*> \brief \b SLAED2 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/
*
*> \htmlonly
*> Download SLAED2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaed2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slaed2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaed2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMBDA, W,
* Q2, INDX, INDXC, INDXP, COLTYP, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDQ, N, N1
* REAL RHO
* ..
* .. Array Arguments ..
* INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
* $ INDXQ( * )
* REAL D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
* $ W( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \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 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.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is 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.
*> \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 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.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is 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.
*> \endverbatim
*>
*> \param[out] DLAMBDA
*> \verbatim
*> DLAMBDA is REAL array, dimension (N)
*> A copy of the first K eigenvalues which will be used by
*> SLAED3 to form the secular equation.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is REAL array, dimension (N)
*> The first k values of the final deflation-altered z-vector
*> which will be passed to SLAED3.
*> \endverbatim
*>
*> \param[out] Q2
*> \verbatim
*> Q2 is 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.
*> \endverbatim
*>
*> \param[out] INDX
*> \verbatim
*> INDX is INTEGER array, dimension (N)
*> The permutation used to sort the contents of DLAMBDA 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.
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee
*>
* =====================================================================
SUBROUTINE SLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMBDA, W,
$ Q2, INDX, INDXC, INDXP, COLTYP, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDQ, N, N1
REAL RHO
* ..
* .. Array Arguments ..
INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
$ INDXQ( * )
REAL D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
$ W( * ), Z( * )
* ..
*
* =====================================================================
*
* .. 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
DLAMBDA( I ) = D( INDXQ( I ) )
20 CONTINUE
CALL SLAMRG( N1, N2, DLAMBDA, 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 )
DLAMBDA( J ) = D( I )
IQ2 = IQ2 + N
40 CONTINUE
CALL SLACPY( 'A', N, N, Q2, N, Q, LDQ )
CALL SCOPY( N, DLAMBDA, 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
DLAMBDA( 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
DLAMBDA( 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 DLAMBDA
* and Q2 respectively. The eigenvalues/vectors which were not
* 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
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.
*
IF( K.LT.N ) THEN
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 )
END IF
*
* 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