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*> \brief \b DSTEBZ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEDC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstedc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstedc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstedc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
* LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPZ
* INTEGER INFO, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEDC computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric tridiagonal matrix using the divide and conquer method.
*> The eigenvectors of a full or band real symmetric matrix can also be
*> found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this
*> matrix to tridiagonal form.
*>
*> This code makes very mild assumptions about floating point
*> arithmetic. It will work on machines with a guard digit in
*> add/subtract, or on those binary machines without guard digits
*> which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
*> It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none. See DLAED3 for details.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': Compute eigenvalues only.
*> = 'I': Compute eigenvectors of tridiagonal matrix also.
*> = 'V': Compute eigenvectors of original dense symmetric
*> matrix also. On entry, Z contains the orthogonal
*> matrix used to reduce the original matrix to
*> tridiagonal form.
*> \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 diagonal elements of the tridiagonal matrix.
*> On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the subdiagonal elements of the tridiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> On entry, if COMPZ = 'V', then Z contains the orthogonal
*> matrix used in the reduction to tridiagonal form.
*> On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
*> orthonormal eigenvectors of the original symmetric matrix,
*> and if COMPZ = 'I', Z contains the orthonormal eigenvectors
*> of the symmetric tridiagonal matrix.
*> If COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1.
*> If eigenvectors are desired, then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array,
*> dimension (LWORK)
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
*> If COMPZ = 'V' and N > 1 then LWORK must be at least
*> ( 1 + 3*N + 2*N*lg N + 4*N**2 ),
*> where lg( N ) = smallest integer k such
*> that 2**k >= N.
*> If COMPZ = 'I' and N > 1 then LWORK must be at least
*> ( 1 + 4*N + N**2 ).
*> Note that for COMPZ = 'I' or 'V', then if N is less than or
*> equal to the minimum divide size, usually 25, then LWORK need
*> only be max(1,2*(N-1)).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
*> If COMPZ = 'V' and N > 1 then LIWORK must be at least
*> ( 6 + 6*N + 5*N*lg N ).
*> If COMPZ = 'I' and N > 1 then LIWORK must be at least
*> ( 3 + 5*N ).
*> Note that for COMPZ = 'I' or 'V', then if N is less than or
*> equal to the minimum divide size, usually 25, then LIWORK
*> need only be 1.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal size of the IWORK array,
*> returns this value as the first entry of the IWORK array, and
*> no error message related to LIWORK is issued by XERBLA.
*> \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 November 2011
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee
*>
* =====================================================================
SUBROUTINE DSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
$ LIWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN,
$ LWMIN, M, SMLSIZ, START, STOREZ, STRTRW
DOUBLE PRECISION EPS, ORGNRM, P, TINY
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, ILAENV, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLAED0, DLASCL, DLASET, DLASRT,
$ DSTEQR, DSTERF, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, LOG, MAX, MOD, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ICOMPZ = 0
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ICOMPZ = 2
ELSE
ICOMPZ = -1
END IF
IF( ICOMPZ.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ( LDZ.LT.1 ) .OR.
$ ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
INFO = -6
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Compute the workspace requirements
*
SMLSIZ = ILAENV( 9, 'DSTEDC', ' ', 0, 0, 0, 0 )
IF( N.LE.1 .OR. ICOMPZ.EQ.0 ) THEN
LIWMIN = 1
LWMIN = 1
ELSE IF( N.LE.SMLSIZ ) THEN
LIWMIN = 1
LWMIN = 2*( N - 1 )
ELSE
LGN = INT( LOG( DBLE( N ) )/LOG( TWO ) )
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( ICOMPZ.EQ.1 ) THEN
LWMIN = 1 + 3*N + 2*N*LGN + 4*N**2
LIWMIN = 6 + 6*N + 5*N*LGN
ELSE IF( ICOMPZ.EQ.2 ) THEN
LWMIN = 1 + 4*N + N**2
LIWMIN = 3 + 5*N
END IF
END IF
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT. LQUERY ) THEN
INFO = -8
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT. LQUERY ) THEN
INFO = -10
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEDC', -INFO )
RETURN
ELSE IF (LQUERY) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( N.EQ.1 ) THEN
IF( ICOMPZ.NE.0 )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* If the following conditional clause is removed, then the routine
* will use the Divide and Conquer routine to compute only the
* eigenvalues, which requires (3N + 3N**2) real workspace and
* (2 + 5N + 2N lg(N)) integer workspace.
* Since on many architectures DSTERF is much faster than any other
* algorithm for finding eigenvalues only, it is used here
* as the default. If the conditional clause is removed, then
* information on the size of workspace needs to be changed.
*
* If COMPZ = 'N', use DSTERF to compute the eigenvalues.
*
IF( ICOMPZ.EQ.0 ) THEN
CALL DSTERF( N, D, E, INFO )
GO TO 50
END IF
*
* If N is smaller than the minimum divide size (SMLSIZ+1), then
* solve the problem with another solver.
*
IF( N.LE.SMLSIZ ) THEN
*
CALL DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
ELSE
*
* If COMPZ = 'V', the Z matrix must be stored elsewhere for later
* use.
*
IF( ICOMPZ.EQ.1 ) THEN
STOREZ = 1 + N*N
ELSE
STOREZ = 1
END IF
*
IF( ICOMPZ.EQ.2 ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
END IF
*
* Scale.
*
ORGNRM = DLANST( 'M', N, D, E )
IF( ORGNRM.EQ.ZERO )
$ GO TO 50
*
EPS = DLAMCH( 'Epsilon' )
*
START = 1
*
* while ( START <= N )
*
10 CONTINUE
IF( START.LE.N ) THEN
*
* Let FINISH be the position of the next subdiagonal entry
* such that E( FINISH ) <= TINY or FINISH = N if no such
* subdiagonal exists. The matrix identified by the elements
* between START and FINISH constitutes an independent
* sub-problem.
*
FINISH = START
20 CONTINUE
IF( FINISH.LT.N ) THEN
TINY = EPS*SQRT( ABS( D( FINISH ) ) )*
$ SQRT( ABS( D( FINISH+1 ) ) )
IF( ABS( E( FINISH ) ).GT.TINY ) THEN
FINISH = FINISH + 1
GO TO 20
END IF
END IF
*
* (Sub) Problem determined. Compute its size and solve it.
*
M = FINISH - START + 1
IF( M.EQ.1 ) THEN
START = FINISH + 1
GO TO 10
END IF
IF( M.GT.SMLSIZ ) THEN
*
* Scale.
*
ORGNRM = DLANST( 'M', M, D( START ), E( START ) )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M, 1, D( START ), M,
$ INFO )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M-1, 1, E( START ),
$ M-1, INFO )
*
IF( ICOMPZ.EQ.1 ) THEN
STRTRW = 1
ELSE
STRTRW = START
END IF
CALL DLAED0( ICOMPZ, N, M, D( START ), E( START ),
$ Z( STRTRW, START ), LDZ, WORK( 1 ), N,
$ WORK( STOREZ ), IWORK, INFO )
IF( INFO.NE.0 ) THEN
INFO = ( INFO / ( M+1 )+START-1 )*( N+1 ) +
$ MOD( INFO, ( M+1 ) ) + START - 1
GO TO 50
END IF
*
* Scale back.
*
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, M, 1, D( START ), M,
$ INFO )
*
ELSE
IF( ICOMPZ.EQ.1 ) THEN
*
* Since QR won't update a Z matrix which is larger than
* the length of D, we must solve the sub-problem in a
* workspace and then multiply back into Z.
*
CALL DSTEQR( 'I', M, D( START ), E( START ), WORK, M,
$ WORK( M*M+1 ), INFO )
CALL DLACPY( 'A', N, M, Z( 1, START ), LDZ,
$ WORK( STOREZ ), N )
CALL DGEMM( 'N', 'N', N, M, M, ONE,
$ WORK( STOREZ ), N, WORK, M, ZERO,
$ Z( 1, START ), LDZ )
ELSE IF( ICOMPZ.EQ.2 ) THEN
CALL DSTEQR( 'I', M, D( START ), E( START ),
$ Z( START, START ), LDZ, WORK, INFO )
ELSE
CALL DSTERF( M, D( START ), E( START ), INFO )
END IF
IF( INFO.NE.0 ) THEN
INFO = START*( N+1 ) + FINISH
GO TO 50
END IF
END IF
*
START = FINISH + 1
GO TO 10
END IF
*
* endwhile
*
* If the problem split any number of times, then the eigenvalues
* will not be properly ordered. Here we permute the eigenvalues
* (and the associated eigenvectors) into ascending order.
*
IF( M.NE.N ) THEN
IF( ICOMPZ.EQ.0 ) THEN
*
* Use Quick Sort
*
CALL DLASRT( 'I', N, D, INFO )
*
ELSE
*
* Use Selection Sort to minimize swaps of eigenvectors
*
DO 40 II = 2, N
I = II - 1
K = I
P = D( I )
DO 30 J = II, N
IF( D( J ).LT.P ) THEN
K = J
P = D( J )
END IF
30 CONTINUE
IF( K.NE.I ) THEN
D( K ) = D( I )
D( I ) = P
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
END IF
40 CONTINUE
END IF
END IF
END IF
*
50 CONTINUE
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DSTEDC
*
END
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