text: *> \brief \b DLAED2 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
  *
  *  =========== DOCUMENTATION ===========
  *
  * Online html documentation available at
  *            http://www.netlib.org/lapack/explore-html/
  *
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  *> [TXT]</a>
  *> \endhtmlonly
  *
  *  Definition:
  *  ===========
  *
  *       SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
  *                          Q2, INDX, INDXC, INDXP, COLTYP, INFO )
  *
  *       .. Scalar Arguments ..
  *       INTEGER            INFO, K, LDQ, N, N1
  *       DOUBLE PRECISION   RHO
  *       ..
  *       .. Array Arguments ..
  *       INTEGER            COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
  *      $                   INDXQ( * )
  *       DOUBLE PRECISION   D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
  *      $                   W( * ), Z( * )
  *       ..
  *
  *
  *> \par Purpose:
  *  =============
  *>
  *> \verbatim
  *>
  *> DLAED2 merges the two sets of eigenvalues together into a single
  *> sorted set.  Then it tries to deflate the size of the problem.
  *> There are two ways in which deflation can occur:  when two or more
  *> eigenvalues are close together or if there is a tiny entry in the
  *> Z vector.  For each such occurrence the order of the related secular
  *> equation problem is reduced by one.
  *> \endverbatim
  *
  *  Arguments:
  *  ==========
  *
  *> \param[out] K
  *> \verbatim
  *>          K is INTEGER
  *>         The number of non-deflated eigenvalues, and the order of the
  *>         related secular equation. 0 <= K <=N.
  *> \endverbatim
  *>
  *> \param[in] N
  *> \verbatim
  *>          N is INTEGER
  *>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
  *> \endverbatim
  *>
  *> \param[in] N1
  *> \verbatim
  *>          N1 is INTEGER
  *>         The location of the last eigenvalue in the leading sub-matrix.
  *>         min(1,N) <= N1 <= N/2.
  *> \endverbatim
  *>
  *> \param[in,out] D
  *> \verbatim
  *>          D is DOUBLE PRECISION array, dimension (N)
  *>         On entry, D contains the eigenvalues of the two submatrices to
  *>         be combined.
  *>         On exit, D contains the trailing (N-K) updated eigenvalues
  *>         (those which were deflated) sorted into increasing order.
  *> \endverbatim
  *>
  *> \param[in,out] Q
  *> \verbatim
  *>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
  *>         On entry, Q contains the eigenvectors of two submatrices in
  *>         the two square blocks with corners at (1,1), (N1,N1)
  *>         and (N1+1, N1+1), (N,N).
  *>         On exit, Q contains the trailing (N-K) updated eigenvectors
  *>         (those which were deflated) in its last N-K columns.
  *> \endverbatim
  *>
  *> \param[in] LDQ
  *> \verbatim
  *>          LDQ is INTEGER
  *>         The leading dimension of the array Q.  LDQ >= max(1,N).
  *> \endverbatim
  *>
  *> \param[in,out] INDXQ
  *> \verbatim
  *>          INDXQ is INTEGER array, dimension (N)
  *>         The permutation which separately sorts the two sub-problems
  *>         in D into ascending order.  Note that elements in the second
  *>         half of this permutation must first have N1 added to their
  *>         values. Destroyed on exit.
  *> \endverbatim
  *>
  *> \param[in,out] RHO
  *> \verbatim
  *>          RHO is DOUBLE PRECISION
  *>         On entry, the off-diagonal element associated with the rank-1
  *>         cut which originally split the two submatrices which are now
  *>         being recombined.
  *>         On exit, RHO has been modified to the value required by
  *>         DLAED3.
  *> \endverbatim
  *>
  *> \param[in] Z
  *> \verbatim
  *>          Z is DOUBLE PRECISION array, dimension (N)
  *>         On entry, Z contains the updating vector (the last
  *>         row of the first sub-eigenvector matrix and the first row of
  *>         the second sub-eigenvector matrix).
  *>         On exit, the contents of Z have been destroyed by the updating
  *>         process.
  *> \endverbatim
  *>
  *> \param[out] DLAMDA
  *> \verbatim
  *>          DLAMDA 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
  *>
  *> \param[out] W
  *> \verbatim
  *>          W is DOUBLE PRECISION array, dimension (N)
  *>         The first k values of the final deflation-altered z-vector
  *>         which will be passed to DLAED3.
  *> \endverbatim
  *>
  *> \param[out] Q2
  *> \verbatim
  *>          Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
  *>         A copy of the first K eigenvectors which will be used by
  *>         DLAED3 in a matrix multiply (DGEMM) to solve for the new
  *>         eigenvectors.
  *> \endverbatim
  *>
  *> \param[out] INDX
  *> \verbatim
  *>          INDX is INTEGER array, dimension (N)
  *>         The permutation used to sort the contents of DLAMDA into
  *>         ascending order.
  *> \endverbatim
  *>
  *> \param[out] INDXC
  *> \verbatim
  *>          INDXC is INTEGER array, dimension (N)
  *>         The permutation used to arrange the columns of the deflated
  *>         Q matrix into three groups:  the first group contains non-zero
  *>         elements only at and above N1, the second contains
  *>         non-zero elements only below N1, and the third is dense.
  *> \endverbatim
  *>
  *> \param[out] INDXP
  *> \verbatim
  *>          INDXP is INTEGER array, dimension (N)
  *>         The permutation used to place deflated values of D at the end
  *>         of the array.  INDXP(1:K) points to the nondeflated D-values
  *>         and INDXP(K+1:N) points to the deflated eigenvalues.
  *> \endverbatim
  *>
  *> \param[out] COLTYP
  *> \verbatim
  *>          COLTYP is INTEGER array, dimension (N)
  *>         During execution, a label which will indicate which of the
  *>         following types a column in the Q2 matrix is:
  *>         1 : non-zero in the upper half only;
  *>         2 : dense;
  *>         3 : non-zero in the lower half only;
  *>         4 : deflated.
  *>         On exit, COLTYP(i) is the number of columns of type i,
  *>         for i=1 to 4 only.
  *> \endverbatim
  *>
  *> \param[out] INFO
  *> \verbatim
  *>          INFO is INTEGER
  *>          = 0:  successful exit.
  *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
  *> \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 Contributors:
  *  ==================
  *>
  *> Jeff Rutter, Computer Science Division, University of California
  *> at Berkeley, USA \n
  *>  Modified by Francoise Tisseur, University of Tennessee
  *>
  *  =====================================================================
        SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
       $                   Q2, INDX, INDXC, INDXP, COLTYP, 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, K, LDQ, N, N1
        DOUBLE PRECISION   RHO
  *     ..
  *     .. Array Arguments ..
        INTEGER            COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
       $                   INDXQ( * )
        DOUBLE PRECISION   D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
       $                   W( * ), Z( * )
  *     ..
  *
  *  =====================================================================
  *
  *     .. Parameters ..
        DOUBLE PRECISION   MONE, ZERO, ONE, TWO, EIGHT
        PARAMETER          ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
       $                   TWO = 2.0D0, EIGHT = 8.0D0 )
  *     ..
  *     .. Local Arrays ..
        INTEGER            CTOT( 4 ), PSM( 4 )
  *     ..
  *     .. Local Scalars ..
        INTEGER            CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1,
       $                   N2, NJ, PJ
        DOUBLE PRECISION   C, EPS, S, T, TAU, TOL
  *     ..
  *     .. External Functions ..
        INTEGER            IDAMAX
        DOUBLE PRECISION   DLAMCH, DLAPY2
        EXTERNAL           IDAMAX, DLAMCH, DLAPY2
  *     ..
  *     .. External Subroutines ..
        EXTERNAL           DCOPY, DLACPY, DLAMRG, DROT, DSCAL, XERBLA
  *     ..
  *     .. Intrinsic Functions ..
        INTRINSIC          ABS, MAX, MIN, SQRT
  *     ..
  *     .. Executable Statements ..
  *
  *     Test the input parameters.
  *
        INFO = 0
  *
        IF( N.LT.0 ) THEN
           INFO = -2
        ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
           INFO = -6
        ELSE IF( MIN( 1, ( N / 2 ) ).GT.N1 .OR. ( N / 2 ).LT.N1 ) THEN
           INFO = -3
        END IF
        IF( INFO.NE.0 ) THEN
           CALL XERBLA( 'DLAED2', -INFO )
           RETURN
        END IF
  *
  *     Quick return if possible
  *
        IF( N.EQ.0 )
       $   RETURN
  *
        N2 = N - N1
        N1P1 = N1 + 1
  *
        IF( RHO.LT.ZERO ) THEN
           CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
        END IF
  *
  *     Normalize z so that norm(z) = 1.  Since z is the concatenation of
  *     two normalized vectors, norm2(z) = sqrt(2).
  *
        T = ONE / SQRT( TWO )
        CALL DSCAL( N, T, Z, 1 )
  *
  *     RHO = ABS( norm(z)**2 * RHO )
  *
        RHO = ABS( TWO*RHO )
  *
  *     Sort the eigenvalues into increasing order
  *
        DO 10 I = N1P1, N
           INDXQ( I ) = INDXQ( I ) + N1
     10 CONTINUE
  *
  *     re-integrate the deflated parts from the last pass
  *
        DO 20 I = 1, N
           DLAMDA( I ) = D( INDXQ( I ) )
     20 CONTINUE
        CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDXC )
        DO 30 I = 1, N
           INDX( I ) = INDXQ( INDXC( I ) )
     30 CONTINUE
  *
  *     Calculate the allowable deflation tolerance
  *
        IMAX = IDAMAX( N, Z, 1 )
        JMAX = IDAMAX( N, D, 1 )
        EPS = DLAMCH( 'Epsilon' )
        TOL = EIGHT*EPS*MAX( ABS( D( JMAX ) ), ABS( Z( IMAX ) ) )
  *
  *     If the rank-1 modifier is small enough, no more needs to be done
  *     except to reorganize Q so that its columns correspond with the
  *     elements in D.
  *
        IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
           K = 0
           IQ2 = 1
           DO 40 J = 1, N
              I = INDX( J )
              CALL DCOPY( N, Q( 1, I ), 1, Q2( IQ2 ), 1 )
              DLAMDA( J ) = D( I )
              IQ2 = IQ2 + N
     40    CONTINUE
           CALL DLACPY( 'A', N, N, Q2, N, Q, LDQ )
           CALL DCOPY( N, DLAMDA, 1, D, 1 )
           GO TO 190
        END IF
  *
  *     If there are multiple eigenvalues then the problem deflates.  Here
  *     the number of equal eigenvalues are found.  As each equal
  *     eigenvalue is found, an elementary reflector is computed to rotate
  *     the corresponding eigensubspace so that the corresponding
  *     components of Z are zero in this new basis.
  *
        DO 50 I = 1, N1
           COLTYP( I ) = 1
     50 CONTINUE
        DO 60 I = N1P1, N
           COLTYP( I ) = 3
     60 CONTINUE
  *
  *
        K = 0
        K2 = N + 1
        DO 70 J = 1, N
           NJ = INDX( J )
           IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
  *
  *           Deflate due to small z component.
  *
              K2 = K2 - 1
              COLTYP( NJ ) = 4
              INDXP( K2 ) = NJ
              IF( J.EQ.N )
       $         GO TO 100
           ELSE
              PJ = NJ
              GO TO 80
           END IF
     70 CONTINUE
     80 CONTINUE
        J = J + 1
        NJ = INDX( J )
        IF( J.GT.N )
       $   GO TO 100
        IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
  *
  *        Deflate due to small z component.
  *
           K2 = K2 - 1
           COLTYP( NJ ) = 4
           INDXP( K2 ) = NJ
        ELSE
  *
  *        Check if eigenvalues are close enough to allow deflation.
  *
           S = Z( PJ )
           C = Z( NJ )
  *
  *        Find sqrt(a**2+b**2) without overflow or
  *        destructive underflow.
  *
           TAU = DLAPY2( C, S )
           T = D( NJ ) - D( PJ )
           C = C / TAU
           S = -S / TAU
           IF( ABS( T*C*S ).LE.TOL ) THEN
  *
  *           Deflation is possible.
  *
              Z( NJ ) = TAU
              Z( PJ ) = ZERO
              IF( COLTYP( NJ ).NE.COLTYP( PJ ) )
       $         COLTYP( NJ ) = 2
              COLTYP( PJ ) = 4
              CALL DROT( N, Q( 1, PJ ), 1, Q( 1, NJ ), 1, C, S )
              T = D( PJ )*C**2 + D( NJ )*S**2
              D( NJ ) = D( PJ )*S**2 + D( NJ )*C**2
              D( PJ ) = T
              K2 = K2 - 1
              I = 1
     90       CONTINUE
              IF( K2+I.LE.N ) THEN
                 IF( D( PJ ).LT.D( INDXP( K2+I ) ) ) THEN
                    INDXP( K2+I-1 ) = INDXP( K2+I )
                    INDXP( K2+I ) = PJ
                    I = I + 1
                    GO TO 90
                 ELSE
                    INDXP( K2+I-1 ) = PJ
                 END IF
              ELSE
                 INDXP( K2+I-1 ) = PJ
              END IF
              PJ = NJ
           ELSE
              K = K + 1
              DLAMDA( K ) = D( PJ )
              W( K ) = Z( PJ )
              INDXP( K ) = PJ
              PJ = NJ
           END IF
        END IF
        GO TO 80
    100 CONTINUE
  *
  *     Record the last eigenvalue.
  *
        K = K + 1
        DLAMDA( K ) = D( PJ )
        W( K ) = Z( PJ )
        INDXP( K ) = PJ
  *
  *     Count up the total number of the various types of columns, then
  *     form a permutation which positions the four column types into
  *     four uniform groups (although one or more of these groups may be
  *     empty).
  *
        DO 110 J = 1, 4
           CTOT( J ) = 0
    110 CONTINUE
        DO 120 J = 1, N
           CT = COLTYP( J )
           CTOT( CT ) = CTOT( CT ) + 1
    120 CONTINUE
  *
  *     PSM(*) = Position in SubMatrix (of types 1 through 4)
  *
        PSM( 1 ) = 1
        PSM( 2 ) = 1 + CTOT( 1 )
        PSM( 3 ) = PSM( 2 ) + CTOT( 2 )
        PSM( 4 ) = PSM( 3 ) + CTOT( 3 )
        K = N - CTOT( 4 )
  *
  *     Fill out the INDXC array so that the permutation which it induces
  *     will place all type-1 columns first, all type-2 columns next,
  *     then all type-3's, and finally all type-4's.
  *
        DO 130 J = 1, N
           JS = INDXP( J )
           CT = COLTYP( JS )
           INDX( PSM( CT ) ) = JS
           INDXC( PSM( CT ) ) = J
           PSM( CT ) = PSM( CT ) + 1
    130 CONTINUE
  *
  *     Sort the eigenvalues and corresponding eigenvectors into DLAMDA
  *     and Q2 respectively.  The eigenvalues/vectors which were not
  *     deflated go into the first K slots of DLAMDA and Q2 respectively,
  *     while those which were deflated go into the last N - K slots.
  *
        I = 1
        IQ1 = 1
        IQ2 = 1 + ( CTOT( 1 )+CTOT( 2 ) )*N1
        DO 140 J = 1, CTOT( 1 )
           JS = INDX( I )
           CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
           Z( I ) = D( JS )
           I = I + 1
           IQ1 = IQ1 + N1
    140 CONTINUE
  *
        DO 150 J = 1, CTOT( 2 )
           JS = INDX( I )
           CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
           CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
           Z( I ) = D( JS )
           I = I + 1
           IQ1 = IQ1 + N1
           IQ2 = IQ2 + N2
    150 CONTINUE
  *
        DO 160 J = 1, CTOT( 3 )
           JS = INDX( I )
           CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
           Z( I ) = D( JS )
           I = I + 1
           IQ2 = IQ2 + N2
    160 CONTINUE
  *
        IQ1 = IQ2
        DO 170 J = 1, CTOT( 4 )
           JS = INDX( I )
           CALL DCOPY( N, Q( 1, JS ), 1, Q2( IQ2 ), 1 )
           IQ2 = IQ2 + N
           Z( I ) = D( JS )
           I = I + 1
    170 CONTINUE
  *
  *     The deflated eigenvalues and their corresponding vectors go back
  *     into the last N - K slots of D and Q respectively.
  *
        IF( K.LT.N ) THEN
           CALL DLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N,
       $                Q( 1, K+1 ), LDQ )
           CALL DCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 )
        END IF
  *
  *     Copy CTOT into COLTYP for referencing in DLAED3.
  *
        DO 180 J = 1, 4
           COLTYP( J ) = CTOT( J )
    180 CONTINUE
  *
    190 CONTINUE
        RETURN
  *
  *     End of DLAED2
  *
        END
  