SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, \$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) * * -- LAPACK driver routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. CHARACTER JOBVL, JOBVR INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), \$ B( LDB, * ), BETA( * ), VL( LDVL, * ), \$ VR( LDVR, * ), WORK( * ) * .. * * Purpose * ======= * * This routine is deprecated and has been replaced by routine DGGEV. * * DGEGV computes the eigenvalues and, optionally, the left and/or right * eigenvectors of a real 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 -- see * "Further Details", below. * * Arguments * ========= * * JOBVL (input) CHARACTER*1 * = 'N': do not compute the left generalized eigenvectors; * = 'V': compute the left generalized eigenvectors (returned * in VL). * * JOBVR (input) CHARACTER*1 * = 'N': do not compute the right generalized eigenvectors; * = 'V': compute the right generalized eigenvectors (returned * in VR). * * N (input) INTEGER * The order of the matrices A, B, VL, and VR. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA, N) * On entry, the matrix A. * If JOBVL = 'V' or JOBVR = 'V', then on exit A * contains the real 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 * blocks from the Schur form will be correct. See DGGHRD and * DHGEQZ for details. * * LDA (input) INTEGER * The leading dimension of A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION 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 those elements of * B corresponding to the diagonal blocks from the Schur form of * A will be correct. See DGGHRD and DHGEQZ for details. * * LDB (input) INTEGER * The leading dimension of B. LDB >= max(1,N). * * ALPHAR (output) DOUBLE PRECISION array, dimension (N) * The real parts of each scalar alpha defining an eigenvalue of * GNEP. * * ALPHAI (output) DOUBLE PRECISION array, dimension (N) * The imaginary parts of each scalar alpha defining an * eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th * eigenvalue is real; if positive, then the j-th and * (j+1)-st eigenvalues are a complex conjugate pair, with * ALPHAI(j+1) = -ALPHAI(j). * * BETA (output) DOUBLE PRECISION array, dimension (N) * The scalars beta that define the eigenvalues of GNEP. * * Together, the quantities alpha = (ALPHAR(j),ALPHAI(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. * * VL (output) DOUBLE PRECISION 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. * If the j-th eigenvalue is real, then u(j) = VL(:,j). * If the j-th and (j+1)-st eigenvalues form a complex conjugate * pair, then * u(j) = VL(:,j) + i*VL(:,j+1) * and * u(j+1) = VL(:,j) - i*VL(:,j+1). * * 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'. * * LDVL (input) INTEGER * The leading dimension of the matrix VL. LDVL >= 1, and * if JOBVL = 'V', LDVL >= N. * * VR (output) DOUBLE PRECISION 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. * If the j-th eigenvalue is real, then x(j) = VR(:,j). * If the j-th and (j+1)-st eigenvalues form a complex conjugate * pair, then * x(j) = VR(:,j) + i*VR(:,j+1) * and * x(j+1) = VR(:,j) - i*VR(:,j+1). * * Each eigenvector is scaled so that its largest component has * abs(real part) + abs(imag. part) = 1, except for eigenvalues * corresponding to an eigenvalue with alpha = beta = 0, which * are set to zero. * Not referenced if JOBVR = 'N'. * * LDVR (input) INTEGER * The leading dimension of the matrix VR. LDVR >= 1, and * if JOBVR = 'V', LDVR >= N. * * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK >= max(1,8*N). * For good performance, LWORK must generally be larger. * To compute the optimal value of LWORK, call ILAENV to get * blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: * NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR; * The optimal LWORK is: * 2*N + MAX( 6*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. * * INFO (output) 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 ALPHAR(j), ALPHAI(j), and BETA(j) * should be correct for j=INFO+1,...,N. * > N: errors that usually indicate LAPACK problems: * =N+1: error return from DGGBAL * =N+2: error return from DGEQRF * =N+3: error return from DORMQR * =N+4: error return from DORGQR * =N+5: error return from DGGHRD * =N+6: error return from DHGEQZ (other than failed * iteration) * =N+7: error return from DTGEVC * =N+8: error return from DGGBAK (computing VL) * =N+9: error return from DGGBAK (computing VR) * =N+10: error return from DLASCL (various calls) * * Further Details * =============== * * Balancing * --------- * * This driver calls DGGBAL 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, DGGBAK 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 real Schur * form[*] of the "balanced" versions of A and B. If no eigenvectors * are computed, then only the diagonal blocks will be correct. * * [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations", * by Golub & van Loan, pub. by Johns Hopkins U. Press. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) * .. * .. Local Scalars .. LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY CHARACTER CHTEMP INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO, \$ IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT, \$ LWKMIN, LWKOPT, NB, NB1, NB2, NB3 DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM, \$ BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN, \$ SALFAI, SALFAR, SBETA, SCALE, TEMP * .. * .. Local Arrays .. LOGICAL LDUMMA( 1 ) * .. * .. External Subroutines .. EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY, \$ DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE * .. * .. Intrinsic Functions .. INTRINSIC ABS, INT, MAX * .. * .. 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( 8*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 = -12 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN INFO = -14 ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN INFO = -16 END IF * IF( INFO.EQ.0 ) THEN NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 ) NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 ) NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 ) NB = MAX( NB1, NB2, NB3 ) LOPT = 2*N + MAX( 6*N, N*( NB+1 ) ) WORK( 1 ) = LOPT END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEGV ', -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 ONEPLS = ONE + ( 4*EPS ) * * Scale A * ANRM = DLANGE( 'M', N, N, A, LDA, WORK ) 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 DLASCL( '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 = DLANGE( 'M', N, N, B, LDB, WORK ) 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 DLASCL( '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 * Workspace layout: (8*N words -- "work" requires 6*N words) * left_permutation, right_permutation, work... * ILEFT = 1 IRIGHT = N + 1 IWORK = IRIGHT + N CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ), \$ WORK( IRIGHT ), WORK( IWORK ), IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 1 GO TO 120 END IF * * Reduce B to triangular form, and initialize VL and/or VR * Workspace layout: ("work..." must have at least N words) * left_permutation, right_permutation, tau, work... * IROWS = IHI + 1 - ILO IF( ILV ) THEN ICOLS = N + 1 - ILO ELSE ICOLS = IROWS END IF ITAU = IWORK IWORK = ITAU + IROWS CALL DGEQRF( 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 120 END IF * CALL DORMQR( 'L', 'T', 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 120 END IF * IF( ILVL ) THEN CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL ) CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, \$ VL( ILO+1, ILO ), LDVL ) CALL DORGQR( 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 120 END IF END IF * IF( ILVR ) \$ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR ) * * Reduce to generalized Hessenberg form * IF( ILV ) THEN * * Eigenvectors requested -- work on whole matrix. * CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, \$ LDVL, VR, LDVR, IINFO ) ELSE CALL DGGHRD( '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 120 END IF * * Perform QZ algorithm * Workspace layout: ("work..." must have at least 1 word) * left_permutation, right_permutation, work... * IWORK = ITAU IF( ILV ) THEN CHTEMP = 'S' ELSE CHTEMP = 'E' END IF CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, \$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, \$ WORK( IWORK ), LWORK+1-IWORK, 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 120 END IF * IF( ILV ) THEN * * Compute Eigenvectors (DTGEVC requires 6*N words of workspace) * IF( ILVL ) THEN IF( ILVR ) THEN CHTEMP = 'B' ELSE CHTEMP = 'L' END IF ELSE CHTEMP = 'R' END IF * CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL, \$ VR, LDVR, N, IN, WORK( IWORK ), IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 7 GO TO 120 END IF * * Undo balancing on VL and VR, rescale * IF( ILVL ) THEN CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ), \$ WORK( IRIGHT ), N, VL, LDVL, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 8 GO TO 120 END IF DO 50 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) \$ GO TO 50 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 10 JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) ) 10 CONTINUE ELSE DO 20 JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+ \$ ABS( VL( JR, JC+1 ) ) ) 20 CONTINUE END IF IF( TEMP.LT.SAFMIN ) \$ GO TO 50 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 30 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP 30 CONTINUE ELSE DO 40 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP 40 CONTINUE END IF 50 CONTINUE END IF IF( ILVR ) THEN CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ), \$ WORK( IRIGHT ), N, VR, LDVR, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 9 GO TO 120 END IF DO 100 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) \$ GO TO 100 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 60 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) ) 60 CONTINUE ELSE DO 70 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+ \$ ABS( VR( JR, JC+1 ) ) ) 70 CONTINUE END IF IF( TEMP.LT.SAFMIN ) \$ GO TO 100 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 80 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP 80 CONTINUE ELSE DO 90 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP 90 CONTINUE END IF 100 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 110 JC = 1, N ABSAR = ABS( ALPHAR( JC ) ) ABSAI = ABS( ALPHAI( JC ) ) ABSB = ABS( BETA( JC ) ) SALFAR = ANRM*ALPHAR( JC ) SALFAI = ANRM*ALPHAI( JC ) SBETA = BNRM*BETA( JC ) ILIMIT = .FALSE. SCALE = ONE * * Check for significant underflow in ALPHAI * IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE. \$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN ILIMIT = .TRUE. SCALE = ( ONEPLS*SAFMIN / ANRM1 ) / \$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAI ) * ELSE IF( SALFAI.EQ.ZERO ) THEN * * If insignificant underflow in ALPHAI, then make the * conjugate eigenvalue real. * IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN ALPHAI( JC-1 ) = ZERO ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN ALPHAI( JC+1 ) = ZERO END IF END IF * * Check for significant underflow in ALPHAR * IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE. \$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN ILIMIT = .TRUE. SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) / \$ MAX( ONEPLS*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, ( ONEPLS*SAFMIN / BNRM1 ) / \$ MAX( ONEPLS*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 ALPHAR, ALPHAI, BETA if necessary. * IF( ILIMIT ) THEN SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM SBETA = ( SCALE*BETA( JC ) )*BNRM END IF ALPHAR( JC ) = SALFAR ALPHAI( JC ) = SALFAI BETA( JC ) = SBETA 110 CONTINUE * 120 CONTINUE WORK( 1 ) = LWKOPT * RETURN * * End of DGEGV * END