dlasq2.f

*> \brief \b DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASQ2( N, Z, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASQ2 computes all the eigenvalues of the symmetric positive
*> definite tridiagonal matrix associated with the qd array Z to high
*> relative accuracy are computed to high relative accuracy, in the
*> absence of denormalization, underflow and overflow.
*>
*> To see the relation of Z to the tridiagonal matrix, let L be a
*> unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
*> let U be an upper bidiagonal matrix with 1's above and diagonal
*> Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
*> symmetric tridiagonal to which it is similar.
*>
*> Note : DLASQ2 defines a logical variable, IEEE, which is true
*> on machines which follow ieee-754 floating-point standard in their
*> handling of infinities and NaNs, and false otherwise. This variable
*> is passed to DLASQ3.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>        The number of rows and columns in the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( 4*N )
*>        On entry Z holds the qd array. On exit, entries 1 to N hold
*>        the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
*>        trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
*>        N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
*>        holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
*>        shifts that failed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>        = 0: successful exit
*>        < 0: if the i-th argument is a scalar and had an illegal
*>             value, then INFO = -i, if the i-th argument is an
*>             array and the j-entry had an illegal value, then
*>             INFO = -(i*100+j)
*>        > 0: the algorithm failed
*>              = 1, a split was marked by a positive value in E
*>              = 2, current block of Z not diagonalized after 100*N
*>                   iterations (in inner while loop).  On exit Z holds
*>                   a qd array with the same eigenvalues as the given Z.
*>              = 3, termination criterion of outer while loop not met
*>                   (program created more than N unreduced blocks)
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup auxOTHERcomputational
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Local Variables: I0:N0 defines a current unreduced segment of Z.
*>  The shifts are accumulated in SIGMA. Iteration count is in ITER.
*>  Ping-pong is controlled by PP (alternates between 0 and 1).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLASQ2( N, Z, INFO )
*
*  -- LAPACK computational routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   CBIAS
      PARAMETER          ( CBIAS = 1.50D0 )
      DOUBLE PRECISION   ZERO, HALF, ONE, TWO, FOUR, HUNDRD
      PARAMETER          ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
     $                     TWO = 2.0D0, FOUR = 4.0D0, HUNDRD = 100.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            IEEE
      INTEGER            I0, I1, I4, IINFO, IPN4, ITER, IWHILA, IWHILB,
     $                   K, KMIN, N0, N1, NBIG, NDIV, NFAIL, PP, SPLT,
     $                   TTYPE
      DOUBLE PRECISION   D, DEE, DEEMIN, DESIG, DMIN, DMIN1, DMIN2, DN,
     $                   DN1, DN2, E, EMAX, EMIN, EPS, G, OLDEMN, QMAX,
     $                   QMIN, S, SAFMIN, SIGMA, T, TAU, TEMP, TOL,
     $                   TOL2, TRACE, ZMAX, TEMPE, TEMPQ
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASQ3, DLASRT, XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH, ILAENV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*     (in case DLASQ2 is not called by DLASQ1)
*
      INFO = 0
      EPS = DLAMCH( 'Precision' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      TOL = EPS*HUNDRD
      TOL2 = TOL**2
*
      IF( N.LT.0 ) THEN
         INFO = -1
         CALL XERBLA( 'DLASQ2', 1 )
         RETURN
      ELSE IF( N.EQ.0 ) THEN
         RETURN
      ELSE IF( N.EQ.1 ) THEN
*
*        1-by-1 case.
*
         IF( Z( 1 ).LT.ZERO ) THEN
            INFO = -201
            CALL XERBLA( 'DLASQ2', 2 )
         END IF
         RETURN
      ELSE IF( N.EQ.2 ) THEN
*
*        2-by-2 case.
*
         IF( Z( 2 ).LT.ZERO .OR. Z( 3 ).LT.ZERO ) THEN
            INFO = -2
            CALL XERBLA( 'DLASQ2', 2 )
            RETURN
         ELSE IF( Z( 3 ).GT.Z( 1 ) ) THEN
            D = Z( 3 )
            Z( 3 ) = Z( 1 )
            Z( 1 ) = D
         END IF
         Z( 5 ) = Z( 1 ) + Z( 2 ) + Z( 3 )
         IF( Z( 2 ).GT.Z( 3 )*TOL2 ) THEN
            T = HALF*( ( Z( 1 )-Z( 3 ) )+Z( 2 ) )
            S = Z( 3 )*( Z( 2 ) / T )
            IF( S.LE.T ) THEN
               S = Z( 3 )*( Z( 2 ) / ( T*( ONE+SQRT( ONE+S / T ) ) ) )
            ELSE
               S = Z( 3 )*( Z( 2 ) / ( T+SQRT( T )*SQRT( T+S ) ) )
            END IF
            T = Z( 1 ) + ( S+Z( 2 ) )
            Z( 3 ) = Z( 3 )*( Z( 1 ) / T )
            Z( 1 ) = T
         END IF
         Z( 2 ) = Z( 3 )
         Z( 6 ) = Z( 2 ) + Z( 1 )
         RETURN
      END IF
*
*     Check for negative data and compute sums of q's and e's.
*
      Z( 2*N ) = ZERO
      EMIN = Z( 2 )
      QMAX = ZERO
      ZMAX = ZERO
      D = ZERO
      E = ZERO
*
      DO 10 K = 1, 2*( N-1 ), 2
         IF( Z( K ).LT.ZERO ) THEN
            INFO = -( 200+K )
            CALL XERBLA( 'DLASQ2', 2 )
            RETURN
         ELSE IF( Z( K+1 ).LT.ZERO ) THEN
            INFO = -( 200+K+1 )
            CALL XERBLA( 'DLASQ2', 2 )
            RETURN
         END IF
         D = D + Z( K )
         E = E + Z( K+1 )
         QMAX = MAX( QMAX, Z( K ) )
         EMIN = MIN( EMIN, Z( K+1 ) )
         ZMAX = MAX( QMAX, ZMAX, Z( K+1 ) )
   10 CONTINUE
      IF( Z( 2*N-1 ).LT.ZERO ) THEN
         INFO = -( 200+2*N-1 )
         CALL XERBLA( 'DLASQ2', 2 )
         RETURN
      END IF
      D = D + Z( 2*N-1 )
      QMAX = MAX( QMAX, Z( 2*N-1 ) )
      ZMAX = MAX( QMAX, ZMAX )
*
*     Check for diagonality.
*
      IF( E.EQ.ZERO ) THEN
         DO 20 K = 2, N
            Z( K ) = Z( 2*K-1 )
   20    CONTINUE
         CALL DLASRT( 'D', N, Z, IINFO )
         Z( 2*N-1 ) = D
         RETURN
      END IF
*
      TRACE = D + E
*
*     Check for zero data.
*
      IF( TRACE.EQ.ZERO ) THEN
         Z( 2*N-1 ) = ZERO
         RETURN
      END IF
*
*     Check whether the machine is IEEE conformable.
*
      IEEE = ILAENV( 10, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1 .AND.
     $       ILAENV( 11, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1
*
*     Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...).
*
      DO 30 K = 2*N, 2, -2
         Z( 2*K ) = ZERO
         Z( 2*K-1 ) = Z( K )
         Z( 2*K-2 ) = ZERO
         Z( 2*K-3 ) = Z( K-1 )
   30 CONTINUE
*
      I0 = 1
      N0 = N
*
*     Reverse the qd-array, if warranted.
*
      IF( CBIAS*Z( 4*I0-3 ).LT.Z( 4*N0-3 ) ) THEN
         IPN4 = 4*( I0+N0 )
         DO 40 I4 = 4*I0, 2*( I0+N0-1 ), 4
            TEMP = Z( I4-3 )
            Z( I4-3 ) = Z( IPN4-I4-3 )
            Z( IPN4-I4-3 ) = TEMP
            TEMP = Z( I4-1 )
            Z( I4-1 ) = Z( IPN4-I4-5 )
            Z( IPN4-I4-5 ) = TEMP
   40    CONTINUE
      END IF
*
*     Initial split checking via dqd and Li's test.
*
      PP = 0
*
      DO 80 K = 1, 2
*
         D = Z( 4*N0+PP-3 )
         DO 50 I4 = 4*( N0-1 ) + PP, 4*I0 + PP, -4
            IF( Z( I4-1 ).LE.TOL2*D ) THEN
               Z( I4-1 ) = -ZERO
               D = Z( I4-3 )
            ELSE
               D = Z( I4-3 )*( D / ( D+Z( I4-1 ) ) )
            END IF
   50    CONTINUE
*
*        dqd maps Z to ZZ plus Li's test.
*
         EMIN = Z( 4*I0+PP+1 )
         D = Z( 4*I0+PP-3 )
         DO 60 I4 = 4*I0 + PP, 4*( N0-1 ) + PP, 4
            Z( I4-2*PP-2 ) = D + Z( I4-1 )
            IF( Z( I4-1 ).LE.TOL2*D ) THEN
               Z( I4-1 ) = -ZERO
               Z( I4-2*PP-2 ) = D
               Z( I4-2*PP ) = ZERO
               D = Z( I4+1 )
            ELSE IF( SAFMIN*Z( I4+1 ).LT.Z( I4-2*PP-2 ) .AND.
     $               SAFMIN*Z( I4-2*PP-2 ).LT.Z( I4+1 ) ) THEN
               TEMP = Z( I4+1 ) / Z( I4-2*PP-2 )
               Z( I4-2*PP ) = Z( I4-1 )*TEMP
               D = D*TEMP
            ELSE
               Z( I4-2*PP ) = Z( I4+1 )*( Z( I4-1 ) / Z( I4-2*PP-2 ) )
               D = Z( I4+1 )*( D / Z( I4-2*PP-2 ) )
            END IF
            EMIN = MIN( EMIN, Z( I4-2*PP ) )
   60    CONTINUE
         Z( 4*N0-PP-2 ) = D
*
*        Now find qmax.
*
         QMAX = Z( 4*I0-PP-2 )
         DO 70 I4 = 4*I0 - PP + 2, 4*N0 - PP - 2, 4
            QMAX = MAX( QMAX, Z( I4 ) )
   70    CONTINUE
*
*        Prepare for the next iteration on K.
*
         PP = 1 - PP
   80 CONTINUE
*
*     Initialise variables to pass to DLASQ3.
*
      TTYPE = 0
      DMIN1 = ZERO
      DMIN2 = ZERO
      DN    = ZERO
      DN1   = ZERO
      DN2   = ZERO
      G     = ZERO
      TAU   = ZERO
*
      ITER = 2
      NFAIL = 0
      NDIV = 2*( N0-I0 )
*
      DO 160 IWHILA = 1, N + 1
         IF( N0.LT.1 )
     $      GO TO 170
*
*        While array unfinished do
*
*        E(N0) holds the value of SIGMA when submatrix in I0:N0
*        splits from the rest of the array, but is negated.
*
         DESIG = ZERO
         IF( N0.EQ.N ) THEN
            SIGMA = ZERO
         ELSE
            SIGMA = -Z( 4*N0-1 )
         END IF
         IF( SIGMA.LT.ZERO ) THEN
            INFO = 1
            RETURN
         END IF
*
*        Find last unreduced submatrix's top index I0, find QMAX and
*        EMIN. Find Gershgorin-type bound if Q's much greater than E's.
*
         EMAX = ZERO
         IF( N0.GT.I0 ) THEN
            EMIN = ABS( Z( 4*N0-5 ) )
         ELSE
            EMIN = ZERO
         END IF
         QMIN = Z( 4*N0-3 )
         QMAX = QMIN
         DO 90 I4 = 4*N0, 8, -4
            IF( Z( I4-5 ).LE.ZERO )
     $         GO TO 100
            IF( QMIN.GE.FOUR*EMAX ) THEN
               QMIN = MIN( QMIN, Z( I4-3 ) )
               EMAX = MAX( EMAX, Z( I4-5 ) )
            END IF
            QMAX = MAX( QMAX, Z( I4-7 )+Z( I4-5 ) )
            EMIN = MIN( EMIN, Z( I4-5 ) )
   90    CONTINUE
         I4 = 4
*
  100    CONTINUE
         I0 = I4 / 4
         PP = 0
*
         IF( N0-I0.GT.1 ) THEN
            DEE = Z( 4*I0-3 )
            DEEMIN = DEE
            KMIN = I0
            DO 110 I4 = 4*I0+1, 4*N0-3, 4
               DEE = Z( I4 )*( DEE /( DEE+Z( I4-2 ) ) )
               IF( DEE.LE.DEEMIN ) THEN
                  DEEMIN = DEE
                  KMIN = ( I4+3 )/4
               END IF
  110       CONTINUE
            IF( (KMIN-I0)*2.LT.N0-KMIN .AND.
     $         DEEMIN.LE.HALF*Z(4*N0-3) ) THEN
               IPN4 = 4*( I0+N0 )
               PP = 2
               DO 120 I4 = 4*I0, 2*( I0+N0-1 ), 4
                  TEMP = Z( I4-3 )
                  Z( I4-3 ) = Z( IPN4-I4-3 )
                  Z( IPN4-I4-3 ) = TEMP
                  TEMP = Z( I4-2 )
                  Z( I4-2 ) = Z( IPN4-I4-2 )
                  Z( IPN4-I4-2 ) = TEMP
                  TEMP = Z( I4-1 )
                  Z( I4-1 ) = Z( IPN4-I4-5 )
                  Z( IPN4-I4-5 ) = TEMP
                  TEMP = Z( I4 )
                  Z( I4 ) = Z( IPN4-I4-4 )
                  Z( IPN4-I4-4 ) = TEMP
  120          CONTINUE
            END IF
         END IF
*
*        Put -(initial shift) into DMIN.
*
         DMIN = -MAX( ZERO, QMIN-TWO*SQRT( QMIN )*SQRT( EMAX ) )
*
*        Now I0:N0 is unreduced.
*        PP = 0 for ping, PP = 1 for pong.
*        PP = 2 indicates that flipping was applied to the Z array and
*               and that the tests for deflation upon entry in DLASQ3
*               should not be performed.
*
         NBIG = 100*( N0-I0+1 )
         DO 140 IWHILB = 1, NBIG
            IF( I0.GT.N0 )
     $         GO TO 150
*
*           While submatrix unfinished take a good dqds step.
*
            CALL DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL,
     $                   ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
     $                   DN2, G, TAU )
*
            PP = 1 - PP
*
*           When EMIN is very small check for splits.
*
            IF( PP.EQ.0 .AND. N0-I0.GE.3 ) THEN
               IF( Z( 4*N0 ).LE.TOL2*QMAX .OR.
     $             Z( 4*N0-1 ).LE.TOL2*SIGMA ) THEN
                  SPLT = I0 - 1
                  QMAX = Z( 4*I0-3 )
                  EMIN = Z( 4*I0-1 )
                  OLDEMN = Z( 4*I0 )
                  DO 130 I4 = 4*I0, 4*( N0-3 ), 4
                     IF( Z( I4 ).LE.TOL2*Z( I4-3 ) .OR.
     $                   Z( I4-1 ).LE.TOL2*SIGMA ) THEN
                        Z( I4-1 ) = -SIGMA
                        SPLT = I4 / 4
                        QMAX = ZERO
                        EMIN = Z( I4+3 )
                        OLDEMN = Z( I4+4 )
                     ELSE
                        QMAX = MAX( QMAX, Z( I4+1 ) )
                        EMIN = MIN( EMIN, Z( I4-1 ) )
                        OLDEMN = MIN( OLDEMN, Z( I4 ) )
                     END IF
  130             CONTINUE
                  Z( 4*N0-1 ) = EMIN
                  Z( 4*N0 ) = OLDEMN
                  I0 = SPLT + 1
               END IF
            END IF
*
  140    CONTINUE
*
         INFO = 2
*
*        Maximum number of iterations exceeded, restore the shift
*        SIGMA and place the new d's and e's in a qd array.
*        This might need to be done for several blocks
*
         I1 = I0
         N1 = N0
 145     CONTINUE
         TEMPQ = Z( 4*I0-3 )
         Z( 4*I0-3 ) = Z( 4*I0-3 ) + SIGMA
         DO K = I0+1, N0
            TEMPE = Z( 4*K-5 )
            Z( 4*K-5 ) = Z( 4*K-5 ) * (TEMPQ / Z( 4*K-7 ))
            TEMPQ = Z( 4*K-3 )
            Z( 4*K-3 ) = Z( 4*K-3 ) + SIGMA + TEMPE - Z( 4*K-5 )
         END DO
*
*        Prepare to do this on the previous block if there is one
*
         IF( I1.GT.1 ) THEN
            N1 = I1-1
            DO WHILE( ( I1.GE.2 ) .AND. ( Z(4*I1-5).GE.ZERO ) )
               I1 = I1 - 1
            END DO
            SIGMA = -Z(4*N1-1)
            GO TO 145
         END IF

         DO K = 1, N
            Z( 2*K-1 ) = Z( 4*K-3 )
*
*        Only the block 1..N0 is unfinished.  The rest of the e's
*        must be essentially zero, although sometimes other data
*        has been stored in them.
*
            IF( K.LT.N0 ) THEN
               Z( 2*K ) = Z( 4*K-1 )
            ELSE
               Z( 2*K ) = 0
            END IF
         END DO
         RETURN
*
*        end IWHILB
*
  150    CONTINUE
*
  160 CONTINUE
*
      INFO = 3
      RETURN
*
*     end IWHILA
*
  170 CONTINUE
*
*     Move q's to the front.
*
      DO 180 K = 2, N
         Z( K ) = Z( 4*K-3 )
  180 CONTINUE
*
*     Sort and compute sum of eigenvalues.
*
      CALL DLASRT( 'D', N, Z, IINFO )
*
      E = ZERO
      DO 190 K = N, 1, -1
         E = E + Z( K )
  190 CONTINUE
*
*     Store trace, sum(eigenvalues) and information on performance.
*
      Z( 2*N+1 ) = TRACE
      Z( 2*N+2 ) = E
      Z( 2*N+3 ) = DBLE( ITER )
      Z( 2*N+4 ) = DBLE( NDIV ) / DBLE( N**2 )
      Z( 2*N+5 ) = HUNDRD*NFAIL / DBLE( ITER )
      RETURN
*
*     End of DLASQ2
*
      END