SUBROUTINE MB03VY( N, P, ILO, IHI, A, LDA1, LDA2, TAU, LDTAU,
$ DWORK, LDWORK, INFO )
C
C PURPOSE
C
C To generate the real orthogonal matrices Q_1, Q_2, ..., Q_p,
C which are defined as the product of ihi-ilo elementary reflectors
C of order n, as returned by SLICOT Library routine MB03VD:
C
C Q_j = H_j(ilo) H_j(ilo+1) . . . H_j(ihi-1).
C
C ARGUMENTS
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the matrices Q_1, Q_2, ..., Q_p. N >= 0.
C
C P (input) INTEGER
C The number p of transformation matrices. P >= 1.
C
C ILO (input) INTEGER
C IHI (input) INTEGER
C The values of the indices ilo and ihi, respectively, used
C in the previous call of the SLICOT Library routine MB03VD.
C 1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.
C
C A (input/output) DOUBLE PRECISION array, dimension
C (LDA1,LDA2,N)
C On entry, the leading N-by-N strictly lower triangular
C part of A(*,*,j) must contain the vectors which define the
C elementary reflectors used for reducing A_j, as returned
C by SLICOT Library routine MB03VD, j = 1, ..., p.
C On exit, the leading N-by-N part of A(*,*,j) contains the
C N-by-N orthogonal matrix Q_j, j = 1, ..., p.
C
C LDA1 INTEGER
C The first leading dimension of the array A.
C LDA1 >= max(1,N).
C
C LDA2 INTEGER
C The second leading dimension of the array A.
C LDA2 >= max(1,N).
C
C TAU (input) DOUBLE PRECISION array, dimension (LDTAU,P)
C The leading N-1 elements in the j-th column must contain
C the scalar factors of the elementary reflectors used to
C form the matrix Q_j, as returned by SLICOT Library routine
C MB03VD.
C
C LDTAU INTEGER
C The leading dimension of the array TAU.
C LDTAU >= max(1,N-1).
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK. LDWORK >= MAX(1,N).
C For optimum performance LDWORK should be larger.
C
C If LDWORK = -1, then a workspace query is assumed;
C the routine only calculates the optimal size of the
C DWORK array, returns this value as the first entry of
C the DWORK array, and no error message related to LDWORK
C is issued by XERBLA.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C METHOD
C
C Each matrix Q_j is generated as the product of the elementary
C reflectors used for reducing A_j. Standard LAPACK routines for
C Hessenberg and QR decompositions are used.
C
C REFERENCES
C
C [1] Bojanczyk, A.W., Golub, G. and Van Dooren, P.
C The periodic Schur decomposition: algorithms and applications.
C Proc. of the SPIE Conference (F.T. Luk, Ed.), 1770, pp. 31-42,
C 1992.
C
C [2] Sreedhar, J. and Van Dooren, P.
C Periodic Schur form and some matrix equations.
C Proc. of the Symposium on the Mathematical Theory of Networks
C and Systems (MTNS'93), Regensburg, Germany (U. Helmke,
C R. Mennicken and J. Saurer, Eds.), Vol. 1, pp. 339-362, 1994.
C
C NUMERICAL ASPECTS
C
C The algorithm is numerically stable.
C
C CONTRIBUTOR
C
C V. Sima, Katholieke Univ. Leuven, Belgium, and A. Varga,
C German Aerospace Center, DLR Oberpfaffenhofen, February 1999.
C Partly based on the routine PSHTR by A. Varga
C (DLR Oberpfaffenhofen), November 26, 1995.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, Feb. 2004,
C July 2012, June 2022.
C
C KEYWORDS
C
C Hessenberg form, orthogonal transformation, periodic systems,
C similarity transformation, triangular form.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
C
C .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA1, LDA2, LDTAU, LDWORK, N, P
C ..
C .. Array Arguments ..
DOUBLE PRECISION A( LDA1, LDA2, * ), DWORK( * ), TAU( LDTAU, * )
C ..
C .. Local Scalars ..
LOGICAL LQUERY
INTEGER J, NH, WRKOPT
C ..
C .. External Subroutines ..
EXTERNAL DLASET, DORGHR, DORGQR, XERBLA
C ..
C .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
C ..
C .. Executable Statements ..
C
C Test the input scalar arguments.
C
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( P.LT.1 ) THEN
INFO = -2
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -3
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -4
ELSE IF( LDA1.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDA2.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDTAU.LT.MAX( 1, N-1 ) ) THEN
INFO = -9
ELSE
NH = IHI - ILO + 1
LQUERY = LDWORK.EQ.-1
IF( LQUERY ) THEN
CALL DORGHR( N, ILO, IHI, A, LDA1, TAU, DWORK, -1, INFO )
WRKOPT = MAX( 1, N, INT( DWORK( 1 ) ) )
IF ( NH.GT.1 ) THEN
CALL DORGQR( NH, NH, NH-1, A, LDA1, TAU, DWORK, -1, INFO)
WRKOPT = MAX( WRKOPT, INT( DWORK( 1 ) ) )
END IF
END IF
IF( LDWORK.LT.MAX( 1, N ) .AND. .NOT. LQUERY )
$ INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'MB03VY', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( N.EQ.0 ) THEN
DWORK( 1 ) = ONE
RETURN
ELSE IF( LQUERY ) THEN
DWORK( 1 ) = WRKOPT
RETURN
END IF
C
C Generate the orthogonal matrix Q_1.
C
CALL DORGHR( N, ILO, IHI, A, LDA1, TAU, DWORK, LDWORK, INFO )
WRKOPT = INT( DWORK( 1 ) )
C
DO 20 J = 2, P
C
C Generate the orthogonal matrix Q_j.
C Set the first ILO-1 and the last N-IHI rows and columns of Q_j
C to those of the unit matrix.
C
CALL DLASET( 'Full', N, ILO-1, ZERO, ONE, A( 1, 1, J ), LDA1 )
CALL DLASET( 'Full', ILO-1, NH, ZERO, ZERO, A( 1, ILO, J ),
$ LDA1 )
IF ( NH.GT.1 ) THEN
CALL DORGQR( NH, NH, NH-1, A( ILO, ILO, J ), LDA1,
$ TAU( ILO, J ), DWORK, LDWORK, INFO )
ELSE
A( ILO, ILO, J ) = ONE
END IF
IF ( IHI.LT.N ) THEN
CALL DLASET( 'Full', N-IHI, NH, ZERO, ZERO,
$ A( IHI+1, ILO, J ), LDA1 )
CALL DLASET( 'Full', IHI, N-IHI, ZERO, ZERO,
$ A( 1, IHI+1, J ), LDA1 )
CALL DLASET( 'Full', N-IHI, N-IHI, ZERO, ONE,
$ A( IHI+1, IHI+1, J ), LDA1 )
END IF
20 CONTINUE
C
DWORK( 1 ) = MAX( WRKOPT, INT( DWORK( 1 ) ) )
RETURN
C
C *** Last line of MB03VY ***
END