SUBROUTINE MB03YT( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
$ CSR, SNR )
C
C PURPOSE
C
C To compute the periodic Schur factorization of a real 2-by-2
C matrix pair (A,B) where B is upper triangular. This routine
C computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
C SNR such that
C
C 1) if the pair (A,B) has two real eigenvalues, then
C
C [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
C [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
C
C [ b11 b12 ] := [ CSR SNR ] [ b11 b12 ] [ CSL -SNL ]
C [ 0 b22 ] [ -SNR CSR ] [ 0 b22 ] [ SNL CSL ],
C
C 2) if the pair (A,B) has a pair of complex conjugate eigenvalues,
C then
C
C [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
C [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
C
C [ b11 0 ] := [ CSR SNR ] [ b11 b12 ] [ CSL -SNL ]
C [ 0 b22 ] [ -SNR CSR ] [ 0 b22 ] [ SNL CSL ].
C
C This is a modified version of the LAPACK routine DLAGV2 for
C computing the real, generalized Schur decomposition of a
C two-by-two matrix pencil.
C
C ARGUMENTS
C
C Input/Output Parameters
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,2)
C On entry, the leading 2-by-2 part of this array must
C contain the matrix A.
C On exit, the leading 2-by-2 part of this array contains
C the matrix A of the pair in periodic Schur form.
C
C LDA INTEGER
C The leading dimension of the array A. LDA >= 2.
C
C B (input/output) DOUBLE PRECISION array, dimension (LDB,2)
C On entry, the leading 2-by-2 part of this array must
C contain the upper triangular matrix B.
C On exit, the leading 2-by-2 part of this array contains
C the matrix B of the pair in periodic Schur form.
C
C LDB INTEGER
C The leading dimension of the array B. LDB >= 2.
C
C ALPHAR (output) DOUBLE PRECISION array, dimension (2)
C ALPHAI (output) DOUBLE PRECISION array, dimension (2)
C BETA (output) DOUBLE PRECISION array, dimension (2)
C (ALPHAR(k)+i*ALPHAI(k))*BETA(k) are the eigenvalues of the
C pair (A,B), k=1,2, i = sqrt(-1). ALPHAI(1) >= 0.
C
C CSL (output) DOUBLE PRECISION
C The cosine of the first rotation matrix.
C
C SNL (output) DOUBLE PRECISION
C The sine of the first rotation matrix.
C
C CSR (output) DOUBLE PRECISION
C The cosine of the second rotation matrix.
C
C SNR (output) DOUBLE PRECISION
C The sine of the second rotation matrix.
C
C REFERENCES
C
C [1] Van Loan, C.
C Generalized Singular Values with Algorithms and Applications.
C Ph. D. Thesis, University of Michigan, 1973.
C
C CONTRIBUTORS
C
C D. Kressner, Technical Univ. Berlin, Germany, and
C P. Benner, Technical Univ. Chemnitz, Germany, December 2003.
C
C REVISIONS
C
C V. Sima, June 2008 (SLICOT version of the HAPACK routine DLAPV2).
C V. Sima, July 2008, May 2009.
C
C KEYWORDS
C
C Eigenvalue, periodic Schur form
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
INTEGER LDA, LDB
DOUBLE PRECISION CSL, CSR, SNL, SNR
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHAI(2), ALPHAR(2), B(LDB,*),
$ BETA(2)
C .. Local Scalars ..
DOUBLE PRECISION ANORM, BNORM, H1, H2, H3, QQ, R, RR, SAFMIN,
$ SCALE1, SCALE2, T, ULP, WI, WR1, WR2
C .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL DLAMCH, DLAPY2
C .. External Subroutines ..
EXTERNAL DLAG2, DLARTG, DLASV2, DROT
C .. Intrinsic Functions ..
INTRINSIC ABS, MAX
C
C .. Executable Statements ..
C
SAFMIN = DLAMCH( 'S' )
ULP = DLAMCH( 'P' )
C
C Scale A.
C
ANORM = MAX( ABS( A(1,1) ) + ABS( A(2,1) ),
$ ABS( A(1,2) ) + ABS( A(2,2) ), SAFMIN )
A(1,1) = A(1,1) / ANORM
A(1,2) = A(1,2) / ANORM
A(2,1) = A(2,1) / ANORM
A(2,2) = A(2,2) / ANORM
C
C Scale B.
C
BNORM = MAX( ABS( B(1,1) ), ABS( B(1,2) ) + ABS( B(2,2) ), SAFMIN)
B(1,1) = B(1,1) / BNORM
B(1,2) = B(1,2) / BNORM
B(2,2) = B(2,2) / BNORM
C
C Check if A can be deflated.
C
IF ( ABS( A(2,1) ).LE.ULP ) THEN
CSL = ONE
SNL = ZERO
CSR = ONE
SNR = ZERO
WI = ZERO
A(2,1) = ZERO
B(2,1) = ZERO
C
C Check if B is singular.
C
ELSE IF ( ABS( B(1,1) ).LE.ULP ) THEN
CALL DLARTG( A(2,2), A(2,1), CSR, SNR, T )
SNR = -SNR
CALL DROT( 2, A(1,1), 1, A(1,2), 1, CSR, SNR )
CALL DROT( 2, B(1,1), LDB, B(2,1), LDB, CSR, SNR )
CSL = ONE
SNL = ZERO
WI = ZERO
A(2,1) = ZERO
B(1,1) = ZERO
B(2,1) = ZERO
ELSE IF( ABS( B(2,2) ).LE.ULP ) THEN
CALL DLARTG( A(1,1), A(2,1), CSL, SNL, R )
CSR = ONE
SNR = ZERO
WI = ZERO
CALL DROT( 2, A(1,1), LDA, A(2,1), LDA, CSL, SNL )
CALL DROT( 2, B(1,1), 1, B(1,2), 1, CSL, SNL )
A(2,1) = ZERO
B(2,1) = ZERO
B(2,2) = ZERO
ELSE
C
C B is nonsingular, first compute the eigenvalues of A / adj(B).
C
R = B(1,1)
B(1,1) = B(2,2)
B(2,2) = R
B(1,2) = -B(1,2)
CALL DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2,
$ WI )
C
IF( WI.EQ.ZERO ) THEN
C
C Two real eigenvalues, compute s*A-w*B.
C
H1 = SCALE1*A(1,1) - WR1*B(1,1)
H2 = SCALE1*A(1,2) - WR1*B(1,2)
H3 = SCALE1*A(2,2) - WR1*B(2,2)
C
RR = DLAPY2( H1, H2 )
QQ = DLAPY2( SCALE1*A(2,1), H3 )
C
IF ( RR.GT.QQ ) THEN
C
C Find right rotation matrix to zero 1,1 element of
C (sA - wB).
C
CALL DLARTG( H2, H1, CSR, SNR, T )
C
ELSE
C
C Find right rotation matrix to zero 2,1 element of
C (sA - wB).
C
CALL DLARTG( H3, SCALE1*A(2,1), CSR, SNR, T )
C
END IF
C
SNR = -SNR
CALL DROT( 2, A(1,1), 1, A(1,2), 1, CSR, SNR )
CALL DROT( 2, B(1,1), 1, B(1,2), 1, CSR, SNR )
C
C Compute inf norms of A and B.
C
H1 = MAX( ABS( A(1,1) ) + ABS( A(1,2) ),
$ ABS( A(2,1) ) + ABS( A(2,2) ) )
H2 = MAX( ABS( B(1,1) ) + ABS( B(1,2) ),
$ ABS( B(2,1) ) + ABS( B(2,2) ) )
C
IF( ( SCALE1*H1 ).GE.ABS( WR1 )*H2 ) THEN
C
C Find left rotation matrix Q to zero out B(2,1).
C
CALL DLARTG( B(1,1), B(2,1), CSL, SNL, R )
C
ELSE
C
C Find left rotation matrix Q to zero out A(2,1).
C
CALL DLARTG( A(1,1), A(2,1), CSL, SNL, R )
C
END IF
C
CALL DROT( 2, A(1,1), LDA, A(2,1), LDA, CSL, SNL )
CALL DROT( 2, B(1,1), LDB, B(2,1), LDB, CSL, SNL )
C
A(2,1) = ZERO
B(2,1) = ZERO
C
C Re-adjoint B.
C
R = B(1,1)
B(1,1) = B(2,2)
B(2,2) = R
B(1,2) = -B(1,2)
C
ELSE
C
C A pair of complex conjugate eigenvalues:
C first compute the SVD of the matrix adj(B).
C
R = B(1,1)
B(1,1) = B(2,2)
B(2,2) = R
B(1,2) = -B(1,2)
CALL DLASV2( B(1,1), B(1,2), B(2,2), R, T, SNL, CSL,
$ SNR, CSR )
C
C Form (A,B) := Q(A,adj(B))Z' where Q is left rotation matrix
C and Z is right rotation matrix computed from DLASV2.
C
CALL DROT( 2, A(1,1), LDA, A(2,1), LDA, CSL, SNL )
CALL DROT( 2, B(1,1), LDB, B(2,1), LDB, CSR, SNR )
CALL DROT( 2, A(1,1), 1, A(1,2), 1, CSR, SNR )
CALL DROT( 2, B(1,1), 1, B(1,2), 1, CSL, SNL )
C
B(2,1) = ZERO
B(1,2) = ZERO
END IF
C
END IF
C
C Unscaling
C
R = B(1,1)
T = B(2,2)
A(1,1) = ANORM*A(1,1)
A(2,1) = ANORM*A(2,1)
A(1,2) = ANORM*A(1,2)
A(2,2) = ANORM*A(2,2)
B(1,1) = BNORM*B(1,1)
B(2,1) = BNORM*B(2,1)
B(1,2) = BNORM*B(1,2)
B(2,2) = BNORM*B(2,2)
C
IF( WI.EQ.ZERO ) THEN
ALPHAR(1) = A(1,1)
ALPHAR(2) = A(2,2)
ALPHAI(1) = ZERO
ALPHAI(2) = ZERO
BETA(1) = B(1,1)
BETA(2) = B(2,2)
ELSE
WR1 = ANORM*WR1
WI = ANORM*WI
IF ( ABS( WR1 ).GT.ONE .OR. WI.GT.ONE ) THEN
WR1 = WR1*R
WI = WI*R
R = ONE
END IF
IF ( ABS( WR1 ).GT.ONE .OR. ABS( WI ).GT.ONE ) THEN
WR1 = WR1*T
WI = WI*T
T = ONE
END IF
ALPHAR(1) = ( WR1 / SCALE1 )*R*T
ALPHAI(1) = ABS( ( WI / SCALE1 )*R*T )
ALPHAR(2) = ALPHAR(1)
ALPHAI(2) = -ALPHAI(1)
BETA(1) = BNORM
BETA(2) = BNORM
END IF
RETURN
C *** Last line of MB03YT ***
END