SUBROUTINE MB03QY( N, L, A, LDA, U, LDU, E1, E2, INFO )
C
C PURPOSE
C
C To compute the eigenvalues of a selected 2-by-2 diagonal block
C of an upper quasi-triangular matrix, to reduce the selected block
C to the standard form and to split the block in the case of real
C eigenvalues by constructing an orthogonal transformation UT.
C This transformation is applied to A (by similarity) and to
C another matrix U from the right.
C
C ARGUMENTS
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the matrices A and UT. N >= 2.
C
C L (input) INTEGER
C Specifies the position of the block. 1 <= L < 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 upper quasi-triangular matrix A whose
C selected 2-by-2 diagonal block is to be processed.
C On exit, the leading N-by-N part of this array contains
C the upper quasi-triangular matrix A after its selected
C block has been splitt and/or put in the LAPACK standard
C form.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= N.
C
C U (input/output) DOUBLE PRECISION array, dimension (LDU,N)
C On entry, the leading N-by-N part of this array must
C contain a transformation matrix U.
C On exit, the leading N-by-N part of this array contains
C U*UT, where UT is the transformation matrix used to
C split and/or standardize the selected block.
C
C LDU INTEGER
C The leading dimension of array U. LDU >= N.
C
C E1, E2 (output) DOUBLE PRECISION
C E1 and E2 contain either the real eigenvalues or the real
C and positive imaginary parts, respectively, of the complex
C eigenvalues of the selected 2-by-2 diagonal block of A.
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 Let A1 = ( A(L,L) A(L,L+1) )
C ( A(L+1,L) A(L+1,L+1) )
C be the specified 2-by-2 diagonal block of matrix A.
C If the eigenvalues of A1 are complex, then they are computed and
C stored in E1 and E2, where the real part is stored in E1 and the
C positive imaginary part in E2. The 2-by-2 block is reduced if
C necessary to the standard form, such that A(L,L) = A(L+1,L+1), and
C A(L,L+1) and A(L+1,L) have oposite signs. If the eigenvalues are
C real, the 2-by-2 block is reduced to an upper triangular form such
C that ABS(A(L,L)) >= ABS(A(L+1,L+1)).
C In both cases, an orthogonal rotation U1' is constructed such that
C U1'*A1*U1 has the appropriate form. Let UT be an extension of U1
C to an N-by-N orthogonal matrix, using identity submatrices. Then A
C is replaced by UT'*A*UT and the contents of array U is U * UT.
C
C CONTRIBUTOR
C
C A. Varga, German Aerospace Center, DLR Oberpfaffenhofen,
C March 1998. Based on the RASP routine SPLITB.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Eigenvalues, orthogonal transformation, real Schur form,
C similarity transformation.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
C .. Scalar Arguments ..
INTEGER INFO, L, LDA, LDU, N
DOUBLE PRECISION E1, E2
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), U(LDU,*)
C .. Local Scalars ..
INTEGER L1
DOUBLE PRECISION EW1, EW2, CS, SN
C .. External Subroutines ..
EXTERNAL DLANV2, DROT, XERBLA
C .. Executable Statements ..
C
INFO = 0
C
C Test the input scalar arguments.
C
IF( N.LT.2 ) THEN
INFO = -1
ELSE IF( L.LT.1 .OR. L.GE.N ) THEN
INFO = -2
ELSE IF( LDA.LT.N ) THEN
INFO = -4
ELSE IF( LDU.LT.N ) THEN
INFO = -6
END IF
C
IF( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'MB03QY', -INFO )
RETURN
END IF
C
C Compute the eigenvalues and the elements of the Givens
C transformation.
C
L1 = L + 1
CALL DLANV2( A(L,L), A(L,L1), A(L1,L), A(L1,L1), E1, E2,
$ EW1, EW2, CS, SN )
IF( E2.EQ.ZERO ) E2 = EW1
C
C Apply the transformation to A.
C
IF( L1.LT.N )
$ CALL DROT( N-L1, A(L,L1+1), LDA, A(L1,L1+1), LDA, CS, SN )
CALL DROT( L-1, A(1,L), 1, A(1,L1), 1, CS, SN )
C
C Accumulate the transformation in U.
C
CALL DROT( N, U(1,L), 1, U(1,L1), 1, CS, SN )
C
RETURN
C *** Last line of MB03QY ***
END