*> \brief \b ZTGEVC * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZTGEVC + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, * LDVL, VR, LDVR, MM, M, WORK, RWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER HOWMNY, SIDE * INTEGER INFO, LDP, LDS, LDVL, LDVR, M, MM, N * .. * .. Array Arguments .. * LOGICAL SELECT( * ) * DOUBLE PRECISION RWORK( * ) * COMPLEX*16 P( LDP, * ), S( LDS, * ), VL( LDVL, * ), * \$ VR( LDVR, * ), WORK( * ) * .. * * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZTGEVC computes some or all of the right and/or left eigenvectors of *> a pair of complex matrices (S,P), where S and P are upper triangular. *> Matrix pairs of this type are produced by the generalized Schur *> factorization of a complex matrix pair (A,B): *> *> A = Q*S*Z**H, B = Q*P*Z**H *> *> as computed by ZGGHRD + ZHGEQZ. *> *> The right eigenvector x and the left eigenvector y of (S,P) *> corresponding to an eigenvalue w are defined by: *> *> S*x = w*P*x, (y**H)*S = w*(y**H)*P, *> *> where y**H denotes the conjugate tranpose of y. *> The eigenvalues are not input to this routine, but are computed *> directly from the diagonal elements of S and P. *> *> This routine returns the matrices X and/or Y of right and left *> eigenvectors of (S,P), or the products Z*X and/or Q*Y, *> where Z and Q are input matrices. *> If Q and Z are the unitary factors from the generalized Schur *> factorization of a matrix pair (A,B), then Z*X and Q*Y *> are the matrices of right and left eigenvectors of (A,B). *> \endverbatim * * Arguments: * ========== * *> \param[in] SIDE *> \verbatim *> SIDE is CHARACTER*1 *> = 'R': compute right eigenvectors only; *> = 'L': compute left eigenvectors only; *> = 'B': compute both right and left eigenvectors. *> \endverbatim *> *> \param[in] HOWMNY *> \verbatim *> HOWMNY is CHARACTER*1 *> = 'A': compute all right and/or left eigenvectors; *> = 'B': compute all right and/or left eigenvectors, *> backtransformed by the matrices in VR and/or VL; *> = 'S': compute selected right and/or left eigenvectors, *> specified by the logical array SELECT. *> \endverbatim *> *> \param[in] SELECT *> \verbatim *> SELECT is LOGICAL array, dimension (N) *> If HOWMNY='S', SELECT specifies the eigenvectors to be *> computed. The eigenvector corresponding to the j-th *> eigenvalue is computed if SELECT(j) = .TRUE.. *> Not referenced if HOWMNY = 'A' or 'B'. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrices S and P. N >= 0. *> \endverbatim *> *> \param[in] S *> \verbatim *> S is COMPLEX*16 array, dimension (LDS,N) *> The upper triangular matrix S from a generalized Schur *> factorization, as computed by ZHGEQZ. *> \endverbatim *> *> \param[in] LDS *> \verbatim *> LDS is INTEGER *> The leading dimension of array S. LDS >= max(1,N). *> \endverbatim *> *> \param[in] P *> \verbatim *> P is COMPLEX*16 array, dimension (LDP,N) *> The upper triangular matrix P from a generalized Schur *> factorization, as computed by ZHGEQZ. P must have real *> diagonal elements. *> \endverbatim *> *> \param[in] LDP *> \verbatim *> LDP is INTEGER *> The leading dimension of array P. LDP >= max(1,N). *> \endverbatim *> *> \param[in,out] VL *> \verbatim *> VL is COMPLEX*16 array, dimension (LDVL,MM) *> On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must *> contain an N-by-N matrix Q (usually the unitary matrix Q *> of left Schur vectors returned by ZHGEQZ). *> On exit, if SIDE = 'L' or 'B', VL contains: *> if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P); *> if HOWMNY = 'B', the matrix Q*Y; *> if HOWMNY = 'S', the left eigenvectors of (S,P) specified by *> SELECT, stored consecutively in the columns of *> VL, in the same order as their eigenvalues. *> Not referenced if SIDE = 'R'. *> \endverbatim *> *> \param[in] LDVL *> \verbatim *> LDVL is INTEGER *> The leading dimension of array VL. LDVL >= 1, and if *> SIDE = 'L' or 'l' or 'B' or 'b', LDVL >= N. *> \endverbatim *> *> \param[in,out] VR *> \verbatim *> VR is COMPLEX*16 array, dimension (LDVR,MM) *> On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must *> contain an N-by-N matrix Q (usually the unitary matrix Z *> of right Schur vectors returned by ZHGEQZ). *> On exit, if SIDE = 'R' or 'B', VR contains: *> if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P); *> if HOWMNY = 'B', the matrix Z*X; *> if HOWMNY = 'S', the right eigenvectors of (S,P) specified by *> SELECT, stored consecutively in the columns of *> VR, in the same order as their eigenvalues. *> Not referenced if SIDE = 'L'. *> \endverbatim *> *> \param[in] LDVR *> \verbatim *> LDVR is INTEGER *> The leading dimension of the array VR. LDVR >= 1, and if *> SIDE = 'R' or 'B', LDVR >= N. *> \endverbatim *> *> \param[in] MM *> \verbatim *> MM is INTEGER *> The number of columns in the arrays VL and/or VR. MM >= M. *> \endverbatim *> *> \param[out] M *> \verbatim *> M is INTEGER *> The number of columns in the arrays VL and/or VR actually *> used to store the eigenvectors. If HOWMNY = 'A' or 'B', M *> is set to N. Each selected eigenvector occupies one column. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX*16 array, dimension (2*N) *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is DOUBLE PRECISION array, dimension (2*N) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit. *> < 0: if INFO = -i, the i-th argument had an illegal value. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup complex16GEcomputational * * ===================================================================== SUBROUTINE ZTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, \$ LDVL, VR, LDVR, MM, M, WORK, RWORK, INFO ) * * -- LAPACK computational routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. CHARACTER HOWMNY, SIDE INTEGER INFO, LDP, LDS, LDVL, LDVR, M, MM, N * .. * .. Array Arguments .. LOGICAL SELECT( * ) DOUBLE PRECISION RWORK( * ) COMPLEX*16 P( LDP, * ), S( LDS, * ), VL( LDVL, * ), \$ VR( LDVR, * ), WORK( * ) * .. * * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), \$ CONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL COMPL, COMPR, ILALL, ILBACK, ILBBAD, ILCOMP, \$ LSA, LSB INTEGER I, IBEG, IEIG, IEND, IHWMNY, IM, ISIDE, ISRC, \$ J, JE, JR DOUBLE PRECISION ACOEFA, ACOEFF, ANORM, ASCALE, BCOEFA, BIG, \$ BIGNUM, BNORM, BSCALE, DMIN, SAFMIN, SBETA, \$ SCALE, SMALL, TEMP, ULP, XMAX COMPLEX*16 BCOEFF, CA, CB, D, SALPHA, SUM, SUMA, SUMB, X * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH COMPLEX*16 ZLADIV EXTERNAL LSAME, DLAMCH, ZLADIV * .. * .. External Subroutines .. EXTERNAL DLABAD, XERBLA, ZGEMV * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN * .. * .. Statement Functions .. DOUBLE PRECISION ABS1 * .. * .. Statement Function definitions .. ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) ) * .. * .. Executable Statements .. * * Decode and Test the input parameters * IF( LSAME( HOWMNY, 'A' ) ) THEN IHWMNY = 1 ILALL = .TRUE. ILBACK = .FALSE. ELSE IF( LSAME( HOWMNY, 'S' ) ) THEN IHWMNY = 2 ILALL = .FALSE. ILBACK = .FALSE. ELSE IF( LSAME( HOWMNY, 'B' ) ) THEN IHWMNY = 3 ILALL = .TRUE. ILBACK = .TRUE. ELSE IHWMNY = -1 END IF * IF( LSAME( SIDE, 'R' ) ) THEN ISIDE = 1 COMPL = .FALSE. COMPR = .TRUE. ELSE IF( LSAME( SIDE, 'L' ) ) THEN ISIDE = 2 COMPL = .TRUE. COMPR = .FALSE. ELSE IF( LSAME( SIDE, 'B' ) ) THEN ISIDE = 3 COMPL = .TRUE. COMPR = .TRUE. ELSE ISIDE = -1 END IF * INFO = 0 IF( ISIDE.LT.0 ) THEN INFO = -1 ELSE IF( IHWMNY.LT.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDS.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDP.LT.MAX( 1, N ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZTGEVC', -INFO ) RETURN END IF * * Count the number of eigenvectors * IF( .NOT.ILALL ) THEN IM = 0 DO 10 J = 1, N IF( SELECT( J ) ) \$ IM = IM + 1 10 CONTINUE ELSE IM = N END IF * * Check diagonal of B * ILBBAD = .FALSE. DO 20 J = 1, N IF( DIMAG( P( J, J ) ).NE.ZERO ) \$ ILBBAD = .TRUE. 20 CONTINUE * IF( ILBBAD ) THEN INFO = -7 ELSE IF( COMPL .AND. LDVL.LT.N .OR. LDVL.LT.1 ) THEN INFO = -10 ELSE IF( COMPR .AND. LDVR.LT.N .OR. LDVR.LT.1 ) THEN INFO = -12 ELSE IF( MM.LT.IM ) THEN INFO = -13 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZTGEVC', -INFO ) RETURN END IF * * Quick return if possible * M = IM IF( N.EQ.0 ) \$ RETURN * * Machine Constants * SAFMIN = DLAMCH( 'Safe minimum' ) BIG = ONE / SAFMIN CALL DLABAD( SAFMIN, BIG ) ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' ) SMALL = SAFMIN*N / ULP BIG = ONE / SMALL BIGNUM = ONE / ( SAFMIN*N ) * * Compute the 1-norm of each column of the strictly upper triangular * part of A and B to check for possible overflow in the triangular * solver. * ANORM = ABS1( S( 1, 1 ) ) BNORM = ABS1( P( 1, 1 ) ) RWORK( 1 ) = ZERO RWORK( N+1 ) = ZERO DO 40 J = 2, N RWORK( J ) = ZERO RWORK( N+J ) = ZERO DO 30 I = 1, J - 1 RWORK( J ) = RWORK( J ) + ABS1( S( I, J ) ) RWORK( N+J ) = RWORK( N+J ) + ABS1( P( I, J ) ) 30 CONTINUE ANORM = MAX( ANORM, RWORK( J )+ABS1( S( J, J ) ) ) BNORM = MAX( BNORM, RWORK( N+J )+ABS1( P( J, J ) ) ) 40 CONTINUE * ASCALE = ONE / MAX( ANORM, SAFMIN ) BSCALE = ONE / MAX( BNORM, SAFMIN ) * * Left eigenvectors * IF( COMPL ) THEN IEIG = 0 * * Main loop over eigenvalues * DO 140 JE = 1, N IF( ILALL ) THEN ILCOMP = .TRUE. ELSE ILCOMP = SELECT( JE ) END IF IF( ILCOMP ) THEN IEIG = IEIG + 1 * IF( ABS1( S( JE, JE ) ).LE.SAFMIN .AND. \$ ABS( DBLE( P( JE, JE ) ) ).LE.SAFMIN ) THEN * * Singular matrix pencil -- return unit eigenvector * DO 50 JR = 1, N VL( JR, IEIG ) = CZERO 50 CONTINUE VL( IEIG, IEIG ) = CONE GO TO 140 END IF * * Non-singular eigenvalue: * Compute coefficients a and b in * H * y ( a A - b B ) = 0 * TEMP = ONE / MAX( ABS1( S( JE, JE ) )*ASCALE, \$ ABS( DBLE( P( JE, JE ) ) )*BSCALE, SAFMIN ) SALPHA = ( TEMP*S( JE, JE ) )*ASCALE SBETA = ( TEMP*DBLE( P( JE, JE ) ) )*BSCALE ACOEFF = SBETA*ASCALE BCOEFF = SALPHA*BSCALE * * Scale to avoid underflow * LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEFF ).LT.SMALL LSB = ABS1( SALPHA ).GE.SAFMIN .AND. ABS1( BCOEFF ).LT. \$ SMALL * SCALE = ONE IF( LSA ) \$ SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG ) IF( LSB ) \$ SCALE = MAX( SCALE, ( SMALL / ABS1( SALPHA ) )* \$ MIN( BNORM, BIG ) ) IF( LSA .OR. LSB ) THEN SCALE = MIN( SCALE, ONE / \$ ( SAFMIN*MAX( ONE, ABS( ACOEFF ), \$ ABS1( BCOEFF ) ) ) ) IF( LSA ) THEN ACOEFF = ASCALE*( SCALE*SBETA ) ELSE ACOEFF = SCALE*ACOEFF END IF IF( LSB ) THEN BCOEFF = BSCALE*( SCALE*SALPHA ) ELSE BCOEFF = SCALE*BCOEFF END IF END IF * ACOEFA = ABS( ACOEFF ) BCOEFA = ABS1( BCOEFF ) XMAX = ONE DO 60 JR = 1, N WORK( JR ) = CZERO 60 CONTINUE WORK( JE ) = CONE DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN ) * * H * Triangular solve of (a A - b B) y = 0 * * H * (rowwise in (a A - b B) , or columnwise in a A - b B) * DO 100 J = JE + 1, N * * Compute * j-1 * SUM = sum conjg( a*S(k,j) - b*P(k,j) )*x(k) * k=je * (Scale if necessary) * TEMP = ONE / XMAX IF( ACOEFA*RWORK( J )+BCOEFA*RWORK( N+J ).GT.BIGNUM* \$ TEMP ) THEN DO 70 JR = JE, J - 1 WORK( JR ) = TEMP*WORK( JR ) 70 CONTINUE XMAX = ONE END IF SUMA = CZERO SUMB = CZERO * DO 80 JR = JE, J - 1 SUMA = SUMA + DCONJG( S( JR, J ) )*WORK( JR ) SUMB = SUMB + DCONJG( P( JR, J ) )*WORK( JR ) 80 CONTINUE SUM = ACOEFF*SUMA - DCONJG( BCOEFF )*SUMB * * Form x(j) = - SUM / conjg( a*S(j,j) - b*P(j,j) ) * * with scaling and perturbation of the denominator * D = DCONJG( ACOEFF*S( J, J )-BCOEFF*P( J, J ) ) IF( ABS1( D ).LE.DMIN ) \$ D = DCMPLX( DMIN ) * IF( ABS1( D ).LT.ONE ) THEN IF( ABS1( SUM ).GE.BIGNUM*ABS1( D ) ) THEN TEMP = ONE / ABS1( SUM ) DO 90 JR = JE, J - 1 WORK( JR ) = TEMP*WORK( JR ) 90 CONTINUE XMAX = TEMP*XMAX SUM = TEMP*SUM END IF END IF WORK( J ) = ZLADIV( -SUM, D ) XMAX = MAX( XMAX, ABS1( WORK( J ) ) ) 100 CONTINUE * * Back transform eigenvector if HOWMNY='B'. * IF( ILBACK ) THEN CALL ZGEMV( 'N', N, N+1-JE, CONE, VL( 1, JE ), LDVL, \$ WORK( JE ), 1, CZERO, WORK( N+1 ), 1 ) ISRC = 2 IBEG = 1 ELSE ISRC = 1 IBEG = JE END IF * * Copy and scale eigenvector into column of VL * XMAX = ZERO DO 110 JR = IBEG, N XMAX = MAX( XMAX, ABS1( WORK( ( ISRC-1 )*N+JR ) ) ) 110 CONTINUE * IF( XMAX.GT.SAFMIN ) THEN TEMP = ONE / XMAX DO 120 JR = IBEG, N VL( JR, IEIG ) = TEMP*WORK( ( ISRC-1 )*N+JR ) 120 CONTINUE ELSE IBEG = N + 1 END IF * DO 130 JR = 1, IBEG - 1 VL( JR, IEIG ) = CZERO 130 CONTINUE * END IF 140 CONTINUE END IF * * Right eigenvectors * IF( COMPR ) THEN IEIG = IM + 1 * * Main loop over eigenvalues * DO 250 JE = N, 1, -1 IF( ILALL ) THEN ILCOMP = .TRUE. ELSE ILCOMP = SELECT( JE ) END IF IF( ILCOMP ) THEN IEIG = IEIG - 1 * IF( ABS1( S( JE, JE ) ).LE.SAFMIN .AND. \$ ABS( DBLE( P( JE, JE ) ) ).LE.SAFMIN ) THEN * * Singular matrix pencil -- return unit eigenvector * DO 150 JR = 1, N VR( JR, IEIG ) = CZERO 150 CONTINUE VR( IEIG, IEIG ) = CONE GO TO 250 END IF * * Non-singular eigenvalue: * Compute coefficients a and b in * * ( a A - b B ) x = 0 * TEMP = ONE / MAX( ABS1( S( JE, JE ) )*ASCALE, \$ ABS( DBLE( P( JE, JE ) ) )*BSCALE, SAFMIN ) SALPHA = ( TEMP*S( JE, JE ) )*ASCALE SBETA = ( TEMP*DBLE( P( JE, JE ) ) )*BSCALE ACOEFF = SBETA*ASCALE BCOEFF = SALPHA*BSCALE * * Scale to avoid underflow * LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEFF ).LT.SMALL LSB = ABS1( SALPHA ).GE.SAFMIN .AND. ABS1( BCOEFF ).LT. \$ SMALL * SCALE = ONE IF( LSA ) \$ SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG ) IF( LSB ) \$ SCALE = MAX( SCALE, ( SMALL / ABS1( SALPHA ) )* \$ MIN( BNORM, BIG ) ) IF( LSA .OR. LSB ) THEN SCALE = MIN( SCALE, ONE / \$ ( SAFMIN*MAX( ONE, ABS( ACOEFF ), \$ ABS1( BCOEFF ) ) ) ) IF( LSA ) THEN ACOEFF = ASCALE*( SCALE*SBETA ) ELSE ACOEFF = SCALE*ACOEFF END IF IF( LSB ) THEN BCOEFF = BSCALE*( SCALE*SALPHA ) ELSE BCOEFF = SCALE*BCOEFF END IF END IF * ACOEFA = ABS( ACOEFF ) BCOEFA = ABS1( BCOEFF ) XMAX = ONE DO 160 JR = 1, N WORK( JR ) = CZERO 160 CONTINUE WORK( JE ) = CONE DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN ) * * Triangular solve of (a A - b B) x = 0 (columnwise) * * WORK(1:j-1) contains sums w, * WORK(j+1:JE) contains x * DO 170 JR = 1, JE - 1 WORK( JR ) = ACOEFF*S( JR, JE ) - BCOEFF*P( JR, JE ) 170 CONTINUE WORK( JE ) = CONE * DO 210 J = JE - 1, 1, -1 * * Form x(j) := - w(j) / d * with scaling and perturbation of the denominator * D = ACOEFF*S( J, J ) - BCOEFF*P( J, J ) IF( ABS1( D ).LE.DMIN ) \$ D = DCMPLX( DMIN ) * IF( ABS1( D ).LT.ONE ) THEN IF( ABS1( WORK( J ) ).GE.BIGNUM*ABS1( D ) ) THEN TEMP = ONE / ABS1( WORK( J ) ) DO 180 JR = 1, JE WORK( JR ) = TEMP*WORK( JR ) 180 CONTINUE END IF END IF * WORK( J ) = ZLADIV( -WORK( J ), D ) * IF( J.GT.1 ) THEN * * w = w + x(j)*(a S(*,j) - b P(*,j) ) with scaling * IF( ABS1( WORK( J ) ).GT.ONE ) THEN TEMP = ONE / ABS1( WORK( J ) ) IF( ACOEFA*RWORK( J )+BCOEFA*RWORK( N+J ).GE. \$ BIGNUM*TEMP ) THEN DO 190 JR = 1, JE WORK( JR ) = TEMP*WORK( JR ) 190 CONTINUE END IF END IF * CA = ACOEFF*WORK( J ) CB = BCOEFF*WORK( J ) DO 200 JR = 1, J - 1 WORK( JR ) = WORK( JR ) + CA*S( JR, J ) - \$ CB*P( JR, J ) 200 CONTINUE END IF 210 CONTINUE * * Back transform eigenvector if HOWMNY='B'. * IF( ILBACK ) THEN CALL ZGEMV( 'N', N, JE, CONE, VR, LDVR, WORK, 1, \$ CZERO, WORK( N+1 ), 1 ) ISRC = 2 IEND = N ELSE ISRC = 1 IEND = JE END IF * * Copy and scale eigenvector into column of VR * XMAX = ZERO DO 220 JR = 1, IEND XMAX = MAX( XMAX, ABS1( WORK( ( ISRC-1 )*N+JR ) ) ) 220 CONTINUE * IF( XMAX.GT.SAFMIN ) THEN TEMP = ONE / XMAX DO 230 JR = 1, IEND VR( JR, IEIG ) = TEMP*WORK( ( ISRC-1 )*N+JR ) 230 CONTINUE ELSE IEND = 0 END IF * DO 240 JR = IEND + 1, N VR( JR, IEIG ) = CZERO 240 CONTINUE * END IF 250 CONTINUE END IF * RETURN * * End of ZTGEVC * END