*> \brief DGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEEV + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
* LDVR, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVL, JOBVR
* INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
* $ WI( * ), WORK( * ), WR( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEEV computes for an N-by-N real nonsymmetric matrix A, the
*> eigenvalues and, optionally, the left and/or right eigenvectors.
*>
*> The right eigenvector v(j) of A satisfies
*> A * v(j) = lambda(j) * v(j)
*> where lambda(j) is its eigenvalue.
*> The left eigenvector u(j) of A satisfies
*> u(j)**H * A = lambda(j) * u(j)**H
*> where u(j)**H denotes the conjugate-transpose of u(j).
*>
*> The computed eigenvectors are normalized to have Euclidean norm
*> equal to 1 and largest component real.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': left eigenvectors of A are not computed;
*> = 'V': left eigenvectors of A are computed.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': right eigenvectors of A are not computed;
*> = 'V': right eigenvectors of A are computed.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the N-by-N matrix A.
*> On exit, A has been overwritten.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*> WR and WI contain the real and imaginary parts,
*> respectively, of the computed eigenvalues. Complex
*> conjugate pairs of eigenvalues appear consecutively
*> with the eigenvalue having the positive imaginary part
*> first.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is DOUBLE PRECISION 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 JOBVL = 'N', VL is not referenced.
*> 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)-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).
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL. LDVL >= 1; if
*> JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is DOUBLE PRECISION 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 JOBVR = 'N', VR is not referenced.
*> 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)-st 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).
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR. LDVR >= 1; if
*> JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,3*N), and
*> if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, the QR algorithm failed to compute all the
*> eigenvalues, and no eigenvectors have been computed;
*> elements i+1:N of WR and WI contain eigenvalues which
*> have converged.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleGEeigen
*
* =====================================================================
SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
$ LDVR, WORK, LWORK, INFO )
*
* -- LAPACK driver routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
$ WI( * ), WORK( * ), WR( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, SCALEA, WANTVL, WANTVR
CHARACTER SIDE
INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K,
$ MAXWRK, MINWRK, NOUT
DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
$ SN
* ..
* .. Local Arrays ..
LOGICAL SELECT( 1 )
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY,
$ DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX, ILAENV
DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2
EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2,
$ DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
WANTVL = LSAME( JOBVL, 'V' )
WANTVR = LSAME( JOBVR, 'V' )
IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
INFO = -1
ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
INFO = -9
ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
INFO = -11
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.
* HSWORK refers to the workspace preferred by DHSEQR, as
* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
* the worst case.)
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
ELSE
MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
IF( WANTVL ) THEN
MINWRK = 4*N
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
$ 'DORGHR', ' ', N, 1, N, -1 ) )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
$ WORK, -1, INFO )
HSWORK = WORK( 1 )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
MAXWRK = MAX( MAXWRK, 4*N )
ELSE IF( WANTVR ) THEN
MINWRK = 4*N
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
$ 'DORGHR', ' ', N, 1, N, -1 ) )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
$ WORK, -1, INFO )
HSWORK = WORK( 1 )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
MAXWRK = MAX( MAXWRK, 4*N )
ELSE
MINWRK = 3*N
CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, LDVR,
$ WORK, -1, INFO )
HSWORK = WORK( 1 )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEEV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
CALL DLABAD( SMLNUM, BIGNUM )
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
SCALEA = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
SCALEA = .TRUE.
CSCALE = SMLNUM
ELSE IF( ANRM.GT.BIGNUM ) THEN
SCALEA = .TRUE.
CSCALE = BIGNUM
END IF
IF( SCALEA )
$ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
* Balance the matrix
* (Workspace: need N)
*
IBAL = 1
CALL DGEBAL( 'B', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
*
* Reduce to upper Hessenberg form
* (Workspace: need 3*N, prefer 2*N+N*NB)
*
ITAU = IBAL + N
IWRK = ITAU + N
CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
IF( WANTVL ) THEN
*
* Want left eigenvectors
* Copy Householder vectors to VL
*
SIDE = 'L'
CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
*
* Generate orthogonal matrix in VL
* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VL
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
IF( WANTVR ) THEN
*
* Want left and right eigenvectors
* Copy Schur vectors to VR
*
SIDE = 'B'
CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
END IF
*
ELSE IF( WANTVR ) THEN
*
* Want right eigenvectors
* Copy Householder vectors to VR
*
SIDE = 'R'
CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
*
* Generate orthogonal matrix in VR
* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VR
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
ELSE
*
* Compute eigenvalues only
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
END IF
*
* If INFO .NE. 0 from DHSEQR, then quit
*
IF( INFO.NE.0 )
$ GO TO 50
*
IF( WANTVL .OR. WANTVR ) THEN
*
* Compute left and/or right eigenvectors
* (Workspace: need 4*N)
*
CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
$ N, NOUT, WORK( IWRK ), IERR )
END IF
*
IF( WANTVL ) THEN
*
* Undo balancing of left eigenvectors
* (Workspace: need N)
*
CALL DGEBAK( 'B', 'L', N, ILO, IHI, WORK( IBAL ), N, VL, LDVL,
$ IERR )
*
* Normalize left eigenvectors and make largest component real
*
DO 20 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
$ DNRM2( N, VL( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
DO 10 K = 1, N
WORK( IWRK+K-1 ) = VL( K, I )**2 + VL( K, I+1 )**2
10 CONTINUE
K = IDAMAX( N, WORK( IWRK ), 1 )
CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
VL( K, I+1 ) = ZERO
END IF
20 CONTINUE
END IF
*
IF( WANTVR ) THEN
*
* Undo balancing of right eigenvectors
* (Workspace: need N)
*
CALL DGEBAK( 'B', 'R', N, ILO, IHI, WORK( IBAL ), N, VR, LDVR,
$ IERR )
*
* Normalize right eigenvectors and make largest component real
*
DO 40 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
$ DNRM2( N, VR( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
DO 30 K = 1, N
WORK( IWRK+K-1 ) = VR( K, I )**2 + VR( K, I+1 )**2
30 CONTINUE
K = IDAMAX( N, WORK( IWRK ), 1 )
CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
VR( K, I+1 ) = ZERO
END IF
40 CONTINUE
END IF
*
* Undo scaling if necessary
*
50 CONTINUE
IF( SCALEA ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ),
$ MAX( N-INFO, 1 ), IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ),
$ MAX( N-INFO, 1 ), IERR )
IF( INFO.GT.0 ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
$ IERR )
END IF
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
* End of DGEEV
*
END