ -27,7 +27,7 @@
 *       ..
 *       .. Array Arguments ..
 *       INTEGER            CTOT( * ), INDX( * )
-*       DOUBLE PRECISION   D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
+*       DOUBLE PRECISION   D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
 *      $                   S( * ), W( * )
 *       ..
 *
@@ -44,12 +44,6 @@
 *> being combined by the matrix of eigenvectors of the K-by-K system
 *> which is solved here.
 *>
-*> This code makes very mild assumptions about floating point
-*> arithmetic. It will work on machines with a guard digit in
-*> add/subtract, or on those binary machines without guard digits
-*> which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
-*> It could conceivably fail on hexadecimal or decimal machines
-*> without guard digits, but we know of none.
 *> \endverbatim
 *
 *  Arguments:
@@ -104,14 +98,12 @@
 *>          RHO >= 0 required.
 *> \endverbatim
 *>
-*> \param[in,out] DLAMDA
+*> \param[in] DLAMBDA
 *> \verbatim
-*>          DLAMDA is DOUBLE PRECISION array, dimension (K)
+*>          DLAMBDA is DOUBLE PRECISION array, dimension (K)
 *>          The first K elements of this array contain the old roots
 *>          of the deflated updating problem.  These are the poles
-*>          of the secular equation. May be changed on output by
-*>          having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
-*>          Cray-2, or Cray C-90, as described above.
+*>          of the secular equation.
 *> \endverbatim
 *>
 *> \param[in] Q2
@@ -180,7 +172,7 @@
 *>  Modified by Francoise Tisseur, University of Tennessee
 *>
 *  =====================================================================
-      SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
+      SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMBDA, Q2, INDX,
      $                   CTOT, W, S, INFO )
 *
 *  -- LAPACK computational routine --
@@ -193,7 +185,7 @@ SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
 *     ..
 *     .. Array Arguments ..
       INTEGER            CTOT( * ), INDX( * )
-      DOUBLE PRECISION   D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
+      DOUBLE PRECISION   D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
      $                   S( * ), W( * )
 *     ..
 *
@@ -208,8 +200,8 @@ SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
       DOUBLE PRECISION   TEMP
 *     ..
 *     .. External Functions ..
-      DOUBLE PRECISION   DLAMC3, DNRM2
-      EXTERNAL           DLAMC3, DNRM2
+      DOUBLE PRECISION   DNRM2
+      EXTERNAL           DNRM2
 *     ..
 *     .. External Subroutines ..
       EXTERNAL           DCOPY, DGEMM, DLACPY, DLAED4, DLASET, XERBLA
@@ -240,29 +232,9 @@ SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
       IF( K.EQ.0 )
      $   RETURN
 *
-*     Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
-*     be computed with high relative accuracy (barring over/underflow).
-*     This is a problem on machines without a guard digit in
-*     add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
-*     The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
-*     which on any of these machines zeros out the bottommost
-*     bit of DLAMDA(I) if it is 1; this makes the subsequent
-*     subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
-*     occurs. On binary machines with a guard digit (almost all
-*     machines) it does not change DLAMDA(I) at all. On hexadecimal
-*     and decimal machines with a guard digit, it slightly
-*     changes the bottommost bits of DLAMDA(I). It does not account
-*     for hexadecimal or decimal machines without guard digits
-*     (we know of none). We use a subroutine call to compute
-*     2*DLAMBDA(I) to prevent optimizing compilers from eliminating
-*     this code.
-*
-      DO 10 I = 1, K
-         DLAMDA( I ) = DLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I )
-   10 CONTINUE
 *
       DO 20 J = 1, K
-         CALL DLAED4( K, J, DLAMDA, W, Q( 1, J ), RHO, D( J ), INFO )
+         CALL DLAED4( K, J, DLAMBDA, W, Q( 1, J ), RHO, D( J ), INFO )
 *
 *        If the zero finder fails, the computation is terminated.
 *
@@ -293,10 +265,10 @@ SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
       CALL DCOPY( K, Q, LDQ+1, W, 1 )
       DO 60 J = 1, K
          DO 40 I = 1, J - 1
-            W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
+            W( I ) = W( I )*( Q( I, J )/( DLAMBDA( I )-DLAMBDA( J ) ) )
    40    CONTINUE
          DO 50 I = J + 1, K
-            W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
+            W( I ) = W( I )*( Q( I, J )/( DLAMBDA( I )-DLAMBDA( J ) ) )
    50    CONTINUE
    60 CONTINUE
       DO 70 I = 1, K