SUBROUTINE TB01ND( JOBU, UPLO, N, P, A, LDA, C, LDC, U, LDU,
$ DWORK, INFO )
C
C PURPOSE
C
C To reduce the pair (A,C) to lower or upper observer Hessenberg
C form using (and optionally accumulating) unitary state-space
C transformations.
C
C ARGUMENTS
C
C Mode Parameters
C
C JOBU CHARACTER*1
C Indicates whether the user wishes to accumulate in a
C matrix U the unitary state-space transformations for
C reducing the system, as follows:
C = 'N': Do not form U;
C = 'I': U is initialized to the unit matrix and the
C unitary transformation matrix U is returned;
C = 'U': The given matrix U is updated by the unitary
C transformations used in the reduction.
C
C UPLO CHARACTER*1
C Indicates whether the user wishes the pair (A,C) to be
C reduced to upper or lower observer Hessenberg form as
C follows:
C = 'U': Upper observer Hessenberg form;
C = 'L': Lower observer Hessenberg form.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The actual state dimension, i.e. the order of the
C matrix A. N >= 0.
C
C P (input) INTEGER
C The actual output dimension, i.e. the number of rows of
C the matrix C. 0 <= P <= N.
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On entry, the leading N-by-N part of this array must
C contain the state transition matrix A to be transformed.
C On exit, the leading N-by-N part of this array contains
C the transformed state transition matrix U' * A * U.
C The annihilated elements are set to zero.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
C On entry, the leading P-by-N part of this array must
C contain the output matrix C to be transformed.
C On exit, the leading P-by-N part of this array contains
C the transformed output matrix C * U.
C The annihilated elements are set to zero.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C U (input/output) DOUBLE PRECISION array, dimension (LDU,*)
C On entry, if JOBU = 'U', then the leading N-by-N part of
C this array must contain a given matrix U (e.g. from a
C previous call to another SLICOT routine), and on exit, the
C leading N-by-N part of this array contains the product of
C the input matrix U and the state-space transformation
C matrix which reduces the given pair to observer Hessenberg
C form.
C On exit, if JOBU = 'I', then the leading N-by-N part of
C this array contains the matrix of accumulated unitary
C similarity transformations which reduces the given pair
C to observer Hessenberg form.
C If JOBU = 'N', the array U is not referenced and can be
C supplied as a dummy array (i.e. set parameter LDU = 1 and
C declare this array to be U(1,1) in the calling program).
C
C LDU INTEGER
C The leading dimension of array U. If JOBU = 'U' or
C JOBU = 'I', LDU >= MAX(1,N); if JOBU = 'N', LDU >= 1.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (MAX(N,P-1))
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 routine computes a unitary state-space transformation U, which
C reduces the pair (A,C) to one of the following observer Hessenberg
C forms:
C
C N
C |* . . . . . . *|
C |. .|
C |. .|
C |. .| N
C |* .|
C |U'AU| | . .|
C |----| = | . .|
C |CU | | * . . . *|
C -------------------
C | * . . *|
C | . .| P
C | . .|
C | *|
C
C if UPLO = 'U', or
C
C N
C |* |
C |. . |
C |. . | P
C |* . . * |
C |CU | -------------------
C |----| = |* . . . * |
C |U'AU| |. . |
C |. . |
C |. *|
C |. .| N
C |. .|
C |. .|
C |* . . . . . . *|
C
C if UPLO = 'L'.
C
C If P >= N, then the matrix CU is trapezoidal and U'AU is full.
C If P = 0, but N > 0, the array A is unchanged on exit.
C
C REFERENCES
C
C [1] Van Dooren, P. and Verhaegen, M.H.G.
C On the use of unitary state-space transformations.
C In : Contemporary Mathematics on Linear Algebra and its Role
C in Systems Theory, 47, AMS, Providence, 1985.
C
C NUMERICAL ASPECTS
C
C The algorithm requires O((N + P) x N**2) operations and is
C backward stable (see [1]).
C
C CONTRIBUTORS
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, Dec. 1996.
C Supersedes Release 2.0 routine TB01BD by M. Vanbegin, and
C P. Van Dooren, Philips Research Laboratory, Brussels, Belgium.
C
C REVISIONS
C
C February 1997, April 2021.
C
C KEYWORDS
C
C Controllability, observer Hessenberg form, orthogonal
C transformation, unitary transformation.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDC, LDU, N, P
CHARACTER JOBU, UPLO
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), C(LDC,*), DWORK(*), U(LDU,*)
C .. Local Scalars ..
LOGICAL LJOBA, LJOBI, LUPLO
INTEGER II, J, N1, NJ, P1, PAR1, PAR2, PAR3, PAR4, PAR5,
$ PAR6
DOUBLE PRECISION DZ
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DLARFG, DLASET, DLATZM, XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX, MIN
C .. Executable Statements ..
C
INFO = 0
LUPLO = LSAME( UPLO, 'U' )
LJOBI = LSAME( JOBU, 'I' )
LJOBA = LJOBI.OR.LSAME( JOBU, 'U' )
C
C Test the input scalar arguments.
C
IF( .NOT.LJOBA .AND. .NOT.LSAME( JOBU, 'N' ) ) THEN
INFO = -1
ELSE IF( .NOT.LUPLO .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( P.LT.0 .OR. P.GT.N ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
INFO = -8
ELSE IF( .NOT.LJOBA .AND. LDU.LT.1 .OR.
$ LJOBA .AND. LDU.LT.MAX( 1, N ) ) THEN
INFO = -10
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return
C
CALL XERBLA( 'TB01ND', -INFO )
RETURN
END IF
C
IF ( LJOBI ) THEN
C
C Initialize U to the identity matrix.
C
CALL DLASET( 'Full', N, N, ZERO, ONE, U, LDU )
END IF
C
C Quick return if possible.
C
IF ( N.EQ.0 .OR. P.EQ.0 )
$ RETURN
C
P1 = P + 1
N1 = N - 1
C
C Perform transformations involving both C and A.
C
DO 20 J = 1, MIN( P, N1 )
NJ = N - J
IF ( LUPLO ) THEN
PAR1 = P - J + 1
PAR2 = NJ + 1
PAR3 = 1
PAR4 = P - J
PAR5 = NJ
ELSE
PAR1 = J
PAR2 = J
PAR3 = J + 1
PAR4 = P
PAR5 = N
END IF
C
CALL DLARFG( NJ+1, C(PAR1,PAR2), C(PAR1,PAR3), LDC, DZ )
C
C Update A.
C
CALL DLATZM( 'Left', NJ+1, N, C(PAR1,PAR3), LDC, DZ, A(PAR2,1),
$ A(PAR3,1), LDA, DWORK )
CALL DLATZM( 'Right', N, NJ+1, C(PAR1,PAR3), LDC, DZ,
$ A(1,PAR2), A(1,PAR3), LDA, DWORK )
C
IF ( LJOBA ) THEN
C
C Update U.
C
CALL DLATZM( 'Right', N, NJ+1, C(PAR1,PAR3), LDC, DZ,
$ U(1,PAR2), U(1,PAR3), LDU, DWORK )
END IF
C
IF ( J.NE.P ) THEN
C
C Update C.
C
CALL DLATZM( 'Right', PAR4-PAR3+1, NJ+1, C(PAR1,PAR3), LDC,
$ DZ, C(PAR3,PAR2), C(PAR3,PAR3), LDC, DWORK )
END IF
C
DO 10 II = PAR3, PAR5
C(PAR1,II) = ZERO
10 CONTINUE
C
20 CONTINUE
C
DO 40 J = P1, N1
C
C Perform next transformations only involving A.
C
NJ = N - J
IF ( LUPLO ) THEN
PAR1 = N + P1 - J
PAR2 = NJ + 1
PAR3 = 1
PAR4 = NJ
PAR5 = 1
PAR6 = N + P - J
ELSE
PAR1 = J - P
PAR2 = J
PAR3 = J + 1
PAR4 = N
PAR5 = J - P + 1
PAR6 = N
END IF
C
IF ( NJ.GT.0 ) THEN
C
CALL DLARFG( NJ+1, A(PAR1,PAR2), A(PAR1,PAR3), LDA, DZ )
C
C Update A.
C
CALL DLATZM( 'Left', NJ+1, N, A(PAR1,PAR3), LDA, DZ,
$ A(PAR2,1), A(PAR3,1), LDA, DWORK )
CALL DLATZM( 'Right', PAR6-PAR5+1, NJ+1, A(PAR1,PAR3), LDA,
$ DZ, A(PAR5,PAR2), A(PAR5,PAR3), LDA, DWORK )
C
IF ( LJOBA ) THEN
C
C Update U.
C
CALL DLATZM( 'Right', N, NJ+1, A(PAR1,PAR3), LDA, DZ,
$ U(1,PAR2), U(1,PAR3), LDU, DWORK )
END IF
C
DO 30 II = PAR3, PAR4
A(PAR1,II) = ZERO
30 CONTINUE
C
END IF
C
40 CONTINUE
C
RETURN
C *** Last line of TB01ND ***
END