*> \brief SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SGEESX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, * WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, * IWORK, LIWORK, BWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBVS, SENSE, SORT * INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM * REAL RCONDE, RCONDV * .. * .. Array Arguments .. * LOGICAL BWORK( * ) * INTEGER IWORK( * ) * REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), * \$ WR( * ) * .. * .. Function Arguments .. * LOGICAL SELECT * EXTERNAL SELECT * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SGEESX computes for an N-by-N real nonsymmetric matrix A, the *> eigenvalues, the real Schur form T, and, optionally, the matrix of *> Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T). *> *> Optionally, it also orders the eigenvalues on the diagonal of the *> real Schur form so that selected eigenvalues are at the top left; *> computes a reciprocal condition number for the average of the *> selected eigenvalues (RCONDE); and computes a reciprocal condition *> number for the right invariant subspace corresponding to the *> selected eigenvalues (RCONDV). The leading columns of Z form an *> orthonormal basis for this invariant subspace. *> *> For further explanation of the reciprocal condition numbers RCONDE *> and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where *> these quantities are called s and sep respectively). *> *> A real matrix is in real Schur form if it is upper quasi-triangular *> with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in *> the form *> [ a b ] *> [ c a ] *> *> where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc). *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBVS *> \verbatim *> JOBVS is CHARACTER*1 *> = 'N': Schur vectors are not computed; *> = 'V': Schur vectors are computed. *> \endverbatim *> *> \param[in] SORT *> \verbatim *> SORT is CHARACTER*1 *> Specifies whether or not to order the eigenvalues on the *> diagonal of the Schur form. *> = 'N': Eigenvalues are not ordered; *> = 'S': Eigenvalues are ordered (see SELECT). *> \endverbatim *> *> \param[in] SELECT *> \verbatim *> SELECT is a LOGICAL FUNCTION of two REAL arguments *> SELECT must be declared EXTERNAL in the calling subroutine. *> If SORT = 'S', SELECT is used to select eigenvalues to sort *> to the top left of the Schur form. *> If SORT = 'N', SELECT is not referenced. *> An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if *> SELECT(WR(j),WI(j)) is true; i.e., if either one of a *> complex conjugate pair of eigenvalues is selected, then both *> are. Note that a selected complex eigenvalue may no longer *> satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since *> ordering may change the value of complex eigenvalues *> (especially if the eigenvalue is ill-conditioned); in this *> case INFO may be set to N+3 (see INFO below). *> \endverbatim *> *> \param[in] SENSE *> \verbatim *> SENSE is CHARACTER*1 *> Determines which reciprocal condition numbers are computed. *> = 'N': None are computed; *> = 'E': Computed for average of selected eigenvalues only; *> = 'V': Computed for selected right invariant subspace only; *> = 'B': Computed for both. *> If SENSE = 'E', 'V' or 'B', SORT must equal 'S'. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is REAL array, dimension (LDA, N) *> On entry, the N-by-N matrix A. *> On exit, A is overwritten by its real Schur form T. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[out] SDIM *> \verbatim *> SDIM is INTEGER *> If SORT = 'N', SDIM = 0. *> If SORT = 'S', SDIM = number of eigenvalues (after sorting) *> for which SELECT is true. (Complex conjugate *> pairs for which SELECT is true for either *> eigenvalue count as 2.) *> \endverbatim *> *> \param[out] WR *> \verbatim *> WR is REAL array, dimension (N) *> \endverbatim *> *> \param[out] WI *> \verbatim *> WI is REAL array, dimension (N) *> WR and WI contain the real and imaginary parts, respectively, *> of the computed eigenvalues, in the same order that they *> appear on the diagonal of the output Schur form T. Complex *> conjugate pairs of eigenvalues appear consecutively with the *> eigenvalue having the positive imaginary part first. *> \endverbatim *> *> \param[out] VS *> \verbatim *> VS is REAL array, dimension (LDVS,N) *> If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur *> vectors. *> If JOBVS = 'N', VS is not referenced. *> \endverbatim *> *> \param[in] LDVS *> \verbatim *> LDVS is INTEGER *> The leading dimension of the array VS. LDVS >= 1, and if *> JOBVS = 'V', LDVS >= N. *> \endverbatim *> *> \param[out] RCONDE *> \verbatim *> RCONDE is REAL *> If SENSE = 'E' or 'B', RCONDE contains the reciprocal *> condition number for the average of the selected eigenvalues. *> Not referenced if SENSE = 'N' or 'V'. *> \endverbatim *> *> \param[out] RCONDV *> \verbatim *> RCONDV is REAL *> If SENSE = 'V' or 'B', RCONDV contains the reciprocal *> condition number for the selected right invariant subspace. *> Not referenced if SENSE = 'N' or 'E'. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL 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). *> Also, if SENSE = 'E' or 'V' or 'B', *> LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of *> selected eigenvalues computed by this routine. Note that *> N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only *> returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or *> 'B' this may not be large enough. *> For good performance, LWORK must generally be larger. *> *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates upper bounds on the optimal sizes of the *> arrays WORK and IWORK, returns these values as the first *> entries of the WORK and IWORK arrays, and no error messages *> related to LWORK or LIWORK are issued by XERBLA. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. *> \endverbatim *> *> \param[in] LIWORK *> \verbatim *> LIWORK is INTEGER *> The dimension of the array IWORK. *> LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM). *> Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is *> only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this *> may not be large enough. *> *> If LIWORK = -1, then a workspace query is assumed; the *> routine only calculates upper bounds on the optimal sizes of *> the arrays WORK and IWORK, returns these values as the first *> entries of the WORK and IWORK arrays, and no error messages *> related to LWORK or LIWORK are issued by XERBLA. *> \endverbatim *> *> \param[out] BWORK *> \verbatim *> BWORK is LOGICAL array, dimension (N) *> Not referenced if SORT = '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, and i is *> <= N: the QR algorithm failed to compute all the *> eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI *> contain those eigenvalues which have converged; if *> JOBVS = 'V', VS contains the transformation which *> reduces A to its partially converged Schur form. *> = N+1: the eigenvalues could not be reordered because some *> eigenvalues were too close to separate (the problem *> is very ill-conditioned); *> = N+2: after reordering, roundoff changed values of some *> complex eigenvalues so that leading eigenvalues in *> the Schur form no longer satisfy SELECT=.TRUE. This *> could also be caused by underflow due to scaling. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date June 2016 * *> \ingroup realGEeigen * * ===================================================================== SUBROUTINE SGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, \$ WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, \$ IWORK, LIWORK, BWORK, INFO ) * * -- 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 JOBVS, SENSE, SORT INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM REAL RCONDE, RCONDV * .. * .. Array Arguments .. LOGICAL BWORK( * ) INTEGER IWORK( * ) REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), \$ WR( * ) * .. * .. Function Arguments .. LOGICAL SELECT EXTERNAL SELECT * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) * .. * .. Local Scalars .. LOGICAL CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTSB, \$ WANTSE, WANTSN, WANTST, WANTSV, WANTVS INTEGER HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL, \$ IHI, ILO, INXT, IP, ITAU, IWRK, LWRK, LIWRK, \$ MAXWRK, MINWRK REAL ANRM, BIGNUM, CSCALE, EPS, SMLNUM * .. * .. Local Arrays .. REAL DUM( 1 ) * .. * .. External Subroutines .. EXTERNAL SCOPY, SGEBAK, SGEBAL, SGEHRD, SHSEQR, SLABAD, \$ SLACPY, SLASCL, SORGHR, SSWAP, STRSEN, XERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV REAL SLAMCH, SLANGE EXTERNAL LSAME, ILAENV, SLAMCH, SLANGE * .. * .. Intrinsic Functions .. INTRINSIC MAX, SQRT * .. * .. Executable Statements .. * * Test the input arguments * INFO = 0 WANTVS = LSAME( JOBVS, 'V' ) WANTST = LSAME( SORT, 'S' ) WANTSN = LSAME( SENSE, 'N' ) WANTSE = LSAME( SENSE, 'E' ) WANTSV = LSAME( SENSE, 'V' ) WANTSB = LSAME( SENSE, 'B' ) LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) * IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN INFO = -1 ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR. \$ ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN INFO = -12 END IF * * Compute workspace * (Note: Comments in the code beginning "RWorkspace:" describe the * minimal amount of real workspace needed at that point in the * code, as well as the preferred amount for good performance. * IWorkspace refers to integer workspace. * NB refers to the optimal block size for the immediately * following subroutine, as returned by ILAENV. * HSWORK refers to the workspace preferred by SHSEQR, as * calculated below. HSWORK is computed assuming ILO=1 and IHI=N, * the worst case. * If SENSE = 'E', 'V' or 'B', then the amount of workspace needed * depends on SDIM, which is computed by the routine STRSEN later * in the code.) * IF( INFO.EQ.0 ) THEN LIWRK = 1 IF( N.EQ.0 ) THEN MINWRK = 1 LWRK = 1 ELSE MAXWRK = 2*N + N*ILAENV( 1, 'SGEHRD', ' ', N, 1, N, 0 ) MINWRK = 3*N * CALL SHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS, LDVS, \$ WORK, -1, IEVAL ) HSWORK = WORK( 1 ) * IF( .NOT.WANTVS ) THEN MAXWRK = MAX( MAXWRK, N + HSWORK ) ELSE MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1, \$ 'SORGHR', ' ', N, 1, N, -1 ) ) MAXWRK = MAX( MAXWRK, N + HSWORK ) END IF LWRK = MAXWRK IF( .NOT.WANTSN ) \$ LWRK = MAX( LWRK, N + ( N*N )/2 ) IF( WANTSV .OR. WANTSB ) \$ LIWRK = ( N*N )/4 END IF IWORK( 1 ) = LIWRK WORK( 1 ) = LWRK * IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -16 ELSE IF( LIWORK.LT.1 .AND. .NOT.LQUERY ) THEN INFO = -18 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGEESX', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) THEN SDIM = 0 RETURN END IF * * 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 = SLANGE( '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 SLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) * * Permute the matrix to make it more nearly triangular * (RWorkspace: need N) * IBAL = 1 CALL SGEBAL( 'P', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR ) * * Reduce to upper Hessenberg form * (RWorkspace: need 3*N, prefer 2*N+N*NB) * ITAU = N + IBAL IWRK = N + ITAU CALL SGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), \$ LWORK-IWRK+1, IERR ) * IF( WANTVS ) THEN * * Copy Householder vectors to VS * CALL SLACPY( 'L', N, N, A, LDA, VS, LDVS ) * * Generate orthogonal matrix in VS * (RWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB) * CALL SORGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ), \$ LWORK-IWRK+1, IERR ) END IF * SDIM = 0 * * Perform QR iteration, accumulating Schur vectors in VS if desired * (RWorkspace: need N+1, prefer N+HSWORK (see comments) ) * IWRK = ITAU CALL SHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, WR, WI, VS, LDVS, \$ WORK( IWRK ), LWORK-IWRK+1, IEVAL ) IF( IEVAL.GT.0 ) \$ INFO = IEVAL * * Sort eigenvalues if desired * IF( WANTST .AND. INFO.EQ.0 ) THEN IF( SCALEA ) THEN CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR ) CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WI, N, IERR ) END IF DO 10 I = 1, N BWORK( I ) = SELECT( WR( I ), WI( I ) ) 10 CONTINUE * * Reorder eigenvalues, transform Schur vectors, and compute * reciprocal condition numbers * (RWorkspace: if SENSE is not 'N', need N+2*SDIM*(N-SDIM) * otherwise, need N ) * (IWorkspace: if SENSE is 'V' or 'B', need SDIM*(N-SDIM) * otherwise, need 0 ) * CALL STRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, WR, WI, \$ SDIM, RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1, \$ IWORK, LIWORK, ICOND ) IF( .NOT.WANTSN ) \$ MAXWRK = MAX( MAXWRK, N+2*SDIM*( N-SDIM ) ) IF( ICOND.EQ.-15 ) THEN * * Not enough real workspace * INFO = -16 ELSE IF( ICOND.EQ.-17 ) THEN * * Not enough integer workspace * INFO = -18 ELSE IF( ICOND.GT.0 ) THEN * * STRSEN failed to reorder or to restore standard Schur form * INFO = ICOND + N END IF END IF * IF( WANTVS ) THEN * * Undo balancing * (RWorkspace: need N) * CALL SGEBAK( 'P', 'R', N, ILO, IHI, WORK( IBAL ), N, VS, LDVS, \$ IERR ) END IF * IF( SCALEA ) THEN * * Undo scaling for the Schur form of A * CALL SLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR ) CALL SCOPY( N, A, LDA+1, WR, 1 ) IF( ( WANTSV .OR. WANTSB ) .AND. INFO.EQ.0 ) THEN DUM( 1 ) = RCONDV CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR ) RCONDV = DUM( 1 ) END IF IF( CSCALE.EQ.SMLNUM ) THEN * * If scaling back towards underflow, adjust WI if an * offdiagonal element of a 2-by-2 block in the Schur form * underflows. * IF( IEVAL.GT.0 ) THEN I1 = IEVAL + 1 I2 = IHI - 1 CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N, \$ IERR ) ELSE IF( WANTST ) THEN I1 = 1 I2 = N - 1 ELSE I1 = ILO I2 = IHI - 1 END IF INXT = I1 - 1 DO 20 I = I1, I2 IF( I.LT.INXT ) \$ GO TO 20 IF( WI( I ).EQ.ZERO ) THEN INXT = I + 1 ELSE IF( A( I+1, I ).EQ.ZERO ) THEN WI( I ) = ZERO WI( I+1 ) = ZERO ELSE IF( A( I+1, I ).NE.ZERO .AND. A( I, I+1 ).EQ. \$ ZERO ) THEN WI( I ) = ZERO WI( I+1 ) = ZERO IF( I.GT.1 ) \$ CALL SSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ), 1 ) IF( N.GT.I+1 ) \$ CALL SSWAP( N-I-1, A( I, I+2 ), LDA, \$ A( I+1, I+2 ), LDA ) CALL SSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 ) A( I, I+1 ) = A( I+1, I ) A( I+1, I ) = ZERO END IF INXT = I + 2 END IF 20 CONTINUE END IF CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N-IEVAL, 1, \$ WI( IEVAL+1 ), MAX( N-IEVAL, 1 ), IERR ) END IF * IF( WANTST .AND. INFO.EQ.0 ) THEN * * Check if reordering successful * LASTSL = .TRUE. LST2SL = .TRUE. SDIM = 0 IP = 0 DO 30 I = 1, N CURSL = SELECT( WR( I ), WI( I ) ) IF( WI( I ).EQ.ZERO ) THEN IF( CURSL ) \$ SDIM = SDIM + 1 IP = 0 IF( CURSL .AND. .NOT.LASTSL ) \$ INFO = N + 2 ELSE IF( IP.EQ.1 ) THEN * * Last eigenvalue of conjugate pair * CURSL = CURSL .OR. LASTSL LASTSL = CURSL IF( CURSL ) \$ SDIM = SDIM + 2 IP = -1 IF( CURSL .AND. .NOT.LST2SL ) \$ INFO = N + 2 ELSE * * First eigenvalue of conjugate pair * IP = 1 END IF END IF LST2SL = LASTSL LASTSL = CURSL 30 CONTINUE END IF * WORK( 1 ) = MAXWRK IF( WANTSV .OR. WANTSB ) THEN IWORK( 1 ) = SDIM*(N-SDIM) ELSE IWORK( 1 ) = 1 END IF * RETURN * * End of SGEESX * END