*> \brief \b CLAED8 used by CSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CLAED8 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/claed8.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/claed8.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/claed8.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMBDA,
* Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR,
* GIVCOL, GIVNUM, INFO )
*
* .. Scalar Arguments ..
* INTEGER CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ
* REAL RHO
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
* $ INDXQ( * ), PERM( * )
* REAL D( * ), DLAMBDA( * ), GIVNUM( 2, * ), W( * ),
* $ Z( * )
* COMPLEX Q( LDQ, * ), Q2( LDQ2, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CLAED8 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 element 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
*> Contains the number of non-deflated eigenvalues.
*> This is the order of the related secular equation.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*> QSIZ is INTEGER
*> The dimension of the unitary matrix used to reduce
*> the dense or band matrix to tridiagonal form.
*> QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is COMPLEX array, dimension (LDQ,N)
*> On entry, Q contains the eigenvectors of the partially solved
*> system which has been previously updated in matrix
*> multiplies with other partially solved eigensystems.
*> 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] 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] RHO
*> \verbatim
*> RHO is REAL
*> Contains the off diagonal element associated with the rank-1
*> cut which originally split the two submatrices which are now
*> being recombined. RHO is modified during the computation to
*> the value required by SLAED3.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*> CUTPNT is INTEGER
*> Contains the location of the last eigenvalue in the leading
*> sub-matrix. MIN(1,N) <= CUTPNT <= N.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is REAL array, dimension (N)
*> On input this vector contains the updating vector (the last
*> row of the first sub-eigenvector matrix and the first row of
*> the second sub-eigenvector matrix). The contents of Z are
*> destroyed during the updating process.
*> \endverbatim
*>
*> \param[out] DLAMBDA
*> \verbatim
*> DLAMBDA is REAL array, dimension (N)
*> Contains a copy of the first K eigenvalues which will be used
*> by SLAED3 to form the secular equation.
*> \endverbatim
*>
*> \param[out] Q2
*> \verbatim
*> Q2 is COMPLEX array, dimension (LDQ2,N)
*> If ICOMPQ = 0, Q2 is not referenced. Otherwise,
*> Contains a copy of the first K eigenvectors which will be used
*> by SLAED7 in a matrix multiply (SGEMM) to update the new
*> eigenvectors.
*> \endverbatim
*>
*> \param[in] LDQ2
*> \verbatim
*> LDQ2 is INTEGER
*> The leading dimension of the array Q2. LDQ2 >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is REAL array, dimension (N)
*> This will hold the first k values of the final
*> deflation-altered z-vector and will be passed to SLAED3.
*> \endverbatim
*>
*> \param[out] INDXP
*> \verbatim
*> INDXP is INTEGER array, dimension (N)
*> This will contain the permutation used to place deflated
*> values of D at the end of the array. On output INDXP(1:K)
*> points to the nondeflated D-values and INDXP(K+1:N)
*> points to the deflated eigenvalues.
*> \endverbatim
*>
*> \param[out] INDX
*> \verbatim
*> INDX is INTEGER array, dimension (N)
*> This will contain the permutation used to sort the contents of
*> D into ascending order.
*> \endverbatim
*>
*> \param[in] INDXQ
*> \verbatim
*> INDXQ is INTEGER array, dimension (N)
*> This contains 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 CUTPNT
*> added to their values in order to be accurate.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*> PERM is INTEGER array, dimension (N)
*> Contains the permutations (from deflation and sorting) to be
*> applied to each eigenblock.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER
*> Contains the number of Givens rotations which took place in
*> this subproblem.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension (2, N)
*> Each pair of numbers indicates a pair of columns to take place
*> in a Givens rotation.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*> GIVNUM is REAL array, dimension (2, N)
*> Each number indicates the S value to be used in the
*> corresponding Givens rotation.
*> \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 complexOTHERcomputational
*
* =====================================================================
SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMBDA,
$ Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR,
$ GIVCOL, GIVNUM, 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 CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ
REAL RHO
* ..
* .. Array Arguments ..
INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
$ INDXQ( * ), PERM( * )
REAL D( * ), DLAMBDA( * ), GIVNUM( 2, * ), W( * ),
$ Z( * )
COMPLEX Q( LDQ, * ), Q2( LDQ2, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL MONE, ZERO, ONE, TWO, EIGHT
PARAMETER ( MONE = -1.0E0, ZERO = 0.0E0, ONE = 1.0E0,
$ TWO = 2.0E0, EIGHT = 8.0E0 )
* ..
* .. Local Scalars ..
INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
REAL C, EPS, S, T, TAU, TOL
* ..
* .. External Functions ..
INTEGER ISAMAX
REAL SLAMCH, SLAPY2
EXTERNAL ISAMAX, SLAMCH, SLAPY2
* ..
* .. External Subroutines ..
EXTERNAL CCOPY, CLACPY, CSROT, SCOPY, SLAMRG, 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( QSIZ.LT.N ) THEN
INFO = -3
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN
INFO = -8
ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CLAED8', -INFO )
RETURN
END IF
*
* Need to initialize GIVPTR to O here in case of quick exit
* to prevent an unspecified code behavior (usually sigfault)
* when IWORK array on entry to *stedc is not zeroed
* (or at least some IWORK entries which used in *laed7 for GIVPTR).
*
GIVPTR = 0
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
N1 = CUTPNT
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
*
T = ONE / SQRT( TWO )
DO 10 J = 1, N
INDX( J ) = J
10 CONTINUE
CALL SSCAL( N, T, Z, 1 )
RHO = ABS( TWO*RHO )
*
* Sort the eigenvalues into increasing order
*
DO 20 I = CUTPNT + 1, N
INDXQ( I ) = INDXQ( I ) + CUTPNT
20 CONTINUE
DO 30 I = 1, N
DLAMBDA( I ) = D( INDXQ( I ) )
W( I ) = Z( INDXQ( I ) )
30 CONTINUE
I = 1
J = CUTPNT + 1
CALL SLAMRG( N1, N2, DLAMBDA, 1, 1, INDX )
DO 40 I = 1, N
D( I ) = DLAMBDA( INDX( I ) )
Z( I ) = W( INDX( I ) )
40 CONTINUE
*
* Calculate the allowable deflation tolerance
*
IMAX = ISAMAX( N, Z, 1 )
JMAX = ISAMAX( N, D, 1 )
EPS = SLAMCH( 'Epsilon' )
TOL = EIGHT*EPS*ABS( D( JMAX ) )
*
* 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
DO 50 J = 1, N
PERM( J ) = INDXQ( INDX( J ) )
CALL CCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
50 CONTINUE
CALL CLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ), LDQ )
RETURN
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.
*
K = 0
K2 = N + 1
DO 60 J = 1, N
IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
INDXP( K2 ) = J
IF( J.EQ.N )
$ GO TO 100
ELSE
JLAM = J
GO TO 70
END IF
60 CONTINUE
70 CONTINUE
J = J + 1
IF( J.GT.N )
$ GO TO 90
IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
INDXP( K2 ) = J
ELSE
*
* Check if eigenvalues are close enough to allow deflation.
*
S = Z( JLAM )
C = Z( J )
*
* Find sqrt(a**2+b**2) without overflow or
* destructive underflow.
*
TAU = SLAPY2( C, S )
T = D( J ) - D( JLAM )
C = C / TAU
S = -S / TAU
IF( ABS( T*C*S ).LE.TOL ) THEN
*
* Deflation is possible.
*
Z( J ) = TAU
Z( JLAM ) = ZERO
*
* Record the appropriate Givens rotation
*
GIVPTR = GIVPTR + 1
GIVCOL( 1, GIVPTR ) = INDXQ( INDX( JLAM ) )
GIVCOL( 2, GIVPTR ) = INDXQ( INDX( J ) )
GIVNUM( 1, GIVPTR ) = C
GIVNUM( 2, GIVPTR ) = S
CALL CSROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1,
$ Q( 1, INDXQ( INDX( J ) ) ), 1, C, S )
T = D( JLAM )*C*C + D( J )*S*S
D( J ) = D( JLAM )*S*S + D( J )*C*C
D( JLAM ) = T
K2 = K2 - 1
I = 1
80 CONTINUE
IF( K2+I.LE.N ) THEN
IF( D( JLAM ).LT.D( INDXP( K2+I ) ) ) THEN
INDXP( K2+I-1 ) = INDXP( K2+I )
INDXP( K2+I ) = JLAM
I = I + 1
GO TO 80
ELSE
INDXP( K2+I-1 ) = JLAM
END IF
ELSE
INDXP( K2+I-1 ) = JLAM
END IF
JLAM = J
ELSE
K = K + 1
W( K ) = Z( JLAM )
DLAMBDA( K ) = D( JLAM )
INDXP( K ) = JLAM
JLAM = J
END IF
END IF
GO TO 70
90 CONTINUE
*
* Record the last eigenvalue.
*
K = K + 1
W( K ) = Z( JLAM )
DLAMBDA( K ) = D( JLAM )
INDXP( K ) = JLAM
*
100 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.
*
DO 110 J = 1, N
JP = INDXP( J )
DLAMBDA( J ) = D( JP )
PERM( J ) = INDXQ( INDX( JP ) )
CALL CCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
110 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 SCOPY( N-K, DLAMBDA( K+1 ), 1, D( K+1 ), 1 )
CALL CLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2, Q( 1, K+1 ),
$ LDQ )
END IF
*
RETURN
*
* End of CLAED8
*
END