#include "ESMF_LapackBlas.inc" !> \brief <b> DGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b> ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DGEEV + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeev.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeev.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeev.f"> !> [TXT]</a> !> \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 3.7.0) -- ! -- 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