SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
$ SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
$ LDVSR, WORK, LWORK, BWORK, INFO )
*
* -- LAPACK driver routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOBVSL, JOBVSR, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
* ..
* .. Array Arguments ..
LOGICAL BWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
$ VSR( LDVSR, * ), WORK( * )
* ..
* .. Function Arguments ..
LOGICAL SELCTG
EXTERNAL SELCTG
* ..
*
* Purpose
* =======
*
* DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
* the generalized eigenvalues, the generalized real Schur form (S,T),
* optionally, the left and/or right matrices of Schur vectors (VSL and
* VSR). This gives the generalized Schur factorization
*
* (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
*
* Optionally, it also orders the eigenvalues so that a selected cluster
* of eigenvalues appears in the leading diagonal blocks of the upper
* quasi-triangular matrix S and the upper triangular matrix T.The
* leading columns of VSL and VSR then form an orthonormal basis for the
* corresponding left and right eigenspaces (deflating subspaces).
*
* (If only the generalized eigenvalues are needed, use the driver
* DGGEV instead, which is faster.)
*
* A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
* or a ratio alpha/beta = w, such that A - w*B is singular. It is
* usually represented as the pair (alpha,beta), as there is a
* reasonable interpretation for beta=0 or both being zero.
*
* A pair of matrices (S,T) is in generalized real Schur form if T is
* upper triangular with non-negative diagonal and S is block upper
* triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
* to real generalized eigenvalues, while 2-by-2 blocks of S will be
* "standardized" by making the corresponding elements of T have the
* form:
* [ a 0 ]
* [ 0 b ]
*
* and the pair of corresponding 2-by-2 blocks in S and T will have a
* complex conjugate pair of generalized eigenvalues.
*
*
* Arguments
* =========
*
* JOBVSL (input) CHARACTER*1
* = 'N': do not compute the left Schur vectors;
* = 'V': compute the left Schur vectors.
*
* JOBVSR (input) CHARACTER*1
* = 'N': do not compute the right Schur vectors;
* = 'V': compute the right Schur vectors.
*
* SORT (input) CHARACTER*1
* Specifies whether or not to order the eigenvalues on the
* diagonal of the generalized Schur form.
* = 'N': Eigenvalues are not ordered;
* = 'S': Eigenvalues are ordered (see SELCTG);
*
* SELCTG (external procedure) LOGICAL FUNCTION of three DOUBLE PRECISION arguments
* SELCTG must be declared EXTERNAL in the calling subroutine.
* If SORT = 'N', SELCTG is not referenced.
* If SORT = 'S', SELCTG is used to select eigenvalues to sort
* to the top left of the Schur form.
* An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
* SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
* one of a complex conjugate pair of eigenvalues is selected,
* then both complex eigenvalues are selected.
*
* Note that in the ill-conditioned case, a selected complex
* eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
* BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
* in this case.
*
* N (input) INTEGER
* The order of the matrices A, B, VSL, and VSR. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
* On entry, the first of the pair of matrices.
* On exit, A has been overwritten by its generalized Schur
* form S.
*
* LDA (input) INTEGER
* The leading dimension of A. LDA >= max(1,N).
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
* On entry, the second of the pair of matrices.
* On exit, B has been overwritten by its generalized Schur
* form T.
*
* LDB (input) INTEGER
* The leading dimension of B. LDB >= max(1,N).
*
* SDIM (output) INTEGER
* If SORT = 'N', SDIM = 0.
* If SORT = 'S', SDIM = number of eigenvalues (after sorting)
* for which SELCTG is true. (Complex conjugate pairs for which
* SELCTG is true for either eigenvalue count as 2.)
*
* ALPHAR (output) DOUBLE PRECISION array, dimension (N)
* ALPHAI (output) DOUBLE PRECISION array, dimension (N)
* BETA (output) DOUBLE PRECISION array, dimension (N)
* On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
* be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i,
* and BETA(j),j=1,...,N are the diagonals of the complex Schur
* form (S,T) that would result if the 2-by-2 diagonal blocks of
* the real Schur form of (A,B) were further reduced to
* triangular form using 2-by-2 complex unitary transformations.
* 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.
* 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).
*
* VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N)
* If JOBVSL = 'V', VSL will contain the left Schur vectors.
* Not referenced if JOBVSL = 'N'.
*
* LDVSL (input) INTEGER
* The leading dimension of the matrix VSL. LDVSL >=1, and
* if JOBVSL = 'V', LDVSL >= N.
*
* VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N)
* If JOBVSR = 'V', VSR will contain the right Schur vectors.
* Not referenced if JOBVSR = 'N'.
*
* LDVSR (input) INTEGER
* The leading dimension of the matrix VSR. LDVSR >= 1, and
* if JOBVSR = 'V', LDVSR >= 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.
* If N = 0, LWORK >= 1, else LWORK >= 8*N+16.
* 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.
*
* BWORK (workspace) LOGICAL array, dimension (N)
* Not referenced if SORT = '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. (A,B) are not in Schur
* form, 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 DHGEQZ.
* =N+2: after reordering, roundoff changed values of
* some complex eigenvalues so that leading
* eigenvalues in the Generalized Schur form no
* longer satisfy SELCTG=.TRUE. This could also
* be caused due to scaling.
* =N+3: reordering failed in DTGSEN.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
$ LQUERY, LST2SL, WANTST
INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
$ ILO, IP, IRIGHT, IROWS, ITAU, IWRK, MAXWRK,
$ MINWRK
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
$ PVSR, SAFMAX, SAFMIN, SMLNUM
* ..
* .. Local Arrays ..
INTEGER IDUM( 1 )
DOUBLE PRECISION DIF( 2 )
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
$ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGSEN,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVSL, 'N' ) ) THEN
IJOBVL = 1
ILVSL = .FALSE.
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
IJOBVL = 2
ILVSL = .TRUE.
ELSE
IJOBVL = -1
ILVSL = .FALSE.
END IF
*
IF( LSAME( JOBVSR, 'N' ) ) THEN
IJOBVR = 1
ILVSR = .FALSE.
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
IJOBVR = 2
ILVSR = .TRUE.
ELSE
IJOBVR = -1
ILVSR = .FALSE.
END IF
*
WANTST = LSAME( SORT, 'S' )
*
* 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( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
INFO = -15
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
INFO = -17
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.)
*
IF( INFO.EQ.0 ) THEN
IF( N.GT.0 )THEN
MINWRK = MAX( 8*N, 6*N + 16 )
MAXWRK = MINWRK - N +
$ N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 )
MAXWRK = MAX( MAXWRK, MINWRK - N +
$ N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, -1 ) )
IF( ILVSL ) THEN
MAXWRK = MAX( MAXWRK, MINWRK - N +
$ N*ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) )
END IF
ELSE
MINWRK = 1
MAXWRK = 1
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
$ INFO = -19
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGES ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
SDIM = 0
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
CALL DLABAD( SAFMIN, SAFMAX )
SMLNUM = SQRT( SAFMIN ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( '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 DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = DLANGE( '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 DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
* Permute the matrix to make it more nearly triangular
* (Workspace: need 6*N + 2*N space for storing balancing factors)
*
ILEFT = 1
IRIGHT = N + 1
IWRK = IRIGHT + N
CALL DGGBAL( '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
ICOLS = N + 1 - ILO
ITAU = IWRK
IWRK = ITAU + IROWS
CALL DGEQRF( 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 DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
$ LWORK+1-IWRK, IERR )
*
* Initialize VSL
* (Workspace: need N, prefer N*NB)
*
IF( ILVSL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
IF( IROWS.GT.1 ) THEN
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
END IF
CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
* Initialize VSR
*
IF( ILVSR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
* (Workspace: none needed)
*
CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, IERR )
*
* Perform QZ algorithm, computing Schur vectors if desired
* (Workspace: need N)
*
IWRK = ITAU
CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ 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 50
END IF
*
* Sort eigenvalues ALPHA/BETA if desired
* (Workspace: need 4*N+16 )
*
SDIM = 0
IF( WANTST ) THEN
*
* Undo scaling on eigenvalues before SELCTGing
*
IF( ILASCL ) THEN
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IERR )
END IF
IF( ILBSCL )
$ CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
*
* Select eigenvalues
*
DO 10 I = 1, N
BWORK( I ) = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
10 CONTINUE
*
CALL DTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, ALPHAR,
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PVSL,
$ PVSR, DIF, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1,
$ IERR )
IF( IERR.EQ.1 )
$ INFO = N + 3
*
END IF
*
* Apply back-permutation to VSL and VSR
* (Workspace: none needed)
*
IF( ILVSL )
$ CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSL, LDVSL, IERR )
*
IF( ILVSR )
$ CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSR, LDVSR, IERR )
*
* Check if unscaling would cause over/underflow, if so, rescale
* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
*
IF( ILASCL ) THEN
DO 20 I = 1, N
IF( ALPHAI( I ).NE.ZERO ) THEN
IF( ( ALPHAR( I ) / SAFMAX ).GT.( ANRMTO / ANRM ) .OR.
$ ( SAFMIN / ALPHAR( I ) ).GT.( ANRM / ANRMTO ) ) THEN
WORK( 1 ) = ABS( A( I, I ) / ALPHAR( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
ELSE IF( ( ALPHAI( I ) / SAFMAX ).GT.
$ ( ANRMTO / ANRM ) .OR.
$ ( SAFMIN / ALPHAI( I ) ).GT.( ANRM / ANRMTO ) )
$ THEN
WORK( 1 ) = ABS( A( I, I+1 ) / ALPHAI( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
END IF
END IF
20 CONTINUE
END IF
*
IF( ILBSCL ) THEN
DO 30 I = 1, N
IF( ALPHAI( I ).NE.ZERO ) THEN
IF( ( BETA( I ) / SAFMAX ).GT.( BNRMTO / BNRM ) .OR.
$ ( SAFMIN / BETA( I ) ).GT.( BNRM / BNRMTO ) ) THEN
WORK( 1 ) = ABS( B( I, I ) / BETA( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
END IF
END IF
30 CONTINUE
END IF
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL DLASCL( 'H', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR )
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
END IF
*
IF( WANTST ) THEN
*
* Check if reordering is correct
*
LASTSL = .TRUE.
LST2SL = .TRUE.
SDIM = 0
IP = 0
DO 40 I = 1, N
CURSL = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
IF( ALPHAI( I ).EQ.ZERO ) THEN
IF( CURSL )
$ SDIM = SDIM + 1
IP = 0
IF( CURSL .AND. .NOT.LASTSL )
$ INFO = N + 2
ELSE
IF( IP.EQ.1 ) THEN
*
* Last eigenvalue of conjugate pair
*
CURSL = CURSL .OR. LASTSL
LASTSL = CURSL
IF( CURSL )
$ SDIM = SDIM + 2
IP = -1
IF( CURSL .AND. .NOT.LST2SL )
$ INFO = N + 2
ELSE
*
* First eigenvalue of conjugate pair
*
IP = 1
END IF
END IF
LST2SL = LASTSL
LASTSL = CURSL
40 CONTINUE
*
END IF
*
50 CONTINUE
*
WORK( 1 ) = MAXWRK
*
RETURN
*
* End of DGGES
*
END