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*> \brief \b DLAED8 used by sstedc. 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 DLAED8 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed8.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed8.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed8.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
* CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
* GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
* .. Scalar Arguments ..
* INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
* $ QSIZ
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
* $ INDXQ( * ), PERM( * )
* DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ),
* $ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED8 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[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> = 0: Compute eigenvalues only.
*> = 1: Compute eigenvectors of original dense symmetric matrix
*> also. On entry, Q contains the orthogonal matrix used
*> to reduce the original matrix to tridiagonal form.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> The number of non-deflated eigenvalues, and 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 orthogonal matrix used to reduce
*> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the eigenvalues of the two submatrices to be
*> combined. On exit, 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)
*> If ICOMPQ = 0, Q is not referenced. Otherwise,
*> 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] 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 CUTPNT added to
*> their values in order to be accurate.
*> \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] CUTPNT
*> \verbatim
*> CUTPNT is INTEGER
*> The location of the last eigenvalue in the leading
*> sub-matrix. min(1,N) <= CUTPNT <= N.
*> \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 are 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] Q2
*> \verbatim
*> Q2 is DOUBLE PRECISION array, dimension (LDQ2,N)
*> If ICOMPQ = 0, Q2 is not referenced. Otherwise,
*> a copy of the first K eigenvectors which will be used by
*> DLAED7 in a matrix multiply (DGEMM) 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 DOUBLE PRECISION array, dimension (N)
*> The first k values of the final deflation-altered z-vector and
*> will be passed to DLAED3.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*> PERM is INTEGER array, dimension (N)
*> The permutations (from deflation and sorting) to be applied
*> to each eigenblock.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER
*> 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 DOUBLE PRECISION array, dimension (2, N)
*> Each number indicates the S value to be used in the
*> corresponding Givens rotation.
*> \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] INDX
*> \verbatim
*> INDX is INTEGER array, dimension (N)
*> The permutation used to sort the contents of D into ascending
*> order.
*> \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 September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
$ CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
$ GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
$ QSIZ
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
$ INDXQ( * ), PERM( * )
DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ),
$ Q( LDQ, * ), Q2( LDQ2, * ), 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 Scalars ..
*
INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
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( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
INFO = -4
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN
INFO = -10
ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN
INFO = -14
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED8', -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 DSCAL( 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 DSCAL( 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
DLAMDA( I ) = D( INDXQ( I ) )
W( I ) = Z( INDXQ( I ) )
30 CONTINUE
I = 1
J = CUTPNT + 1
CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDX )
DO 40 I = 1, N
D( I ) = DLAMDA( INDX( I ) )
Z( I ) = W( INDX( I ) )
40 CONTINUE
*
* Calculate the allowable deflation tolerence
*
IMAX = IDAMAX( N, Z, 1 )
JMAX = IDAMAX( N, D, 1 )
EPS = DLAMCH( '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
IF( ICOMPQ.EQ.0 ) THEN
DO 50 J = 1, N
PERM( J ) = INDXQ( INDX( J ) )
50 CONTINUE
ELSE
DO 60 J = 1, N
PERM( J ) = INDXQ( INDX( J ) )
CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
60 CONTINUE
CALL DLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ),
$ LDQ )
END IF
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 70 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 110
ELSE
JLAM = J
GO TO 80
END IF
70 CONTINUE
80 CONTINUE
J = J + 1
IF( J.GT.N )
$ GO TO 100
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 = DLAPY2( 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
IF( ICOMPQ.EQ.1 ) THEN
CALL DROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1,
$ Q( 1, INDXQ( INDX( J ) ) ), 1, C, S )
END IF
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
90 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 90
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 )
DLAMDA( K ) = D( JLAM )
INDXP( K ) = JLAM
JLAM = J
END IF
END IF
GO TO 80
100 CONTINUE
*
* Record the last eigenvalue.
*
K = K + 1
W( K ) = Z( JLAM )
DLAMDA( K ) = D( JLAM )
INDXP( K ) = JLAM
*
110 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.
*
IF( ICOMPQ.EQ.0 ) THEN
DO 120 J = 1, N
JP = INDXP( J )
DLAMDA( J ) = D( JP )
PERM( J ) = INDXQ( INDX( JP ) )
120 CONTINUE
ELSE
DO 130 J = 1, N
JP = INDXP( J )
DLAMDA( J ) = D( JP )
PERM( J ) = INDXQ( INDX( JP ) )
CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
130 CONTINUE
END IF
*
* 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
IF( ICOMPQ.EQ.0 ) THEN
CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
ELSE
CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
CALL DLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2,
$ Q( 1, K+1 ), LDQ )
END IF
END IF
*
RETURN
*
* End of DLAED8
*
END
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