SUBROUTINE CGEEV( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR,
$ WORK, LWORK, RWORK, INFO )
*
* -- LAPACK driver routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
REAL RWORK( * )
COMPLEX A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
$ W( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* CGEEV computes for an N-by-N complex 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.
*
* Arguments
* =========
*
* JOBVL (input) CHARACTER*1
* = 'N': left eigenvectors of A are not computed;
* = 'V': left eigenvectors of are computed.
*
* JOBVR (input) CHARACTER*1
* = 'N': right eigenvectors of A are not computed;
* = 'V': right eigenvectors of A are computed.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the N-by-N matrix A.
* On exit, A has been overwritten.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* W (output) COMPLEX array, dimension (N)
* W contains the computed eigenvalues.
*
* VL (output) COMPLEX 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.
* u(j) = VL(:,j), the j-th column of VL.
*
* LDVL (input) INTEGER
* The leading dimension of the array VL. LDVL >= 1; if
* JOBVL = 'V', LDVL >= N.
*
* VR (output) COMPLEX 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.
* v(j) = VR(:,j), the j-th column of VR.
*
* LDVR (input) INTEGER
* The leading dimension of the array VR. LDVR >= 1; if
* JOBVR = 'V', LDVR >= N.
*
* WORK (workspace/output) COMPLEX array, dimension (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.
*
* 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) REAL array, dimension (2*N)
*
* INFO (output) 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 and i+1:N of W contain eigenvalues which have
* converged.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, SCALEA, WANTVL, WANTVR
CHARACTER SIDE
INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, IRWORK, ITAU,
$ IWRK, K, MAXB, MAXWRK, MINWRK, NOUT
REAL ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM
COMPLEX TMP
* ..
* .. Local Arrays ..
LOGICAL SELECT( 1 )
REAL DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL CGEBAK, CGEBAL, CGEHRD, CHSEQR, CLACPY, CLASCL,
$ CSCAL, CSSCAL, CTREVC, CUNGHR, SLABAD, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV, ISAMAX
REAL CLANGE, SCNRM2, SLAMCH
EXTERNAL LSAME, ILAENV, ISAMAX, CLANGE, SCNRM2, SLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC AIMAG, CMPLX, CONJG, MAX, MIN, REAL, 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 = -8
ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
INFO = -10
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.
* CWorkspace refers to complex workspace, and RWorkspace to real
* workspace. NB refers to the optimal block size for the
* immediately following subroutine, as returned by ILAENV.
* HSWORK refers to the workspace preferred by CHSEQR, as
* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
* the worst case.)
*
MINWRK = 1
IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) ) THEN
MAXWRK = N + N*ILAENV( 1, 'CGEHRD', ' ', N, 1, N, 0 )
IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
MINWRK = MAX( 1, 2*N )
MAXB = MAX( ILAENV( 8, 'CHSEQR', 'EN', N, 1, N, -1 ), 2 )
K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'CHSEQR', 'EN', N, 1,
$ N, -1 ) ) )
HSWORK = MAX( K*( K+2 ), 2*N )
MAXWRK = MAX( MAXWRK, HSWORK )
ELSE
MINWRK = MAX( 1, 2*N )
MAXWRK = MAX( MAXWRK, N+( N-1 )*
$ ILAENV( 1, 'CUNGHR', ' ', N, 1, N, -1 ) )
MAXB = MAX( ILAENV( 8, 'CHSEQR', 'SV', N, 1, N, -1 ), 2 )
K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'CHSEQR', 'SV', N, 1,
$ N, -1 ) ) )
HSWORK = MAX( K*( K+2 ), 2*N )
MAXWRK = MAX( MAXWRK, HSWORK, 2*N )
END IF
WORK( 1 ) = MAXWRK
END IF
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGEEV ', -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 = CLANGE( '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 CLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
* Balance the matrix
* (CWorkspace: none)
* (RWorkspace: need N)
*
IBAL = 1
CALL CGEBAL( 'B', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR )
*
* Reduce to upper Hessenberg form
* (CWorkspace: need 2*N, prefer N+N*NB)
* (RWorkspace: none)
*
ITAU = 1
IWRK = ITAU + N
CALL CGEHRD( 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 CLACPY( 'L', N, N, A, LDA, VL, LDVL )
*
* Generate unitary matrix in VL
* (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
* (RWorkspace: none)
*
CALL CUNGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VL
* (CWorkspace: need 1, prefer HSWORK (see comments) )
* (RWorkspace: none)
*
IWRK = ITAU
CALL CHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VL, LDVL,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
IF( WANTVR ) THEN
*
* Want left and right eigenvectors
* Copy Schur vectors to VR
*
SIDE = 'B'
CALL CLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
END IF
*
ELSE IF( WANTVR ) THEN
*
* Want right eigenvectors
* Copy Householder vectors to VR
*
SIDE = 'R'
CALL CLACPY( 'L', N, N, A, LDA, VR, LDVR )
*
* Generate unitary matrix in VR
* (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
* (RWorkspace: none)
*
CALL CUNGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VR
* (CWorkspace: need 1, prefer HSWORK (see comments) )
* (RWorkspace: none)
*
IWRK = ITAU
CALL CHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VR, LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
ELSE
*
* Compute eigenvalues only
* (CWorkspace: need 1, prefer HSWORK (see comments) )
* (RWorkspace: none)
*
IWRK = ITAU
CALL CHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, W, VR, LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
END IF
*
* If INFO > 0 from CHSEQR, then quit
*
IF( INFO.GT.0 )
$ GO TO 50
*
IF( WANTVL .OR. WANTVR ) THEN
*
* Compute left and/or right eigenvectors
* (CWorkspace: need 2*N)
* (RWorkspace: need 2*N)
*
IRWORK = IBAL + N
CALL CTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
$ N, NOUT, WORK( IWRK ), RWORK( IRWORK ), IERR )
END IF
*
IF( WANTVL ) THEN
*
* Undo balancing of left eigenvectors
* (CWorkspace: none)
* (RWorkspace: need N)
*
CALL CGEBAK( 'B', 'L', N, ILO, IHI, RWORK( IBAL ), N, VL, LDVL,
$ IERR )
*
* Normalize left eigenvectors and make largest component real
*
DO 20 I = 1, N
SCL = ONE / SCNRM2( N, VL( 1, I ), 1 )
CALL CSSCAL( N, SCL, VL( 1, I ), 1 )
DO 10 K = 1, N
RWORK( IRWORK+K-1 ) = REAL( VL( K, I ) )**2 +
$ AIMAG( VL( K, I ) )**2
10 CONTINUE
K = ISAMAX( N, RWORK( IRWORK ), 1 )
TMP = CONJG( VL( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) )
CALL CSCAL( N, TMP, VL( 1, I ), 1 )
VL( K, I ) = CMPLX( REAL( VL( K, I ) ), ZERO )
20 CONTINUE
END IF
*
IF( WANTVR ) THEN
*
* Undo balancing of right eigenvectors
* (CWorkspace: none)
* (RWorkspace: need N)
*
CALL CGEBAK( 'B', 'R', N, ILO, IHI, RWORK( IBAL ), N, VR, LDVR,
$ IERR )
*
* Normalize right eigenvectors and make largest component real
*
DO 40 I = 1, N
SCL = ONE / SCNRM2( N, VR( 1, I ), 1 )
CALL CSSCAL( N, SCL, VR( 1, I ), 1 )
DO 30 K = 1, N
RWORK( IRWORK+K-1 ) = REAL( VR( K, I ) )**2 +
$ AIMAG( VR( K, I ) )**2
30 CONTINUE
K = ISAMAX( N, RWORK( IRWORK ), 1 )
TMP = CONJG( VR( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) )
CALL CSCAL( N, TMP, VR( 1, I ), 1 )
VR( K, I ) = CMPLX( REAL( VR( K, I ) ), ZERO )
40 CONTINUE
END IF
*
* Undo scaling if necessary
*
50 CONTINUE
IF( SCALEA ) THEN
CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ),
$ MAX( N-INFO, 1 ), IERR )
IF( INFO.GT.0 ) THEN
CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR )
END IF
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
* End of CGEEV
*
END
