*> \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 June 2016 * * @precisions fortran d -> s * *> \ingroup doubleGEeigen * * ===================================================================== SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, $ LDVR, WORK, LWORK, INFO ) implicit none * * -- LAPACK driver routine (version 3.6.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2016 * * .. 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, $ LWORK_TREVC, 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, DTREVC3, $ 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 = INT( WORK(1) ) MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK ) CALL DTREVC3( 'L', 'B', SELECT, N, A, LDA, $ VL, LDVL, VR, LDVR, N, NOUT, $ WORK, -1, IERR ) LWORK_TREVC = INT( WORK(1) ) MAXWRK = MAX( MAXWRK, N + LWORK_TREVC ) 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 = INT( WORK(1) ) MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK ) CALL DTREVC3( 'R', 'B', SELECT, N, A, LDA, $ VL, LDVL, VR, LDVR, N, NOUT, $ WORK, -1, IERR ) LWORK_TREVC = INT( WORK(1) ) MAXWRK = MAX( MAXWRK, N + LWORK_TREVC ) 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 = INT( 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, prefer N + N + 2*N*NB) * CALL DTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, $ N, NOUT, WORK( IWRK ), LWORK-IWRK+1, 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