SUBROUTINE CGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
$ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
*
* -- LAPACK driver routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
REAL RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
$ WORK( * )
* ..
*
* Purpose
* =======
*
* This routine is deprecated and has been replaced by routine CGGEV.
*
* CGEGV 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 -- 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) COMPLEX 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 CGGHRD and CHGEQZ
* for details.
*
* LDA (input) INTEGER
* The leading dimension of A. LDA >= max(1,N).
*
* B (input/output) COMPLEX 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 CGGHRD and CHGEQZ for
* details.
*
* LDB (input) INTEGER
* The leading dimension of B. LDB >= max(1,N).
*
* ALPHA (output) COMPLEX array, dimension (N)
* The complex scalars alpha that define the eigenvalues of
* GNEP.
*
* BETA (output) COMPLEX 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.
*
* VL (output) COMPLEX 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'.
*
* LDVL (input) INTEGER
* The leading dimension of the matrix VL. LDVL >= 1, and
* if JOBVL = 'V', LDVL >= N.
*
* VR (output) COMPLEX 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'.
*
* LDVR (input) INTEGER
* The leading dimension of the matrix VR. LDVR >= 1, and
* if JOBVR = 'V', LDVR >= N.
*
* WORK (workspace/output) COMPLEX 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(1,2*N).
* For good performance, LWORK must generally be larger.
* To compute the optimal value of LWORK, call ILAENV to get
* blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute:
* NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR;
* 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.
*
* RWORK (workspace/output) REAL array, dimension (8*N)
*
* 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 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 CGGBAL
* =N+2: error return from CGEQRF
* =N+3: error return from CUNMQR
* =N+4: error return from CUNGQR
* =N+5: error return from CGGHRD
* =N+6: error return from CHGEQZ (other than failed
* iteration)
* =N+7: error return from CTGEVC
* =N+8: error return from CGGBAK (computing VL)
* =N+9: error return from CGGBAK (computing VR)
* =N+10: error return from CLASCL (various calls)
*
* Further Details
* ===============
*
* Balancing
* ---------
*
* This driver calls CGGBAL 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, CGGBAK 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.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ),
$ CONE = ( 1.0E0, 0.0E0 ) )
* ..
* .. 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
REAL ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
$ BNRM1, BNRM2, EPS, SAFMAX, SAFMIN, SALFAI,
$ SALFAR, SBETA, SCALE, TEMP
COMPLEX X
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL CGEQRF, CGGBAK, CGGBAL, CGGHRD, CHGEQZ, CLACPY,
$ CLASCL, CLASET, CTGEVC, CUNGQR, CUNMQR, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL CLANGE, SLAMCH
EXTERNAL ILAENV, LSAME, CLANGE, SLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, CMPLX, INT, MAX, REAL
* ..
* .. Statement Functions ..
REAL ABS1
* ..
* .. Statement Function definitions ..
ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( 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, 'CGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'CUNMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'CUNGQR', ' ', 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( 'CGEGV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
SAFMIN = SLAMCH( 'S' )
SAFMIN = SAFMIN + SAFMIN
SAFMAX = ONE / SAFMIN
*
* Scale A
*
ANRM = CLANGE( '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 CLASCL( '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 = CLANGE( '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 CLASCL( '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 CGGBAL( '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 CGEQRF( 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 CUNMQR( '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 CLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
CALL CLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
CALL CUNGQR( 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 CLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
*
IF( ILV ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL CGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, IINFO )
ELSE
CALL CGGHRD( '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 CHGEQZ( 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 CTGEVC( 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 CGGBAK( '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 CGGBAK( '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( REAL( ALPHA( JC ) ) )
ABSAI = ABS( AIMAG( ALPHA( JC ) ) )
ABSB = ABS( REAL( BETA( JC ) ) )
SALFAR = ANRM*REAL( ALPHA( JC ) )
SALFAI = ANRM*AIMAG( ALPHA( JC ) )
SBETA = BNRM*REAL( 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*REAL( ALPHA( JC ) ) )*ANRM
SALFAI = ( SCALE*AIMAG( ALPHA( JC ) ) )*ANRM
SBETA = ( SCALE*BETA( JC ) )*BNRM
END IF
ALPHA( JC ) = CMPLX( SALFAR, SALFAI )
BETA( JC ) = SBETA
70 CONTINUE
*
80 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of CGEGV
*
END