```      SUBROUTINE SLAED2( 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
REAL               RHO
*     ..
*     .. Array Arguments ..
INTEGER            COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
\$                   INDXQ( * )
REAL               D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
\$                   W( * ), Z( * )
*     ..
*
*  Purpose
*  =======
*
*  SLAED2 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) REAL 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) REAL 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) REAL
*         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
*         SLAED3.
*
*  Z      (input) REAL 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) REAL array, dimension (N)
*         A copy of the first K eigenvalues which will be used by
*         SLAED3 to form the secular equation.
*
*  W      (output) REAL array, dimension (N)
*         The first k values of the final deflation-altered z-vector
*         which will be passed to SLAED3.
*
*  Q2     (output) REAL array, dimension (N1**2+(N-N1)**2)
*         A copy of the first K eigenvectors which will be used by
*         SLAED3 in a matrix multiply (SGEMM) 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 ..
REAL               MONE, ZERO, ONE, TWO, EIGHT
PARAMETER          ( MONE = -1.0E0, ZERO = 0.0E0, ONE = 1.0E0,
\$                   TWO = 2.0E0, EIGHT = 8.0E0 )
*     ..
*     .. Local Arrays ..
INTEGER            CTOT( 4 ), PSM( 4 )
*     ..
*     .. Local Scalars ..
INTEGER            CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1,
\$                   N2, NJ, PJ
REAL               C, EPS, S, T, TAU, TOL
*     ..
*     .. External Functions ..
INTEGER            ISAMAX
REAL               SLAMCH, SLAPY2
EXTERNAL           ISAMAX, SLAMCH, SLAPY2
*     ..
*     .. External Subroutines ..
EXTERNAL           SCOPY, SLACPY, SLAMRG, SROT, SSCAL, 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( 'SLAED2', -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 SSCAL( 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 SSCAL( 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 SLAMRG( N1, N2, DLAMDA, 1, 1, INDXC )
DO 30 I = 1, N
INDX( I ) = INDXQ( INDXC( I ) )
30 CONTINUE
*
*     Calculate the allowable deflation tolerance
*
IMAX = ISAMAX( N, Z, 1 )
JMAX = ISAMAX( N, D, 1 )
EPS = SLAMCH( '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 SCOPY( N, Q( 1, I ), 1, Q2( IQ2 ), 1 )
DLAMDA( J ) = D( I )
IQ2 = IQ2 + N
40    CONTINUE
CALL SLACPY( 'A', N, N, Q2, N, Q, LDQ )
CALL SCOPY( 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 = SLAPY2( 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 SROT( 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 SCOPY( 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 SCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
CALL SCOPY( 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 SCOPY( 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 SCOPY( 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 SLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N, Q( 1, K+1 ), LDQ )
CALL SCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 )
*
*     Copy CTOT into COLTYP for referencing in SLAED3.
*
DO 180 J = 1, 4
COLTYP( J ) = CTOT( J )
180 CONTINUE
*
190 CONTINUE
RETURN
*
*     End of SLAED2
*
END

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