SUBROUTINE MB03VD( N, P, ILO, IHI, A, LDA1, LDA2, TAU, LDTAU,
$ DWORK, INFO )
C
C PURPOSE
C
C To reduce a product of p real general matrices A = A_1*A_2*...*A_p
C to upper Hessenberg form, H = H_1*H_2*...*H_p, where H_1 is
C upper Hessenberg, and H_2, ..., H_p are upper triangular, by using
C orthogonal similarity transformations on A,
C
C Q_1' * A_1 * Q_2 = H_1,
C Q_2' * A_2 * Q_3 = H_2,
C ...
C Q_p' * A_p * Q_1 = H_p.
C
C ARGUMENTS
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the square matrices A_1, A_2, ..., A_p.
C N >= 0.
C
C P (input) INTEGER
C The number of matrices in the product A_1*A_2*...*A_p.
C P >= 1.
C
C ILO (input) INTEGER
C IHI (input) INTEGER
C It is assumed that all matrices A_j, j = 2, ..., p, are
C already upper triangular in rows and columns 1:ILO-1 and
C IHI+1:N, and A_1 is upper Hessenberg in rows and columns
C 1:ILO-1 and IHI+1:N, with A_1(ILO,ILO-1) = 0 (unless
C ILO = 1), and A_1(IHI+1,IHI) = 0 (unless IHI = N).
C If this is not the case, ILO and IHI should be set to 1
C and N, respectively.
C 1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.
C
C A (input/output) DOUBLE PRECISION array, dimension
C (LDA1,LDA2,P)
C On entry, the leading N-by-N-by-P part of this array must
C contain the matrices of factors to be reduced;
C specifically, A(*,*,j) must contain A_j, j = 1, ..., p.
C On exit, the leading N-by-N upper triangle and the first
C subdiagonal of A(*,*,1) contain the upper Hessenberg
C matrix H_1, and the elements below the first subdiagonal,
C with the first column of the array TAU represent the
C orthogonal matrix Q_1 as a product of elementary
C reflectors. See FURTHER COMMENTS.
C For j > 1, the leading N-by-N upper triangle of A(*,*,j)
C contains the upper triangular matrix H_j, and the elements
C below the diagonal, with the j-th column of the array TAU
C represent the orthogonal matrix Q_j as a product of
C elementary reflectors. See FURTHER COMMENTS.
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 (output) DOUBLE PRECISION array, dimension (LDTAU,P)
C The leading N-1 elements in the j-th column contain the
C scalar factors of the elementary reflectors used to form
C the matrix Q_j, j = 1, ..., P. See FURTHER COMMENTS.
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 (N)
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 The algorithm consists in ihi-ilo major steps. In each such
C step i, ilo <= i <= ihi-1, the subdiagonal elements in the i-th
C column of A_j are annihilated using a Householder transformation
C from the left, which is also applied to A_(j-1) from the right,
C for j = p:-1:2. Then, the elements below the subdiagonal of the
C i-th column of A_1 are annihilated, and the Householder
C transformation is also applied to A_p from the right.
C See FURTHER COMMENTS.
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 FURTHER COMMENTS
C
C Each matrix Q_j is represented as a product of (ihi-ilo)
C elementary reflectors,
C
C Q_j = H_j(ilo) H_j(ilo+1) . . . H_j(ihi-1).
C
C Each H_j(i), i = ilo, ..., ihi-1, has the form
C
C H_j(i) = I - tau_j * v_j * v_j',
C
C where tau_j is a real scalar, and v_j is a real vector with
C v_j(1:i) = 0, v_j(i+1) = 1 and v_j(ihi+1:n) = 0; v_j(i+2:ihi)
C is stored on exit in A_j(i+2:ihi,i), and tau_j in TAU(i,j).
C
C The contents of A_1 are illustrated by the following example
C for n = 7, ilo = 2, and ihi = 6:
C
C on entry on exit
C
C ( a a a a a a a ) ( a h h h h h a )
C ( 0 a a a a a a ) ( 0 h h h h h a )
C ( 0 a a a a a a ) ( 0 h h h h h h )
C ( 0 a a a a a a ) ( 0 v2 h h h h h )
C ( 0 a a a a a a ) ( 0 v2 v3 h h h h )
C ( 0 a a a a a a ) ( 0 v2 v3 v4 h h h )
C ( 0 0 0 0 0 0 a ) ( 0 0 0 0 0 0 a )
C
C where a denotes an element of the original matrix A_1, h denotes
C a modified element of the upper Hessenberg matrix H_1, and vi
C denotes an element of the vector defining H_1(i).
C
C The contents of A_j, j > 1, are illustrated by the following
C example for n = 7, ilo = 2, and ihi = 6:
C
C on entry on exit
C
C ( a a a a a a a ) ( a h h h h h a )
C ( 0 a a a a a a ) ( 0 h h h h h h )
C ( 0 a a a a a a ) ( 0 v2 h h h h h )
C ( 0 a a a a a a ) ( 0 v2 v3 h h h h )
C ( 0 a a a a a a ) ( 0 v2 v3 v4 h h h )
C ( 0 a a a a a a ) ( 0 v2 v3 v4 v5 h h )
C ( 0 0 0 0 0 0 a ) ( 0 0 0 0 0 0 a )
C
C where a denotes an element of the original matrix A_j, h denotes
C a modified element of the upper triangular matrix H_j, and vi
C denotes an element of the vector defining H_j(i). (The element
C (1,2) in A_p is also unchanged for this example.)
C
C Note that for P = 1, the LAPACK Library routine DGEHRD could be
C more efficient on some computer architectures than this routine
C (a BLAS 2 version).
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 PSHESS by A. Varga
C (DLR Oberpfaffenhofen), November 26, 1995.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Hessenberg form, orthogonal transformation, periodic systems,
C similarity transformation, triangular form.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
C ..
C .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA1, LDA2, LDTAU, N, P
C ..
C .. Array Arguments ..
DOUBLE PRECISION A( LDA1, LDA2, * ), DWORK( * ), TAU( LDTAU, * )
C ..
C .. Local Scalars ..
INTEGER I, I1, I2, J, NH
C ..
C .. Local Arrays ..
DOUBLE PRECISION DUMMY( 1 )
C ..
C .. External Subroutines ..
EXTERNAL DCOPY, DLARFG, MB04PY, XERBLA
C ..
C .. Intrinsic Functions ..
INTRINSIC 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
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'MB03VD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
NH = IHI - ILO + 1
IF ( NH.LE.1 )
$ RETURN
C
DUMMY( 1 ) = ZERO
C
DO 20 I = ILO, IHI - 1
I1 = I + 1
I2 = MIN( I+2, N )
C
DO 10 J = P, 2, -1
C
C Set the elements 1:ILO-1 and IHI:N-1 of TAU(*,J) to zero.
C
CALL DCOPY( ILO-1, DUMMY, 0, TAU( 1, J ), 1 )
IF ( IHI.LT.N )
$ CALL DCOPY( N-IHI, DUMMY, 0, TAU( IHI, J ), 1 )
C
C Compute elementary reflector H_j(i) to annihilate
C A_j(i+1:ihi,i).
C
CALL DLARFG( IHI-I+1, A( I, I, J ), A( I1, I, J ), 1,
$ TAU( I, J ) )
C
C Apply H_j(i) to A_(j-1)(1:ihi,i:ihi) from the right.
C
CALL MB04PY( 'Right', IHI, IHI-I+1, A( I1, I, J ),
$ TAU( I, J ), A( 1, I, J-1 ), LDA1, DWORK )
C
C Apply H_j(i) to A_j(i:ihi,i+1:n) from the left.
C
CALL MB04PY( 'Left', IHI-I+1, N-I, A( I1, I, J ),
$ TAU( I, J ), A( I, I1, J ), LDA1, DWORK )
10 CONTINUE
C
C Compute elementary reflector H_1(i) to annihilate
C A_1(i+2:ihi,i).
C
CALL DLARFG( IHI-I, A( I1, I, 1 ), A( I2, I, 1 ), 1,
$ TAU( I, 1 ) )
C
C Apply H_1(i) to A_p(1:ihi,i+1:ihi) from the right.
C
CALL MB04PY( 'Right', IHI, IHI-I, A( I2, I, 1 ), TAU( I, 1 ),
$ A( 1, I1, P ), LDA1, DWORK )
C
C Apply H_1(i) to A_1(i+1:ihi,i+1:n) from the left.
C
CALL MB04PY( 'Left', IHI-I, N-I, A( I2, I, 1 ), TAU( I, 1 ),
$ A( I1, I1, 1 ), LDA1, DWORK )
20 CONTINUE
C
RETURN
C
C *** Last line of MB03VD ***
END