SUBROUTINE MB04TV( UPDATZ, N, NRA, NCA, IFIRA, IFICA, A, LDA, E,
$ LDE, Z, LDZ )
C
C PURPOSE
C
C To reduce a submatrix A(k) of A to upper triangular form by column
C Givens rotations only.
C Here A(k) = A(IFIRA:ma,IFICA:na) where ma = IFIRA - 1 + NRA,
C na = IFICA - 1 + NCA.
C Matrix A(k) is assumed to have full row rank on entry. Hence, no
C pivoting is done during the reduction process. See Algorithm 2.3.1
C and Remark 2.3.4 in [1].
C The constructed column transformations are also applied to matrix
C E(k) = E(1:IFIRA-1,IFICA:na).
C Note that in E columns are transformed with the same column
C indices as in A, but with row indices different from those in A.
C
C ARGUMENTS
C
C Mode Parameters
C
C UPDATZ LOGICAL
C Indicates whether the user wishes to accumulate in a
C matrix Z the orthogonal column transformations, as
C follows:
C = .FALSE.: Do not form Z;
C = .TRUE.: The given matrix Z is updated by the orthogonal
C column transformations used in the reduction.
C
C Input/Output Parameters
C
C N (input) INTEGER
C Number of columns of A and E. N >= 0.
C
C NRA (input) INTEGER
C Number of rows in A to be transformed. 0 <= NRA <= LDA.
C
C NCA (input) INTEGER
C Number of columns in A to be transformed. 0 <= NCA <= N.
C
C IFIRA (input) INTEGER
C Index of the first row in A to be transformed.
C
C IFICA (input) INTEGER
C Index of the first column in A to be transformed.
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On entry, the elements of A(IFIRA:ma,IFICA:na) must
C contain the submatrix A(k) of full row rank to be reduced
C to upper triangular form.
C On exit, it contains the transformed matrix A.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,NRA).
C
C E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
C On entry, the elements of E(1:IFIRA-1,IFICA:na) must
C contain the submatrix E(k).
C On exit, it contains the transformed matrix E.
C
C LDE INTEGER
C The leading dimension of array E. LDE >= MAX(1,IFIRA-1).
C
C Z (input/output) DOUBLE PRECISION array, dimension (LDZ,*)
C On entry, if UPDATZ = .TRUE., then the leading N-by-N
C part of this array must contain a given matrix Z (e.g.
C from a previous call to another SLICOT routine), and on
C exit, the leading N-by-N part of this array contains the
C product of the input matrix Z and the column
C transformation matrix that has transformed the columns of
C the matrices A and E.
C If UPDATZ = .FALSE., the array Z is not referenced and
C can be supplied as a dummy array (i.e. set parameter
C LDZ = 1 and declare this array to be Z(1,1) in the calling
C program).
C
C LDZ INTEGER
C The leading dimension of array Z. If UPDATZ = .TRUE.,
C LDZ >= MAX(1,N); if UPDATZ = .FALSE., LDZ >= 1.
C
C REFERENCES
C
C [1] Beelen, Th.
C New Algorithms for Computing the Kronecker structure of a
C Pencil with Applications to Systems and Control Theory.
C Ph.D.Thesis, Eindhoven University of Technology,
C The Netherlands, 1987.
C
C NUMERICAL ASPECTS
C
C The algorithm is backward stable.
C
C CONTRIBUTOR
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, Apr. 1997.
C Supersedes Release 2.0 routine MB04FV by Th.G.J. Beelen,
C Philips Glass Eindhoven, Holland.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Generalized eigenvalue problem, orthogonal transformation,
C staircase form.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
C .. Scalar Arguments ..
LOGICAL UPDATZ
INTEGER IFICA, IFIRA, LDA, LDE, LDZ, N, NCA, NRA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), E(LDE,*), Z(LDZ,*)
C .. Local Scalars ..
INTEGER I, IFIRA1, J, JPVT
DOUBLE PRECISION SC, SS
C .. External Subroutines ..
EXTERNAL DROT, DROTG
C .. Executable Statements ..
C
IF ( N.LE.0 .OR. NRA.LE.0 .OR. NCA.LE.0 )
$ RETURN
IFIRA1 = IFIRA - 1
JPVT = IFICA + NCA
C
DO 40 I = IFIRA1 + NRA, IFIRA, -1
JPVT = JPVT - 1
C
DO 20 J = JPVT - 1, IFICA, -1
C
C Determine the Givens transformation on columns j and jpvt
C to annihilate A(i,j). Apply the transformation to these
C columns from rows 1 up to i.
C Apply the transformation also to the E-matrix (from rows 1
C up to ifira1).
C Update column transformation matrix Z, if needed.
C
CALL DROTG( A(I,JPVT), A(I,J), SC, SS )
CALL DROT( I-1, A(1,JPVT), 1, A(1,J), 1, SC, SS )
A(I,J) = ZERO
CALL DROT( IFIRA1, E(1,JPVT), 1, E(1,J), 1, SC, SS )
IF( UPDATZ ) CALL DROT( N, Z(1,JPVT), 1, Z(1,J), 1, SC, SS )
20 CONTINUE
C
40 CONTINUE
C
RETURN
C *** Last line of MB04TV ***
END