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*> \brief \b DLAED0 used by sstedc. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed0.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED0( ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER ICOMPQ, INFO, LDQ, LDQS, N, QSIZ
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), Q( LDQ, * ), QSTORE( LDQS, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED0 computes all eigenvalues and corresponding eigenvectors of a
*> symmetric tridiagonal matrix using the divide and conquer method.
*> \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.
*> = 2: Compute eigenvalues and eigenvectors of tridiagonal
*> matrix.
*> \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] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the main diagonal of the tridiagonal matrix.
*> On exit, its eigenvalues.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The off-diagonal elements of the tridiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> On entry, Q must contain an N-by-N orthogonal matrix.
*> If ICOMPQ = 0 Q is not referenced.
*> If ICOMPQ = 1 On entry, Q is a subset of the columns of the
*> orthogonal matrix used to reduce the full
*> matrix to tridiagonal form corresponding to
*> the subset of the full matrix which is being
*> decomposed at this time.
*> If ICOMPQ = 2 On entry, Q will be the identity matrix.
*> On exit, Q contains the eigenvectors of the
*> tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. If eigenvectors are
*> desired, then LDQ >= max(1,N). In any case, LDQ >= 1.
*> \endverbatim
*>
*> \param[out] QSTORE
*> \verbatim
*> QSTORE is DOUBLE PRECISION array, dimension (LDQS, N)
*> Referenced only when ICOMPQ = 1. Used to store parts of
*> the eigenvector matrix when the updating matrix multiplies
*> take place.
*> \endverbatim
*>
*> \param[in] LDQS
*> \verbatim
*> LDQS is INTEGER
*> The leading dimension of the array QSTORE. If ICOMPQ = 1,
*> then LDQS >= max(1,N). In any case, LDQS >= 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array,
*> If ICOMPQ = 0 or 1, the dimension of WORK must be at least
*> 1 + 3*N + 2*N*lg N + 3*N**2
*> ( lg( N ) = smallest integer k
*> such that 2^k >= N )
*> If ICOMPQ = 2, the dimension of WORK must be at least
*> 4*N + N**2.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array,
*> If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
*> 6 + 6*N + 5*N*lg N.
*> ( lg( N ) = smallest integer k
*> such that 2^k >= N )
*> If ICOMPQ = 2, the dimension of IWORK must be at least
*> 3 + 5*N.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: The algorithm failed to compute an eigenvalue while
*> working on the submatrix lying in rows and columns
*> INFO/(N+1) through mod(INFO,N+1).
*> \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 DLAED0( ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS,
$ WORK, IWORK, 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 ICOMPQ, INFO, LDQ, LDQS, N, QSIZ
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E( * ), Q( LDQ, * ), QSTORE( LDQS, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.D0, ONE = 1.D0, TWO = 2.D0 )
* ..
* .. Local Scalars ..
INTEGER CURLVL, CURPRB, CURR, I, IGIVCL, IGIVNM,
$ IGIVPT, INDXQ, IPERM, IPRMPT, IQ, IQPTR, IWREM,
$ J, K, LGN, MATSIZ, MSD2, SMLSIZ, SMM1, SPM1,
$ SPM2, SUBMAT, SUBPBS, TLVLS
DOUBLE PRECISION TEMP
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DLACPY, DLAED1, DLAED7, DSTEQR,
$ XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, LOG, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.2 ) THEN
INFO = -1
ELSE IF( ( ICOMPQ.EQ.1 ) .AND. ( QSIZ.LT.MAX( 0, N ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDQS.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED0', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
SMLSIZ = ILAENV( 9, 'DLAED0', ' ', 0, 0, 0, 0 )
*
* Determine the size and placement of the submatrices, and save in
* the leading elements of IWORK.
*
IWORK( 1 ) = N
SUBPBS = 1
TLVLS = 0
10 CONTINUE
IF( IWORK( SUBPBS ).GT.SMLSIZ ) THEN
DO 20 J = SUBPBS, 1, -1
IWORK( 2*J ) = ( IWORK( J )+1 ) / 2
IWORK( 2*J-1 ) = IWORK( J ) / 2
20 CONTINUE
TLVLS = TLVLS + 1
SUBPBS = 2*SUBPBS
GO TO 10
END IF
DO 30 J = 2, SUBPBS
IWORK( J ) = IWORK( J ) + IWORK( J-1 )
30 CONTINUE
*
* Divide the matrix into SUBPBS submatrices of size at most SMLSIZ+1
* using rank-1 modifications (cuts).
*
SPM1 = SUBPBS - 1
DO 40 I = 1, SPM1
SUBMAT = IWORK( I ) + 1
SMM1 = SUBMAT - 1
D( SMM1 ) = D( SMM1 ) - ABS( E( SMM1 ) )
D( SUBMAT ) = D( SUBMAT ) - ABS( E( SMM1 ) )
40 CONTINUE
*
INDXQ = 4*N + 3
IF( ICOMPQ.NE.2 ) THEN
*
* Set up workspaces for eigenvalues only/accumulate new vectors
* routine
*
TEMP = LOG( DBLE( N ) ) / LOG( TWO )
LGN = INT( TEMP )
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IPRMPT = INDXQ + N + 1
IPERM = IPRMPT + N*LGN
IQPTR = IPERM + N*LGN
IGIVPT = IQPTR + N + 2
IGIVCL = IGIVPT + N*LGN
*
IGIVNM = 1
IQ = IGIVNM + 2*N*LGN
IWREM = IQ + N**2 + 1
*
* Initialize pointers
*
DO 50 I = 0, SUBPBS
IWORK( IPRMPT+I ) = 1
IWORK( IGIVPT+I ) = 1
50 CONTINUE
IWORK( IQPTR ) = 1
END IF
*
* Solve each submatrix eigenproblem at the bottom of the divide and
* conquer tree.
*
CURR = 0
DO 70 I = 0, SPM1
IF( I.EQ.0 ) THEN
SUBMAT = 1
MATSIZ = IWORK( 1 )
ELSE
SUBMAT = IWORK( I ) + 1
MATSIZ = IWORK( I+1 ) - IWORK( I )
END IF
IF( ICOMPQ.EQ.2 ) THEN
CALL DSTEQR( 'I', MATSIZ, D( SUBMAT ), E( SUBMAT ),
$ Q( SUBMAT, SUBMAT ), LDQ, WORK, INFO )
IF( INFO.NE.0 )
$ GO TO 130
ELSE
CALL DSTEQR( 'I', MATSIZ, D( SUBMAT ), E( SUBMAT ),
$ WORK( IQ-1+IWORK( IQPTR+CURR ) ), MATSIZ, WORK,
$ INFO )
IF( INFO.NE.0 )
$ GO TO 130
IF( ICOMPQ.EQ.1 ) THEN
CALL DGEMM( 'N', 'N', QSIZ, MATSIZ, MATSIZ, ONE,
$ Q( 1, SUBMAT ), LDQ, WORK( IQ-1+IWORK( IQPTR+
$ CURR ) ), MATSIZ, ZERO, QSTORE( 1, SUBMAT ),
$ LDQS )
END IF
IWORK( IQPTR+CURR+1 ) = IWORK( IQPTR+CURR ) + MATSIZ**2
CURR = CURR + 1
END IF
K = 1
DO 60 J = SUBMAT, IWORK( I+1 )
IWORK( INDXQ+J ) = K
K = K + 1
60 CONTINUE
70 CONTINUE
*
* Successively merge eigensystems of adjacent submatrices
* into eigensystem for the corresponding larger matrix.
*
* while ( SUBPBS > 1 )
*
CURLVL = 1
80 CONTINUE
IF( SUBPBS.GT.1 ) THEN
SPM2 = SUBPBS - 2
DO 90 I = 0, SPM2, 2
IF( I.EQ.0 ) THEN
SUBMAT = 1
MATSIZ = IWORK( 2 )
MSD2 = IWORK( 1 )
CURPRB = 0
ELSE
SUBMAT = IWORK( I ) + 1
MATSIZ = IWORK( I+2 ) - IWORK( I )
MSD2 = MATSIZ / 2
CURPRB = CURPRB + 1
END IF
*
* Merge lower order eigensystems (of size MSD2 and MATSIZ - MSD2)
* into an eigensystem of size MATSIZ.
* DLAED1 is used only for the full eigensystem of a tridiagonal
* matrix.
* DLAED7 handles the cases in which eigenvalues only or eigenvalues
* and eigenvectors of a full symmetric matrix (which was reduced to
* tridiagonal form) are desired.
*
IF( ICOMPQ.EQ.2 ) THEN
CALL DLAED1( MATSIZ, D( SUBMAT ), Q( SUBMAT, SUBMAT ),
$ LDQ, IWORK( INDXQ+SUBMAT ),
$ E( SUBMAT+MSD2-1 ), MSD2, WORK,
$ IWORK( SUBPBS+1 ), INFO )
ELSE
CALL DLAED7( ICOMPQ, MATSIZ, QSIZ, TLVLS, CURLVL, CURPRB,
$ D( SUBMAT ), QSTORE( 1, SUBMAT ), LDQS,
$ IWORK( INDXQ+SUBMAT ), E( SUBMAT+MSD2-1 ),
$ MSD2, WORK( IQ ), IWORK( IQPTR ),
$ IWORK( IPRMPT ), IWORK( IPERM ),
$ IWORK( IGIVPT ), IWORK( IGIVCL ),
$ WORK( IGIVNM ), WORK( IWREM ),
$ IWORK( SUBPBS+1 ), INFO )
END IF
IF( INFO.NE.0 )
$ GO TO 130
IWORK( I / 2+1 ) = IWORK( I+2 )
90 CONTINUE
SUBPBS = SUBPBS / 2
CURLVL = CURLVL + 1
GO TO 80
END IF
*
* end while
*
* Re-merge the eigenvalues/vectors which were deflated at the final
* merge step.
*
IF( ICOMPQ.EQ.1 ) THEN
DO 100 I = 1, N
J = IWORK( INDXQ+I )
WORK( I ) = D( J )
CALL DCOPY( QSIZ, QSTORE( 1, J ), 1, Q( 1, I ), 1 )
100 CONTINUE
CALL DCOPY( N, WORK, 1, D, 1 )
ELSE IF( ICOMPQ.EQ.2 ) THEN
DO 110 I = 1, N
J = IWORK( INDXQ+I )
WORK( I ) = D( J )
CALL DCOPY( N, Q( 1, J ), 1, WORK( N*I+1 ), 1 )
110 CONTINUE
CALL DCOPY( N, WORK, 1, D, 1 )
CALL DLACPY( 'A', N, N, WORK( N+1 ), N, Q, LDQ )
ELSE
DO 120 I = 1, N
J = IWORK( INDXQ+I )
WORK( I ) = D( J )
120 CONTINUE
CALL DCOPY( N, WORK, 1, D, 1 )
END IF
GO TO 140
*
130 CONTINUE
INFO = SUBMAT*( N+1 ) + SUBMAT + MATSIZ - 1
*
140 CONTINUE
RETURN
*
* End of DLAED0
*
END
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