*> \brief \b DGGHRD
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
* LDQ, Z, LDZ, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPQ, COMPZ
* INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGHRD reduces a pair of real matrices (A,B) to generalized upper
*> Hessenberg form using orthogonal transformations, where A is a
*> general matrix and B is upper triangular. The form of the
*> generalized eigenvalue problem is
*> A*x = lambda*B*x,
*> and B is typically made upper triangular by computing its QR
*> factorization and moving the orthogonal matrix Q to the left side
*> of the equation.
*>
*> This subroutine simultaneously reduces A to a Hessenberg matrix H:
*> Q**T*A*Z = H
*> and transforms B to another upper triangular matrix T:
*> Q**T*B*Z = T
*> in order to reduce the problem to its standard form
*> H*y = lambda*T*y
*> where y = Z**T*x.
*>
*> The orthogonal matrices Q and Z are determined as products of Givens
*> rotations. They may either be formed explicitly, or they may be
*> postmultiplied into input matrices Q1 and Z1, so that
*>
*> Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
*>
*> Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
*>
*> If Q1 is the orthogonal matrix from the QR factorization of B in the
*> original equation A*x = lambda*B*x, then DGGHRD reduces the original
*> problem to generalized Hessenberg form.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPQ
*> \verbatim
*> COMPQ is CHARACTER*1
*> = 'N': do not compute Q;
*> = 'I': Q is initialized to the unit matrix, and the
*> orthogonal matrix Q is returned;
*> = 'V': Q must contain an orthogonal matrix Q1 on entry,
*> and the product Q1*Q is returned.
*> \endverbatim
*>
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': do not compute Z;
*> = 'I': Z is initialized to the unit matrix, and the
*> orthogonal matrix Z is returned;
*> = 'V': Z must contain an orthogonal matrix Z1 on entry,
*> and the product Z1*Z is returned.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*>
*> ILO and IHI mark the rows and columns of A which are to be
*> reduced. It is assumed that A is already upper triangular
*> in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
*> normally set by a previous call to DGGBAL; otherwise they
*> should be set to 1 and N respectively.
*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the N-by-N general matrix to be reduced.
*> On exit, the upper triangle and the first subdiagonal of A
*> are overwritten with the upper Hessenberg matrix H, and the
*> rest is set to zero.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the N-by-N upper triangular matrix B.
*> On exit, the upper triangular matrix T = Q**T B Z. The
*> elements below the diagonal are set to zero.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> On entry, if COMPQ = 'V', the orthogonal matrix Q1,
*> typically from the QR factorization of B.
*> On exit, if COMPQ='I', the orthogonal matrix Q, and if
*> COMPQ = 'V', the product Q1*Q.
*> Not referenced if COMPQ='N'.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q.
*> LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> On entry, if COMPZ = 'V', the orthogonal matrix Z1.
*> On exit, if COMPZ='I', the orthogonal matrix Z, and if
*> COMPZ = 'V', the product Z1*Z.
*> Not referenced if COMPZ='N'.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z.
*> LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
*> \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 doubleOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> This routine reduces A to Hessenberg and B to triangular form by
*> an unblocked reduction, as described in _Matrix_Computations_,
*> by Golub and Van Loan (Johns Hopkins Press.)
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
$ LDQ, Z, LDZ, 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 ..
CHARACTER COMPQ, COMPZ
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ILQ, ILZ
INTEGER ICOMPQ, ICOMPZ, JCOL, JROW
DOUBLE PRECISION C, S, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARTG, DLASET, DROT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Decode COMPQ
*
IF( LSAME( COMPQ, 'N' ) ) THEN
ILQ = .FALSE.
ICOMPQ = 1
ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 2
ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 3
ELSE
ICOMPQ = 0
END IF
*
* Decode COMPZ
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ILZ = .FALSE.
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 2
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 3
ELSE
ICOMPZ = 0
END IF
*
* Test the input parameters.
*
INFO = 0
IF( ICOMPQ.LE.0 ) THEN
INFO = -1
ELSE IF( ICOMPZ.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ILO.LT.1 ) THEN
INFO = -4
ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( ( ILQ .AND. LDQ.LT.N ) .OR. LDQ.LT.1 ) THEN
INFO = -11
ELSE IF( ( ILZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN
INFO = -13
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGHRD', -INFO )
RETURN
END IF
*
* Initialize Q and Z if desired.
*
IF( ICOMPQ.EQ.3 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
IF( ICOMPZ.EQ.3 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
* Zero out lower triangle of B
*
DO 20 JCOL = 1, N - 1
DO 10 JROW = JCOL + 1, N
B( JROW, JCOL ) = ZERO
10 CONTINUE
20 CONTINUE
*
* Reduce A and B
*
DO 40 JCOL = ILO, IHI - 2
*
DO 30 JROW = IHI, JCOL + 2, -1
*
* Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL)
*
TEMP = A( JROW-1, JCOL )
CALL DLARTG( TEMP, A( JROW, JCOL ), C, S,
$ A( JROW-1, JCOL ) )
A( JROW, JCOL ) = ZERO
CALL DROT( N-JCOL, A( JROW-1, JCOL+1 ), LDA,
$ A( JROW, JCOL+1 ), LDA, C, S )
CALL DROT( N+2-JROW, B( JROW-1, JROW-1 ), LDB,
$ B( JROW, JROW-1 ), LDB, C, S )
IF( ILQ )
$ CALL DROT( N, Q( 1, JROW-1 ), 1, Q( 1, JROW ), 1, C, S )
*
* Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1)
*
TEMP = B( JROW, JROW )
CALL DLARTG( TEMP, B( JROW, JROW-1 ), C, S,
$ B( JROW, JROW ) )
B( JROW, JROW-1 ) = ZERO
CALL DROT( IHI, A( 1, JROW ), 1, A( 1, JROW-1 ), 1, C, S )
CALL DROT( JROW-1, B( 1, JROW ), 1, B( 1, JROW-1 ), 1, C,
$ S )
IF( ILZ )
$ CALL DROT( N, Z( 1, JROW ), 1, Z( 1, JROW-1 ), 1, C, S )
30 CONTINUE
40 CONTINUE
*
RETURN
*
* End of DGGHRD
*
END