*> \brief \b SLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLAGV2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, * CSR, SNR ) * * .. Scalar Arguments .. * INTEGER LDA, LDB * REAL CSL, CSR, SNL, SNR * .. * .. Array Arguments .. * REAL A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ), * \$ B( LDB, * ), BETA( 2 ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLAGV2 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. *> *> \endverbatim * * Arguments: * ========== * *> \param[in,out] A *> \verbatim *> A is REAL 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. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> THe leading dimension of the array A. LDA >= 2. *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is REAL 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. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> THe leading dimension of the array B. LDB >= 2. *> \endverbatim *> *> \param[out] ALPHAR *> \verbatim *> ALPHAR is REAL array, dimension (2) *> \endverbatim *> *> \param[out] ALPHAI *> \verbatim *> ALPHAI is REAL array, dimension (2) *> \endverbatim *> *> \param[out] BETA *> \verbatim *> BETA is REAL 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. *> \endverbatim *> *> \param[out] CSL *> \verbatim *> CSL is REAL *> The cosine of the left rotation matrix. *> \endverbatim *> *> \param[out] SNL *> \verbatim *> SNL is REAL *> The sine of the left rotation matrix. *> \endverbatim *> *> \param[out] CSR *> \verbatim *> CSR is REAL *> The cosine of the right rotation matrix. *> \endverbatim *> *> \param[out] SNR *> \verbatim *> SNR is REAL *> The sine of the right rotation matrix. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup realOTHERauxiliary * *> \par Contributors: * ================== *> *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA * * ===================================================================== SUBROUTINE SLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, \$ CSR, SNR ) * * -- LAPACK auxiliary routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER LDA, LDB REAL CSL, CSR, SNL, SNR * .. * .. Array Arguments .. REAL A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ), \$ B( LDB, * ), BETA( 2 ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. REAL ANORM, ASCALE, BNORM, BSCALE, H1, H2, H3, QQ, \$ R, RR, SAFMIN, SCALE1, SCALE2, T, ULP, WI, WR1, \$ WR2 * .. * .. External Subroutines .. EXTERNAL SLAG2, SLARTG, SLASV2, SROT * .. * .. External Functions .. REAL SLAMCH, SLAPY2 EXTERNAL SLAMCH, SLAPY2 * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX * .. * .. Executable Statements .. * SAFMIN = SLAMCH( 'S' ) ULP = SLAMCH( '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 WI = ZERO * * Check if B is singular * ELSE IF( ABS( B( 1, 1 ) ).LE.ULP ) THEN CALL SLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R ) CSR = ONE SNR = ZERO CALL SROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL ) CALL SROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL ) A( 2, 1 ) = ZERO B( 1, 1 ) = ZERO B( 2, 1 ) = ZERO WI = ZERO * ELSE IF( ABS( B( 2, 2 ) ).LE.ULP ) THEN CALL SLARTG( A( 2, 2 ), A( 2, 1 ), CSR, SNR, T ) SNR = -SNR CALL SROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR ) CALL SROT( 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 WI = ZERO * ELSE * * B is nonsingular, first compute the eigenvalues of (A,B) * CALL SLAG2( 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 = SLAPY2( H1, H2 ) QQ = SLAPY2( SCALE1*A( 2, 1 ), H3 ) * IF( RR.GT.QQ ) THEN * * find right rotation matrix to zero 1,1 element of * (sA - wB) * CALL SLARTG( H2, H1, CSR, SNR, T ) * ELSE * * find right rotation matrix to zero 2,1 element of * (sA - wB) * CALL SLARTG( H3, SCALE1*A( 2, 1 ), CSR, SNR, T ) * END IF * SNR = -SNR CALL SROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR ) CALL SROT( 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 SLARTG( B( 1, 1 ), B( 2, 1 ), CSL, SNL, R ) * ELSE * * find left rotation matrix Q to zero out A(2,1) * CALL SLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R ) * END IF * CALL SROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL ) CALL SROT( 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 SLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR, \$ CSR, SNL, CSL ) * * Form (A,B) := Q(A,B)Z**T where Q is left rotation matrix and * Z is right rotation matrix computed from SLASV2 * CALL SROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL ) CALL SROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL ) CALL SROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR ) CALL SROT( 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 SLAGV2 * END