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authorjofret2009-04-28 07:17:00 +0000
committerjofret2009-04-28 07:17:00 +0000
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- SUBROUTINE ZTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL,
- $ LDVL, VR, LDVR, MM, M, WORK, RWORK, INFO )
-*
-* -- LAPACK routine (version 3.1) --
-* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
-* November 2006
-*
-* .. 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( * )
-* ..
-*
-*
-* Purpose
-* =======
-*
-* 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).
-*
-* Arguments
-* =========
-*
-* SIDE (input) CHARACTER*1
-* = 'R': compute right eigenvectors only;
-* = 'L': compute left eigenvectors only;
-* = 'B': compute both right and left eigenvectors.
-*
-* HOWMNY (input) 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.
-*
-* SELECT (input) 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'.
-*
-* N (input) INTEGER
-* The order of the matrices S and P. N >= 0.
-*
-* S (input) COMPLEX*16 array, dimension (LDS,N)
-* The upper triangular matrix S from a generalized Schur
-* factorization, as computed by ZHGEQZ.
-*
-* LDS (input) INTEGER
-* The leading dimension of array S. LDS >= max(1,N).
-*
-* P (input) 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.
-*
-* LDP (input) INTEGER
-* The leading dimension of array P. LDP >= max(1,N).
-*
-* VL (input/output) 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'.
-*
-* LDVL (input) INTEGER
-* The leading dimension of array VL. LDVL >= 1, and if
-* SIDE = 'L' or 'l' or 'B' or 'b', LDVL >= N.
-*
-* VR (input/output) 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'.
-*
-* LDVR (input) INTEGER
-* The leading dimension of the array VR. LDVR >= 1, and if
-* SIDE = 'R' or 'B', LDVR >= N.
-*
-* MM (input) INTEGER
-* The number of columns in the arrays VL and/or VR. MM >= M.
-*
-* M (output) 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.
-*
-* WORK (workspace) COMPLEX*16 array, dimension (2*N)
-*
-* RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
-*
-* INFO (output) INTEGER
-* = 0: successful exit.
-* < 0: if INFO = -i, the i-th argument had an illegal value.
-*
-* =====================================================================
-*
-* .. 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