dlaed2.f

          SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
         $                   Q2, INDX, INDXC, INDXP, COLTYP, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. 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( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  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.
    *
    *  Arguments
    *  =========
    *
    *  K      (output) INTEGER
    *         The number of non-deflated eigenvalues, and the order of the
    *         related secular equation. 0 <= K <=N.
    *
    *  N      (input) INTEGER
    *         The dimension of the symmetric tridiagonal matrix.  N >= 0.
    *
    *  N1     (input) INTEGER
    *         The location of the last eigenvalue in the leading sub-matrix.
    *         min(1,N) <= N1 <= N/2.
    *
    *  D      (input/output) 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.
    *
    *  Q      (input/output) 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.
    *
    *  LDQ    (input) INTEGER
    *         The leading dimension of the array Q.  LDQ >= max(1,N).
    *
    *  INDXQ  (input/output) 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.
    *
    *  RHO    (input/output) 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.
    *
    *  Z      (input) 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.
    *
    *  DLAMDA (output) DOUBLE PRECISION array, dimension (N)
    *         A copy of the first K eigenvalues which will be used by
    *         DLAED3 to form the secular equation.
    *
    *  W      (output) DOUBLE PRECISION array, dimension (N)
    *         The first k values of the final deflation-altered z-vector
    *         which will be passed to DLAED3.
    *
    *  Q2     (output) 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.
    *
    *  INDX   (workspace) INTEGER array, dimension (N)
    *         The permutation used to sort the contents of DLAMDA into
    *         ascending order.
    *
    *  INDXC  (output) 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.
    *
    *  INDXP  (workspace) 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.
    *
    *  COLTYP (workspace/output) 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.
    *
    *  INFO   (output) INTEGER
    *          = 0:  successful exit.
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Jeff Rutter, Computer Science Division, University of California
    *     at Berkeley, USA
    *  Modified by Francoise Tisseur, University of Tennessee.
    *
    *  =====================================================================
    *
    *     .. 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.
    *
          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 )
    *
    *     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