SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
$ CSR, SNR )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER LDA, LDB
DOUBLE PRECISION CSL, CSR, SNL, SNR
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ),
$ B( LDB, * ), BETA( 2 )
* ..
*
* Purpose
* =======
*
* DLAGV2 computes the Generalized Schur factorization of a real 2-by-2
* matrix pencil (A,B) where B is upper triangular. This routine
* computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
* SNR such that
*
* 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
* types), then
*
* [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
* [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
*
* [ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
* [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ],
*
* 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
* then
*
* [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
* [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
*
* [ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
* [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ]
*
* where b11 >= b22 > 0.
*
*
* Arguments
* =========
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA, 2)
* On entry, the 2 x 2 matrix A.
* On exit, A is overwritten by the ``A-part'' of the
* generalized Schur form.
*
* LDA (input) INTEGER
* THe leading dimension of the array A. LDA >= 2.
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB, 2)
* On entry, the upper triangular 2 x 2 matrix B.
* On exit, B is overwritten by the ``B-part'' of the
* generalized Schur form.
*
* LDB (input) INTEGER
* THe leading dimension of the array B. LDB >= 2.
*
* ALPHAR (output) DOUBLE PRECISION array, dimension (2)
* ALPHAI (output) DOUBLE PRECISION array, dimension (2)
* BETA (output) DOUBLE PRECISION array, dimension (2)
* (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
* pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may
* be zero.
*
* CSL (output) DOUBLE PRECISION
* The cosine of the left rotation matrix.
*
* SNL (output) DOUBLE PRECISION
* The sine of the left rotation matrix.
*
* CSR (output) DOUBLE PRECISION
* The cosine of the right rotation matrix.
*
* SNR (output) DOUBLE PRECISION
* The sine of the right rotation matrix.
*
* Further Details
* ===============
*
* Based on contributions by
* Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION ANORM, ASCALE, BNORM, BSCALE, H1, H2, H3, QQ,
$ R, RR, SAFMIN, SCALE1, SCALE2, T, ULP, WI, WR1,
$ WR2
* ..
* .. External Subroutines ..
EXTERNAL DLAG2, DLARTG, DLASV2, DROT
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL DLAMCH, DLAPY2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
SAFMIN = DLAMCH( 'S' )
ULP = DLAMCH( 'P' )
*
* Scale A
*
ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
$ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
ASCALE = ONE / ANORM
A( 1, 1 ) = ASCALE*A( 1, 1 )
A( 1, 2 ) = ASCALE*A( 1, 2 )
A( 2, 1 ) = ASCALE*A( 2, 1 )
A( 2, 2 ) = ASCALE*A( 2, 2 )
*
* Scale B
*
BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 1, 2 ) )+ABS( B( 2, 2 ) ),
$ SAFMIN )
BSCALE = ONE / BNORM
B( 1, 1 ) = BSCALE*B( 1, 1 )
B( 1, 2 ) = BSCALE*B( 1, 2 )
B( 2, 2 ) = BSCALE*B( 2, 2 )
*
* Check if A can be deflated
*
IF( ABS( A( 2, 1 ) ).LE.ULP ) THEN
CSL = ONE
SNL = ZERO
CSR = ONE
SNR = ZERO
A( 2, 1 ) = ZERO
B( 2, 1 ) = ZERO
*
* Check if B is singular
*
ELSE IF( ABS( B( 1, 1 ) ).LE.ULP ) THEN
CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
CSR = ONE
SNR = ZERO
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 )
A( 2, 1 ) = ZERO
B( 1, 1 ) = ZERO
B( 2, 1 ) = ZERO
*
ELSE IF( ABS( B( 2, 2 ) ).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 ), 1, B( 1, 2 ), 1, CSR, SNR )
CSL = ONE
SNL = ZERO
A( 2, 1 ) = ZERO
B( 2, 1 ) = ZERO
B( 2, 2 ) = ZERO
*
ELSE
*
* B is nonsingular, first compute the eigenvalues of (A,B)
*
CALL DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2,
$ WI )
*
IF( WI.EQ.ZERO ) THEN
*
* two real eigenvalues, compute s*A-w*B
*
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 )
*
RR = DLAPY2( H1, H2 )
QQ = DLAPY2( SCALE1*A( 2, 1 ), H3 )
*
IF( RR.GT.QQ ) THEN
*
* find right rotation matrix to zero 1,1 element of
* (sA - wB)
*
CALL DLARTG( H2, H1, CSR, SNR, T )
*
ELSE
*
* find right rotation matrix to zero 2,1 element of
* (sA - wB)
*
CALL DLARTG( H3, SCALE1*A( 2, 1 ), CSR, SNR, T )
*
END IF
*
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 )
*
* compute inf norms of A and B
*
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 ) ) )
*
IF( ( SCALE1*H1 ).GE.ABS( WR1 )*H2 ) THEN
*
* find left rotation matrix Q to zero out B(2,1)
*
CALL DLARTG( B( 1, 1 ), B( 2, 1 ), CSL, SNL, R )
*
ELSE
*
* find left rotation matrix Q to zero out A(2,1)
*
CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
*
END IF
*
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 )
*
A( 2, 1 ) = ZERO
B( 2, 1 ) = ZERO
*
ELSE
*
* a pair of complex conjugate eigenvalues
* first compute the SVD of the matrix B
*
CALL DLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR,
$ CSR, SNL, CSL )
*
* Form (A,B) := Q(A,B)Z' where Q is left rotation matrix and
* Z is right rotation matrix computed from DLASV2
*
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 )
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 )
*
B( 2, 1 ) = ZERO
B( 1, 2 ) = ZERO
*
END IF
*
END IF
*
* Unscaling
*
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 )
*
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
ALPHAR( 1 ) = ANORM*WR1 / SCALE1 / BNORM
ALPHAI( 1 ) = ANORM*WI / SCALE1 / BNORM
ALPHAR( 2 ) = ALPHAR( 1 )
ALPHAI( 2 ) = -ALPHAI( 1 )
BETA( 1 ) = ONE
BETA( 2 ) = ONE
END IF
*
RETURN
*
* End of DLAGV2
*
END