*> \brief ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZGEGV + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, * VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBVL, JOBVR * INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N * .. * .. Array Arguments .. * DOUBLE PRECISION RWORK( * ) * COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), * \$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ), * \$ WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> This routine is deprecated and has been replaced by routine ZGGEV. *> *> ZGEGV computes the eigenvalues and, optionally, the left and/or right *> eigenvectors of a complex matrix pair (A,B). *> Given two square matrices A and B, *> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the *> eigenvalues lambda and corresponding (non-zero) eigenvectors x such *> that *> A*x = lambda*B*x. *> *> An alternate form is to find the eigenvalues mu and corresponding *> eigenvectors y such that *> mu*A*y = B*y. *> *> These two forms are equivalent with mu = 1/lambda and x = y if *> neither lambda nor mu is zero. In order to deal with the case that *> lambda or mu is zero or small, two values alpha and beta are returned *> for each eigenvalue, such that lambda = alpha/beta and *> mu = beta/alpha. *> *> The vectors x and y in the above equations are right eigenvectors of *> the matrix pair (A,B). Vectors u and v satisfying *> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B *> are left eigenvectors of (A,B). *> *> Note: this routine performs "full balancing" on A and B *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBVL *> \verbatim *> JOBVL is CHARACTER*1 *> = 'N': do not compute the left generalized eigenvectors; *> = 'V': compute the left generalized eigenvectors (returned *> in VL). *> \endverbatim *> *> \param[in] JOBVR *> \verbatim *> JOBVR is CHARACTER*1 *> = 'N': do not compute the right generalized eigenvectors; *> = 'V': compute the right generalized eigenvectors (returned *> in VR). *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrices A, B, VL, and VR. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA, N) *> On entry, the matrix A. *> If JOBVL = 'V' or JOBVR = 'V', then on exit A *> contains the Schur form of A from the generalized Schur *> factorization of the pair (A,B) after balancing. If no *> eigenvectors were computed, then only the diagonal elements *> of the Schur form will be correct. See ZGGHRD and ZHGEQZ *> for details. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of A. LDA >= max(1,N). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is COMPLEX*16 array, dimension (LDB, N) *> On entry, the matrix B. *> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the *> upper triangular matrix obtained from B in the generalized *> Schur factorization of the pair (A,B) after balancing. *> If no eigenvectors were computed, then only the diagonal *> elements of B will be correct. See ZGGHRD and ZHGEQZ for *> details. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of B. LDB >= max(1,N). *> \endverbatim *> *> \param[out] ALPHA *> \verbatim *> ALPHA is COMPLEX*16 array, dimension (N) *> The complex scalars alpha that define the eigenvalues of *> GNEP. *> \endverbatim *> *> \param[out] BETA *> \verbatim *> BETA is COMPLEX*16 array, dimension (N) *> The complex scalars beta that define the eigenvalues of GNEP. *> *> Together, the quantities alpha = ALPHA(j) and beta = BETA(j) *> represent the j-th eigenvalue of the matrix pair (A,B), in *> one of the forms lambda = alpha/beta or mu = beta/alpha. *> Since either lambda or mu may overflow, they should not, *> in general, be computed. *> \endverbatim *> *> \param[out] VL *> \verbatim *> VL is COMPLEX*16 array, dimension (LDVL,N) *> If JOBVL = 'V', the left eigenvectors u(j) are stored *> in the columns of VL, in the same order as their eigenvalues. *> Each eigenvector is scaled so that its largest component has *> abs(real part) + abs(imag. part) = 1, except for eigenvectors *> corresponding to an eigenvalue with alpha = beta = 0, which *> are set to zero. *> Not referenced if JOBVL = 'N'. *> \endverbatim *> *> \param[in] LDVL *> \verbatim *> LDVL is INTEGER *> The leading dimension of the matrix VL. LDVL >= 1, and *> if JOBVL = 'V', LDVL >= N. *> \endverbatim *> *> \param[out] VR *> \verbatim *> VR is COMPLEX*16 array, dimension (LDVR,N) *> If JOBVR = 'V', the right eigenvectors x(j) are stored *> in the columns of VR, in the same order as their eigenvalues. *> Each eigenvector is scaled so that its largest component has *> abs(real part) + abs(imag. part) = 1, except for eigenvectors *> corresponding to an eigenvalue with alpha = beta = 0, which *> are set to zero. *> Not referenced if JOBVR = 'N'. *> \endverbatim *> *> \param[in] LDVR *> \verbatim *> LDVR is INTEGER *> The leading dimension of the matrix VR. LDVR >= 1, and *> if JOBVR = 'V', LDVR >= N. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. LWORK >= max(1,2*N). *> For good performance, LWORK must generally be larger. *> To compute the optimal value of LWORK, call ILAENV to get *> blocksizes (for ZGEQRF, ZUNMQR, and ZUNGQR.) Then compute: *> NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR; *> The optimal LWORK is MAX( 2*N, N*(NB+1) ). *> *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates the optimal size of the WORK array, returns *> this value as the first entry of the WORK array, and no error *> message related to LWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is DOUBLE PRECISION array, dimension (8*N) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value. *> =1,...,N: *> The QZ iteration failed. No eigenvectors have been *> calculated, but ALPHA(j) and BETA(j) should be *> correct for j=INFO+1,...,N. *> > N: errors that usually indicate LAPACK problems: *> =N+1: error return from ZGGBAL *> =N+2: error return from ZGEQRF *> =N+3: error return from ZUNMQR *> =N+4: error return from ZUNGQR *> =N+5: error return from ZGGHRD *> =N+6: error return from ZHGEQZ (other than failed *> iteration) *> =N+7: error return from ZTGEVC *> =N+8: error return from ZGGBAK (computing VL) *> =N+9: error return from ZGGBAK (computing VR) *> =N+10: error return from ZLASCL (various calls) *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date November 2011 * *> \ingroup complex16GEeigen * *> \par Further Details: * ===================== *> *> \verbatim *> *> Balancing *> --------- *> *> This driver calls ZGGBAL to both permute and scale rows and columns *> of A and B. The permutations PL and PR are chosen so that PL*A*PR *> and PL*B*R will be upper triangular except for the diagonal blocks *> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as *> possible. The diagonal scaling matrices DL and DR are chosen so *> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to *> one (except for the elements that start out zero.) *> *> After the eigenvalues and eigenvectors of the balanced matrices *> have been computed, ZGGBAK transforms the eigenvectors back to what *> they would have been (in perfect arithmetic) if they had not been *> balanced. *> *> Contents of A and B on Exit *> -------- -- - --- - -- ---- *> *> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or *> both), then on exit the arrays A and B will contain the complex Schur *> form[*] of the "balanced" versions of A and B. If no eigenvectors *> are computed, then only the diagonal blocks will be correct. *> *> [*] In other words, upper triangular form. *> \endverbatim *> * ===================================================================== SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, \$ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO ) * * -- LAPACK driver routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2011 * * .. Scalar Arguments .. CHARACTER JOBVL, JOBVR INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N * .. * .. Array Arguments .. DOUBLE PRECISION RWORK( * ) COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), \$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ), \$ WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ), \$ CONE = ( 1.0D0, 0.0D0 ) ) * .. * .. Local Scalars .. LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY CHARACTER CHTEMP INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO, \$ IN, IRIGHT, IROWS, IRWORK, ITAU, IWORK, JC, JR, \$ LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3 DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM, \$ BNRM1, BNRM2, EPS, SAFMAX, SAFMIN, SALFAI, \$ SALFAR, SBETA, SCALE, TEMP COMPLEX*16 X * .. * .. Local Arrays .. LOGICAL LDUMMA( 1 ) * .. * .. External Subroutines .. EXTERNAL XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD, ZHGEQZ, \$ ZLACPY, ZLASCL, ZLASET, ZTGEVC, ZUNGQR, ZUNMQR * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, ZLANGE EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, MAX * .. * .. Statement Functions .. DOUBLE PRECISION ABS1 * .. * .. Statement Function definitions .. ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) ) * .. * .. Executable Statements .. * * Decode the input arguments * IF( LSAME( JOBVL, 'N' ) ) THEN IJOBVL = 1 ILVL = .FALSE. ELSE IF( LSAME( JOBVL, 'V' ) ) THEN IJOBVL = 2 ILVL = .TRUE. ELSE IJOBVL = -1 ILVL = .FALSE. END IF * IF( LSAME( JOBVR, 'N' ) ) THEN IJOBVR = 1 ILVR = .FALSE. ELSE IF( LSAME( JOBVR, 'V' ) ) THEN IJOBVR = 2 ILVR = .TRUE. ELSE IJOBVR = -1 ILVR = .FALSE. END IF ILV = ILVL .OR. ILVR * * Test the input arguments * LWKMIN = MAX( 2*N, 1 ) LWKOPT = LWKMIN WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) INFO = 0 IF( IJOBVL.LE.0 ) THEN INFO = -1 ELSE IF( IJOBVR.LE.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN INFO = -11 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN INFO = -13 ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN INFO = -15 END IF * IF( INFO.EQ.0 ) THEN NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, N, -1, -1 ) NB2 = ILAENV( 1, 'ZUNMQR', ' ', N, N, N, -1 ) NB3 = ILAENV( 1, 'ZUNGQR', ' ', N, N, N, -1 ) NB = MAX( NB1, NB2, NB3 ) LOPT = MAX( 2*N, N*( NB+1 ) ) WORK( 1 ) = LOPT END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGEGV ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * * Get machine constants * EPS = DLAMCH( 'E' )*DLAMCH( 'B' ) SAFMIN = DLAMCH( 'S' ) SAFMIN = SAFMIN + SAFMIN SAFMAX = ONE / SAFMIN * * Scale A * ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK ) ANRM1 = ANRM ANRM2 = ONE IF( ANRM.LT.ONE ) THEN IF( SAFMAX*ANRM.LT.ONE ) THEN ANRM1 = SAFMIN ANRM2 = SAFMAX*ANRM END IF END IF * IF( ANRM.GT.ZERO ) THEN CALL ZLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 10 RETURN END IF END IF * * Scale B * BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK ) BNRM1 = BNRM BNRM2 = ONE IF( BNRM.LT.ONE ) THEN IF( SAFMAX*BNRM.LT.ONE ) THEN BNRM1 = SAFMIN BNRM2 = SAFMAX*BNRM END IF END IF * IF( BNRM.GT.ZERO ) THEN CALL ZLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 10 RETURN END IF END IF * * Permute the matrix to make it more nearly triangular * Also "balance" the matrix. * ILEFT = 1 IRIGHT = N + 1 IRWORK = IRIGHT + N CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ), \$ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 1 GO TO 80 END IF * * Reduce B to triangular form, and initialize VL and/or VR * IROWS = IHI + 1 - ILO IF( ILV ) THEN ICOLS = N + 1 - ILO ELSE ICOLS = IROWS END IF ITAU = 1 IWORK = ITAU + IROWS CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), \$ WORK( IWORK ), LWORK+1-IWORK, IINFO ) IF( IINFO.GE.0 ) \$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN INFO = N + 2 GO TO 80 END IF * CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, \$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ), \$ LWORK+1-IWORK, IINFO ) IF( IINFO.GE.0 ) \$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN INFO = N + 3 GO TO 80 END IF * IF( ILVL ) THEN CALL ZLASET( 'Full', N, N, CZERO, CONE, VL, LDVL ) CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, \$ VL( ILO+1, ILO ), LDVL ) CALL ZUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL, \$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK, \$ IINFO ) IF( IINFO.GE.0 ) \$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN INFO = N + 4 GO TO 80 END IF END IF * IF( ILVR ) \$ CALL ZLASET( 'Full', N, N, CZERO, CONE, VR, LDVR ) * * Reduce to generalized Hessenberg form * IF( ILV ) THEN * * Eigenvectors requested -- work on whole matrix. * CALL ZGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, \$ LDVL, VR, LDVR, IINFO ) ELSE CALL ZGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA, \$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO ) END IF IF( IINFO.NE.0 ) THEN INFO = N + 5 GO TO 80 END IF * * Perform QZ algorithm * IWORK = ITAU IF( ILV ) THEN CHTEMP = 'S' ELSE CHTEMP = 'E' END IF CALL ZHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, \$ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWORK ), \$ LWORK+1-IWORK, RWORK( IRWORK ), IINFO ) IF( IINFO.GE.0 ) \$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN INFO = IINFO ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN INFO = IINFO - N ELSE INFO = N + 6 END IF GO TO 80 END IF * IF( ILV ) THEN * * Compute Eigenvectors * IF( ILVL ) THEN IF( ILVR ) THEN CHTEMP = 'B' ELSE CHTEMP = 'L' END IF ELSE CHTEMP = 'R' END IF * CALL ZTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL, \$ VR, LDVR, N, IN, WORK( IWORK ), RWORK( IRWORK ), \$ IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 7 GO TO 80 END IF * * Undo balancing on VL and VR, rescale * IF( ILVL ) THEN CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ), \$ RWORK( IRIGHT ), N, VL, LDVL, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 8 GO TO 80 END IF DO 30 JC = 1, N TEMP = ZERO DO 10 JR = 1, N TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) ) 10 CONTINUE IF( TEMP.LT.SAFMIN ) \$ GO TO 30 TEMP = ONE / TEMP DO 20 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP 20 CONTINUE 30 CONTINUE END IF IF( ILVR ) THEN CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ), \$ RWORK( IRIGHT ), N, VR, LDVR, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 9 GO TO 80 END IF DO 60 JC = 1, N TEMP = ZERO DO 40 JR = 1, N TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) ) 40 CONTINUE IF( TEMP.LT.SAFMIN ) \$ GO TO 60 TEMP = ONE / TEMP DO 50 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP 50 CONTINUE 60 CONTINUE END IF * * End of eigenvector calculation * END IF * * Undo scaling in alpha, beta * * Note: this does not give the alpha and beta for the unscaled * problem. * * Un-scaling is limited to avoid underflow in alpha and beta * if they are significant. * DO 70 JC = 1, N ABSAR = ABS( DBLE( ALPHA( JC ) ) ) ABSAI = ABS( DIMAG( ALPHA( JC ) ) ) ABSB = ABS( DBLE( BETA( JC ) ) ) SALFAR = ANRM*DBLE( ALPHA( JC ) ) SALFAI = ANRM*DIMAG( ALPHA( JC ) ) SBETA = BNRM*DBLE( BETA( JC ) ) ILIMIT = .FALSE. SCALE = ONE * * Check for significant underflow in imaginary part of ALPHA * IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE. \$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN ILIMIT = .TRUE. SCALE = ( SAFMIN / ANRM1 ) / MAX( SAFMIN, ANRM2*ABSAI ) END IF * * Check for significant underflow in real part of ALPHA * IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE. \$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN ILIMIT = .TRUE. SCALE = MAX( SCALE, ( SAFMIN / ANRM1 ) / \$ MAX( SAFMIN, ANRM2*ABSAR ) ) END IF * * Check for significant underflow in BETA * IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE. \$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN ILIMIT = .TRUE. SCALE = MAX( SCALE, ( SAFMIN / BNRM1 ) / \$ MAX( SAFMIN, BNRM2*ABSB ) ) END IF * * Check for possible overflow when limiting scaling * IF( ILIMIT ) THEN TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ), \$ ABS( SBETA ) ) IF( TEMP.GT.ONE ) \$ SCALE = SCALE / TEMP IF( SCALE.LT.ONE ) \$ ILIMIT = .FALSE. END IF * * Recompute un-scaled ALPHA, BETA if necessary. * IF( ILIMIT ) THEN SALFAR = ( SCALE*DBLE( ALPHA( JC ) ) )*ANRM SALFAI = ( SCALE*DIMAG( ALPHA( JC ) ) )*ANRM SBETA = ( SCALE*BETA( JC ) )*BNRM END IF ALPHA( JC ) = DCMPLX( SALFAR, SALFAI ) BETA( JC ) = SBETA 70 CONTINUE * 80 CONTINUE WORK( 1 ) = LWKOPT * RETURN * * End of ZGEGV * END