*> \brief CGEEV 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 CGEEV + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CGEEV( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, * WORK, LWORK, RWORK, INFO ) * * .. 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( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> 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. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBVL *> \verbatim *> JOBVL is CHARACTER*1 *> = 'N': left eigenvectors of A are not computed; *> = 'V': left eigenvectors of 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 COMPLEX 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] W *> \verbatim *> W is COMPLEX array, dimension (N) *> W contains the computed eigenvalues. *> \endverbatim *> *> \param[out] VL *> \verbatim *> VL is 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. *> \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 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. *> \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 COMPLEX 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(1,2*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. *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is REAL array, dimension (2*N) *> \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 and i+1:N of W 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 * * @generated from zgeev.f, fortran z -> c, Tue Apr 19 01:47:44 2016 * *> \ingroup complexGEeigen * * ===================================================================== SUBROUTINE CGEEV( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, \$ WORK, LWORK, RWORK, 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 .. REAL RWORK( * ) COMPLEX A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), \$ W( * ), WORK( * ) * .. * * ===================================================================== * * .. 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, LWORK_TREVC, MAXWRK, MINWRK, NOUT REAL ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM COMPLEX TMP * .. * .. Local Arrays .. LOGICAL SELECT( 1 ) REAL DUM( 1 ) * .. * .. External Subroutines .. EXTERNAL SLABAD, XERBLA, CSSCAL, CGEBAK, CGEBAL, CGEHRD, \$ CHSEQR, CLACPY, CLASCL, CSCAL, CTREVC3, CUNGHR * .. * .. External Functions .. LOGICAL LSAME INTEGER ISAMAX, ILAENV REAL SLAMCH, SCNRM2, CLANGE EXTERNAL LSAME, ISAMAX, ILAENV, SLAMCH, SCNRM2, CLANGE * .. * .. Intrinsic Functions .. INTRINSIC REAL, CMPLX, CONJG, AIMAG, 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 = -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.) * IF( INFO.EQ.0 ) THEN IF( N.EQ.0 ) THEN MINWRK = 1 MAXWRK = 1 ELSE MAXWRK = N + N*ILAENV( 1, 'CGEHRD', ' ', N, 1, N, 0 ) MINWRK = 2*N IF( WANTVL ) THEN MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'CUNGHR', \$ ' ', N, 1, N, -1 ) ) CALL CTREVC3( 'L', 'B', SELECT, N, A, LDA, \$ VL, LDVL, VR, LDVR, \$ N, NOUT, WORK, -1, RWORK, -1, IERR ) LWORK_TREVC = INT( WORK(1) ) MAXWRK = MAX( MAXWRK, N + LWORK_TREVC ) CALL CHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL, \$ WORK, -1, INFO ) ELSE IF( WANTVR ) THEN MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'CUNGHR', \$ ' ', N, 1, N, -1 ) ) CALL CTREVC3( 'R', 'B', SELECT, N, A, LDA, \$ VL, LDVL, VR, LDVR, \$ N, NOUT, WORK, -1, RWORK, -1, IERR ) LWORK_TREVC = INT( WORK(1) ) MAXWRK = MAX( MAXWRK, N + LWORK_TREVC ) CALL CHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR, \$ WORK, -1, INFO ) ELSE CALL CHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR, \$ WORK, -1, INFO ) END IF HSWORK = INT( WORK(1) ) MAXWRK = MAX( MAXWRK, HSWORK, MINWRK ) END IF WORK( 1 ) = MAXWRK * IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -12 END IF 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 .NE. 0 from CHSEQR, then quit * IF( INFO.NE.0 ) \$ GO TO 50 * IF( WANTVL .OR. WANTVR ) THEN * * Compute left and/or right eigenvectors * (CWorkspace: need 2*N, prefer N + 2*N*NB) * (RWorkspace: need 2*N) * IRWORK = IBAL + N CALL CTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, \$ N, NOUT, WORK( IWRK ), LWORK-IWRK+1, \$ RWORK( IRWORK ), N, 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