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SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
$ CSR, SNR ) * * -- LAPACK auxiliary routine (version 3.2.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2010 * * .. 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 WI = 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 WI = 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 WI = 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**T 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 |