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SUBROUTINE SGGEV( 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 ..
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* SGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B)
* the generalized eigenvalues, and optionally, the left and/or right
* generalized eigenvectors.
*
* A generalized eigenvalue for a pair of matrices (A,B) is a scalar
* lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
* singular. It is usually represented as the pair (alpha,beta), as
* there is a reasonable interpretation for beta=0, and even for both
* being zero.
*
* The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
* of (A,B) satisfies
*
* A * v(j) = lambda(j) * B * v(j).
*
* The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
* of (A,B) satisfies
*
* u(j)**H * A = lambda(j) * u(j)**H * B .
*
* where u(j)**H is the conjugate-transpose of u(j).
*
*
* Arguments
* =========
*
* JOBVL (input) CHARACTER*1
* = 'N': do not compute the left generalized eigenvectors;
* = 'V': compute the left generalized eigenvectors.
*
* JOBVR (input) CHARACTER*1
* = 'N': do not compute the right generalized eigenvectors;
* = 'V': compute the right generalized eigenvectors.
*
* N (input) INTEGER
* The order of the matrices A, B, VL, and VR. N >= 0.
*
* A (input/output) REAL array, dimension (LDA, N)
* On entry, the matrix A in the pair (A,B).
* On exit, A has been overwritten.
*
* LDA (input) INTEGER
* The leading dimension of A. LDA >= max(1,N).
*
* B (input/output) REAL array, dimension (LDB, N)
* On entry, the matrix B in the pair (A,B).
* On exit, B has been overwritten.
*
* LDB (input) INTEGER
* The leading dimension of B. LDB >= max(1,N).
*
* ALPHAR (output) REAL array, dimension (N)
* ALPHAI (output) REAL array, dimension (N)
* BETA (output) REAL array, dimension (N)
* On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
* be the generalized eigenvalues. 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) negative.
*
* Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
* may easily over- or underflow, and BETA(j) may even be zero.
* Thus, the user should avoid naively computing the ratio
* alpha/beta. However, ALPHAR and ALPHAI will be always less
* than and usually comparable with norm(A) in magnitude, and
* BETA always less than and usually comparable with norm(B).
*
* VL (output) REAL array, dimension (LDVL,N)
* If JOBVL = 'V', the left eigenvectors u(j) are stored one
* after another in the columns of VL, in the same order as
* their eigenvalues. If the j-th eigenvalue is real, then
* u(j) = VL(:,j), the j-th column of VL. If the j-th and
* (j+1)-th 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 the largest component has
* abs(real part)+abs(imag. part)=1.
* 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) REAL array, dimension (LDVR,N)
* If JOBVR = 'V', the right eigenvectors v(j) are stored one
* after another in the columns of VR, in the same order as
* their eigenvalues. If the j-th eigenvalue is real, then
* v(j) = VR(:,j), the j-th column of VR. If the j-th and
* (j+1)-th eigenvalues form a complex conjugate pair, then
* v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
* Each eigenvector is scaled so the largest component has
* abs(real part)+abs(imag. part)=1.
* 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) REAL 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.
*
* 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: =N+1: other than QZ iteration failed in SHGEQZ.
* =N+2: error return from STGEVC.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
CHARACTER CHTEMP
INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
$ IN, IRIGHT, IROWS, ITAU, IWRK, JC, JR, MAXWRK,
$ MINWRK
REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SMLNUM, TEMP
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLABAD,
$ SLACPY, SLASCL, SLASET, SORGQR, SORMQR, STGEVC,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL SLAMCH, SLANGE
EXTERNAL LSAME, ILAENV, SLAMCH, SLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. 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
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
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
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV. The workspace is
* computed assuming ILO = 1 and IHI = N, the worst case.)
*
IF( INFO.EQ.0 ) THEN
MINWRK = MAX( 1, 8*N )
MAXWRK = MAX( 1, N*( 7 +
$ ILAENV( 1, 'SGEQRF', ' ', N, 1, N, 0 ) ) )
MAXWRK = MAX( MAXWRK, N*( 7 +
$ ILAENV( 1, 'SORMQR', ' ', N, 1, N, 0 ) ) )
IF( ILVL ) THEN
MAXWRK = MAX( MAXWRK, N*( 7 +
$ ILAENV( 1, 'SORGQR', ' ', N, 1, N, -1 ) ) )
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
$ INFO = -16
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGGEV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = SLAMCH( 'P' )
SMLNUM = SLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
CALL SLABAD( SMLNUM, BIGNUM )
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = SLANGE( 'M', N, N, A, LDA, WORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
IF( ILASCL )
$ CALL SLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = SLANGE( 'M', N, N, B, LDB, WORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
IF( ILBSCL )
$ CALL SLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
* Permute the matrices A, B to isolate eigenvalues if possible
* (Workspace: need 6*N)
*
ILEFT = 1
IRIGHT = N + 1
IWRK = IRIGHT + N
CALL SGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWRK ), IERR )
*
* Reduce B to triangular form (QR decomposition of B)
* (Workspace: need N, prefer N*NB)
*
IROWS = IHI + 1 - ILO
IF( ILV ) THEN
ICOLS = N + 1 - ILO
ELSE
ICOLS = IROWS
END IF
ITAU = IWRK
IWRK = ITAU + IROWS
CALL SGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
*
* Apply the orthogonal transformation to matrix A
* (Workspace: need N, prefer N*NB)
*
CALL SORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
$ LWORK+1-IWRK, IERR )
*
* Initialize VL
* (Workspace: need N, prefer N*NB)
*
IF( ILVL ) THEN
CALL SLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
IF( IROWS.GT.1 ) THEN
CALL SLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
END IF
CALL SORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
* Initialize VR
*
IF( ILVR )
$ CALL SLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
* (Workspace: none needed)
*
IF( ILV ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL SGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, IERR )
ELSE
CALL SGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
END IF
*
* Perform QZ algorithm (Compute eigenvalues, and optionally, the
* Schur forms and Schur vectors)
* (Workspace: need N)
*
IWRK = ITAU
IF( ILV ) THEN
CHTEMP = 'S'
ELSE
CHTEMP = 'E'
END IF
CALL SHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
IF( IERR.NE.0 ) THEN
IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
INFO = IERR
ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
INFO = IERR - N
ELSE
INFO = N + 1
END IF
GO TO 110
END IF
*
* Compute Eigenvectors
* (Workspace: need 6*N)
*
IF( ILV ) THEN
IF( ILVL ) THEN
IF( ILVR ) THEN
CHTEMP = 'B'
ELSE
CHTEMP = 'L'
END IF
ELSE
CHTEMP = 'R'
END IF
CALL STGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
$ VR, LDVR, N, IN, WORK( IWRK ), IERR )
IF( IERR.NE.0 ) THEN
INFO = N + 2
GO TO 110
END IF
*
* Undo balancing on VL and VR and normalization
* (Workspace: none needed)
*
IF( ILVL ) THEN
CALL SGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VL, LDVL, IERR )
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.SMLNUM )
$ 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 SGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VR, LDVR, IERR )
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.SMLNUM )
$ 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 if necessary
*
IF( ILASCL ) THEN
CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
END IF
*
IF( ILBSCL ) THEN
CALL SLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
END IF
*
110 CONTINUE
*
WORK( 1 ) = MAXWRK
*
RETURN
*
* End of SGGEV
*
END
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