SUBROUTINE CGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL,
$ LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE,
$ RCONDV, 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 BALANC, JOBVL, JOBVR, SENSE
INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
REAL ABNRM
* ..
* .. Array Arguments ..
REAL RCONDE( * ), RCONDV( * ), RWORK( * ),
$ SCALE( * )
COMPLEX A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
$ W( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
* eigenvalues and, optionally, the left and/or right eigenvectors.
*
* Optionally also, it computes a balancing transformation to improve
* the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
* SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
* (RCONDE), and reciprocal condition numbers for the right
* eigenvectors (RCONDV).
*
* 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.
*
* Balancing a matrix means permuting the rows and columns to make it
* more nearly upper triangular, and applying a diagonal similarity
* transformation D * A * D**(-1), where D is a diagonal matrix, to
* make its rows and columns closer in norm and the condition numbers
* of its eigenvalues and eigenvectors smaller. The computed
* reciprocal condition numbers correspond to the balanced matrix.
* Permuting rows and columns will not change the condition numbers
* (in exact arithmetic) but diagonal scaling will. For further
* explanation of balancing, see section 4.10.2 of the LAPACK
* Users' Guide.
*
* Arguments
* =========
*
* BALANC (input) CHARACTER*1
* Indicates how the input matrix should be diagonally scaled
* and/or permuted to improve the conditioning of its
* eigenvalues.
* = 'N': Do not diagonally scale or permute;
* = 'P': Perform permutations to make the matrix more nearly
* upper triangular. Do not diagonally scale;
* = 'S': Diagonally scale the matrix, ie. replace A by
* D*A*D**(-1), where D is a diagonal matrix chosen
* to make the rows and columns of A more equal in
* norm. Do not permute;
* = 'B': Both diagonally scale and permute A.
*
* Computed reciprocal condition numbers will be for the matrix
* after balancing and/or permuting. Permuting does not change
* condition numbers (in exact arithmetic), but balancing does.
*
* JOBVL (input) CHARACTER*1
* = 'N': left eigenvectors of A are not computed;
* = 'V': left eigenvectors of A are computed.
* If SENSE = 'E' or 'B', JOBVL must = 'V'.
*
* JOBVR (input) CHARACTER*1
* = 'N': right eigenvectors of A are not computed;
* = 'V': right eigenvectors of A are computed.
* If SENSE = 'E' or 'B', JOBVR must = 'V'.
*
* SENSE (input) CHARACTER*1
* Determines which reciprocal condition numbers are computed.
* = 'N': None are computed;
* = 'E': Computed for eigenvalues only;
* = 'V': Computed for right eigenvectors only;
* = 'B': Computed for eigenvalues and right eigenvectors.
*
* If SENSE = 'E' or 'B', both left and right eigenvectors
* must also be computed (JOBVL = 'V' and JOBVR = 'V').
*
* 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. If JOBVL = 'V' or
* JOBVR = 'V', A contains the Schur form of the balanced
* version of the matrix A.
*
* 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.
*
* ILO,IHI (output) INTEGER
* ILO and IHI are integer values determined when A was
* balanced. The balanced A(i,j) = 0 if I > J and
* J = 1,...,ILO-1 or I = IHI+1,...,N.
*
* SCALE (output) REAL array, dimension (N)
* Details of the permutations and scaling factors applied
* when balancing A. If P(j) is the index of the row and column
* interchanged with row and column j, and D(j) is the scaling
* factor applied to row and column j, then
* SCALE(J) = P(J), for J = 1,...,ILO-1
* = D(J), for J = ILO,...,IHI
* = P(J) for J = IHI+1,...,N.
* The order in which the interchanges are made is N to IHI+1,
* then 1 to ILO-1.
*
* ABNRM (output) REAL
* The one-norm of the balanced matrix (the maximum
* of the sum of absolute values of elements of any column).
*
* RCONDE (output) REAL array, dimension (N)
* RCONDE(j) is the reciprocal condition number of the j-th
* eigenvalue.
*
* RCONDV (output) REAL array, dimension (N)
* RCONDV(j) is the reciprocal condition number of the j-th
* right eigenvector.
*
* 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. If SENSE = 'N' or 'E',
* LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
* LWORK >= N*N+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 or condition numbers
* have been computed; elements 1:ILO-1 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, WNTSNB, WNTSNE,
$ WNTSNN, WNTSNV
CHARACTER JOB, SIDE
INTEGER HSWORK, I, ICOND, IERR, 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, CTRSNA, CUNGHR, SLABAD,
$ SLASCL, 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' )
WNTSNN = LSAME( SENSE, 'N' )
WNTSNE = LSAME( SENSE, 'E' )
WNTSNV = LSAME( SENSE, 'V' )
WNTSNB = LSAME( SENSE, 'B' )
IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'S' ) .OR.
$ LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) THEN
INFO = -1
ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
INFO = -2
ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
INFO = -3
ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
$ ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
$ WANTVR ) ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
INFO = -10
ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
INFO = -12
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 )
IF( .NOT.( WNTSNN .OR. WNTSNE ) )
$ MINWRK = MAX( MINWRK, N*N+2*N )
MAXB = MAX( ILAENV( 8, 'CHSEQR', 'SN', N, 1, N, -1 ), 2 )
IF( WNTSNN ) THEN
K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'CHSEQR', 'EN', N,
$ 1, N, -1 ) ) )
ELSE
K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'CHSEQR', 'SN', N,
$ 1, N, -1 ) ) )
END IF
HSWORK = MAX( K*( K+2 ), 2*N )
MAXWRK = MAX( MAXWRK, 1, HSWORK )
IF( .NOT.( WNTSNN .OR. WNTSNE ) )
$ MAXWRK = MAX( MAXWRK, N*N+2*N )
ELSE
MINWRK = MAX( 1, 2*N )
IF( .NOT.( WNTSNN .OR. WNTSNE ) )
$ MINWRK = MAX( MINWRK, N*N+2*N )
MAXB = MAX( ILAENV( 8, 'CHSEQR', 'SN', 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, 1, HSWORK )
MAXWRK = MAX( MAXWRK, N+( N-1 )*
$ ILAENV( 1, 'CUNGHR', ' ', N, 1, N, -1 ) )
IF( .NOT.( WNTSNN .OR. WNTSNE ) )
$ MAXWRK = MAX( MAXWRK, N*N+2*N )
MAXWRK = MAX( MAXWRK, 2*N, 1 )
END IF
WORK( 1 ) = MAXWRK
END IF
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -20
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGEEVX', -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]
*
ICOND = 0
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 and compute ABNRM
*
CALL CGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
ABNRM = CLANGE( '1', N, N, A, LDA, DUM )
IF( SCALEA ) THEN
DUM( 1 ) = ABNRM
CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
ABNRM = DUM( 1 )
END IF
*
* 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
* If condition numbers desired, compute Schur form
*
IF( WNTSNN ) THEN
JOB = 'E'
ELSE
JOB = 'S'
END IF
*
* (CWorkspace: need 1, prefer HSWORK (see comments) )
* (RWorkspace: none)
*
IWRK = ITAU
CALL CHSEQR( JOB, '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 N)
*
CALL CTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
$ N, NOUT, WORK( IWRK ), RWORK, IERR )
END IF
*
* Compute condition numbers if desired
* (CWorkspace: need N*N+2*N unless SENSE = 'E')
* (RWorkspace: need 2*N unless SENSE = 'E')
*
IF( .NOT.WNTSNN ) THEN
CALL CTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
$ RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, RWORK,
$ ICOND )
END IF
*
IF( WANTVL ) THEN
*
* Undo balancing of left eigenvectors
*
CALL CGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, 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( K ) = REAL( VL( K, I ) )**2 +
$ AIMAG( VL( K, I ) )**2
10 CONTINUE
K = ISAMAX( N, RWORK, 1 )
TMP = CONJG( VL( K, I ) ) / SQRT( RWORK( K ) )
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
*
CALL CGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, 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( K ) = REAL( VR( K, I ) )**2 +
$ AIMAG( VR( K, I ) )**2
30 CONTINUE
K = ISAMAX( N, RWORK, 1 )
TMP = CONJG( VR( K, I ) ) / SQRT( RWORK( K ) )
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.EQ.0 ) THEN
IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
$ CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
$ IERR )
ELSE
CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR )
END IF
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
* End of CGEEVX
*
END
