424 lines
14 KiB
FortranFixed
424 lines
14 KiB
FortranFixed
SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
|
|
$ LDVR, WORK, LWORK, INFO )
|
|
*
|
|
* -- LAPACK driver routine (version 3.1) --
|
|
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
|
|
* November 2006
|
|
*
|
|
* .. 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( * )
|
|
* ..
|
|
*
|
|
* Purpose
|
|
* =======
|
|
*
|
|
* 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.
|
|
*
|
|
* Arguments
|
|
* =========
|
|
*
|
|
* JOBVL (input) CHARACTER*1
|
|
* = 'N': left eigenvectors of A are not computed;
|
|
* = 'V': left eigenvectors of A 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) DOUBLE PRECISION 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).
|
|
*
|
|
* WR (output) DOUBLE PRECISION array, dimension (N)
|
|
* WI (output) 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.
|
|
*
|
|
* VL (output) 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).
|
|
*
|
|
* LDVL (input) INTEGER
|
|
* The leading dimension of the array VL. LDVL >= 1; if
|
|
* JOBVL = 'V', LDVL >= N.
|
|
*
|
|
* VR (output) 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).
|
|
*
|
|
* LDVR (input) INTEGER
|
|
* The leading dimension of the array VR. LDVR >= 1; if
|
|
* JOBVR = 'V', LDVR >= 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. 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.
|
|
*
|
|
* 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 i+1:N of WR and WI contain eigenvalues which
|
|
* have converged.
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. 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 > 0 from DHSEQR, then quit
|
|
*
|
|
IF( INFO.GT.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
|