SUBROUTINE CGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W, \$ VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK, \$ BWORK, INFO ) * * -- LAPACK driver routine (version 3.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * June 30, 1999 * 8-15-00: Do WS calculations if LWORK = -1 (eca) * * .. Scalar Arguments .. CHARACTER JOBVS, SENSE, SORT INTEGER INFO, LDA, LDVS, LWORK, N, SDIM REAL RCONDE, RCONDV * .. * .. Array Arguments .. LOGICAL BWORK( * ) REAL RWORK( * ) COMPLEX A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * ) * .. * .. Function Arguments .. LOGICAL SELECT EXTERNAL SELECT * .. * * Purpose * ======= * * CGEESX computes for an N-by-N complex nonsymmetric matrix A, the * eigenvalues, the Schur form T, and, optionally, the matrix of Schur * vectors Z. This gives the Schur factorization A = Z*T*(Z**H). * * Optionally, it also orders the eigenvalues on the diagonal of the * 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 complex matrix is in Schur form if it is upper triangular. * * Arguments * ========= * * JOBVS (input) CHARACTER*1 * = 'N': Schur vectors are not computed; * = 'V': Schur vectors are computed. * * SORT (input) 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). * * SELECT (input) LOGICAL FUNCTION of one COMPLEX argument * SELECT must be declared EXTERNAL in the calling subroutine. * If SORT = 'S', SELECT is used to select eigenvalues to order * to the top left of the Schur form. * If SORT = 'N', SELECT is not referenced. * An eigenvalue W(j) is selected if SELECT(W(j)) is true. * * SENSE (input) 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'. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) COMPLEX array, dimension (LDA, N) * On entry, the N-by-N matrix A. * On exit, A is overwritten by its Schur form T. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * SDIM (output) INTEGER * If SORT = 'N', SDIM = 0. * If SORT = 'S', SDIM = number of eigenvalues for which * SELECT is true. * * W (output) COMPLEX array, dimension (N) * W contains the computed eigenvalues, in the same order * that they appear on the diagonal of the output Schur form T. * * VS (output) COMPLEX array, dimension (LDVS,N) * If JOBVS = 'V', VS contains the unitary matrix Z of Schur * vectors. * If JOBVS = 'N', VS is not referenced. * * LDVS (input) INTEGER * The leading dimension of the array VS. LDVS >= 1, and if * JOBVS = 'V', LDVS >= N. * * RCONDE (output) 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'. * * RCONDV (output) 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'. * * WORK (workspace/output) COMPLEX array, dimension (LWORK) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK >= max(1,2*N). * Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM), * where SDIM is the number of selected eigenvalues computed by * this routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. * For good performance, LWORK must generally be larger. * * If LWORK = -1, a workspace query is assumed. The optimal * size for the WORK array is calculated and stored in WORK(1), * and no other work except argument checking is performed. * * RWORK (workspace) REAL array, dimension (N) * * BWORK (workspace) LOGICAL array, dimension (N) * Not referenced if SORT = 'N'. * * INFO (output) 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 W * 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. * * ===================================================================== * * .. Parameters .. INTEGER LQUERV PARAMETER ( LQUERV = -1 ) REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) * .. * .. Local Scalars .. LOGICAL SCALEA, WANTSB, WANTSE, WANTSN, WANTST, \$ WANTSV, WANTVS INTEGER HSWORK, I, IBAL, ICOND, IERR, IEVAL, IHI, ILO, \$ ITAU, IWRK, K, MAXB, MAXWRK, MINWRK REAL ANRM, BIGNUM, CSCALE, EPS, SMLNUM * .. * .. Local Arrays .. REAL DUM( 1 ) * .. * .. External Subroutines .. EXTERNAL CCOPY, CGEBAK, CGEBAL, CGEHRD, CHSEQR, CLACPY, \$ CLASCL, CTRSEN, CUNGHR, SLABAD, SLASCL, XERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV REAL CLANGE, SLAMCH EXTERNAL LSAME, ILAENV, CLANGE, SLAMCH * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, 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' ) 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 = -11 END IF * * Compute workspace * (Note: Comments in the code beginning "Workspace:" describe the * minimal amount of real 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 SENSE = 'E', 'V' or 'B', then the amount of workspace needed * depends on SDIM, which is computed by the routine CTRSEN later * in the code.) * MINWRK = 1 IF( INFO.EQ.0 ) THEN MAXWRK = N + N*ILAENV( 1, 'CGEHRD', ' ', N, 1, N, 0 ) MINWRK = MAX( 1, 2*N ) IF( .NOT.WANTVS ) THEN MAXB = MAX( ILAENV( 8, 'CHSEQR', 'SN', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'CHSEQR', 'SN', N, 1, \$ N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, HSWORK, 1 ) ELSE MAXWRK = MAX( MAXWRK, N+( N-1 )* \$ ILAENV( 1, 'CUNGHR', ' ', N, 1, N, -1 ) ) MAXB = MAX( ILAENV( 8, 'CHSEQR', 'SV', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'CHSEQR', 'SV', N, 1, \$ N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, HSWORK, 1 ) END IF * * Estimate the workspace needed by CTRSEN. * IF( WANTST ) THEN MAXWRK = MAX( MAXWRK, (N*N+1)/2 ) END IF WORK( 1 ) = MAXWRK IF( LWORK.LT.MINWRK .AND. LWORK.NE.LQUERV ) & INFO = -15 END IF * * Quick returns * IF( INFO.NE.0 ) THEN CALL XERBLA( 'CGEESX', -INFO ) RETURN END IF IF( LWORK.EQ.LQUERV ) RETURN 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 = 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 ) * * * Permute the matrix to make it more nearly triangular * (CWorkspace: none) * (RWorkspace: need N) * IBAL = 1 CALL CGEBAL( 'P', 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 = N + ITAU CALL CGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), \$ LWORK-IWRK+1, IERR ) * IF( WANTVS ) THEN * * Copy Householder vectors to VS * CALL CLACPY( 'L', N, N, A, LDA, VS, LDVS ) * * Generate unitary matrix in VS * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) * (RWorkspace: none) * CALL CUNGHR( 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 * (CWorkspace: need 1, prefer HSWORK (see comments) ) * (RWorkspace: none) * IWRK = ITAU CALL CHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, W, 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 ) \$ CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, W, N, IERR ) DO 10 I = 1, N BWORK( I ) = SELECT( W( I ) ) 10 CONTINUE * * Reorder eigenvalues, transform Schur vectors, and compute * reciprocal condition numbers * (CWorkspace: if SENSE is not 'N', need 2*SDIM*(N-SDIM) * otherwise, need none ) * (RWorkspace: none) * CALL CTRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, W, SDIM, \$ RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1, \$ ICOND ) IF( .NOT.WANTSN ) \$ MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) ) IF( ICOND.EQ.-14 ) THEN * * Not enough complex workspace * INFO = -15 END IF END IF * IF( WANTVS ) THEN * * Undo balancing * (CWorkspace: none) * (RWorkspace: need N) * CALL CGEBAK( 'P', 'R', N, ILO, IHI, RWORK( IBAL ), N, VS, LDVS, \$ IERR ) END IF * IF( SCALEA ) THEN * * Undo scaling for the Schur form of A * CALL CLASCL( 'U', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR ) CALL CCOPY( N, A, LDA+1, W, 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 END IF * WORK( 1 ) = MAXWRK RETURN * * End of CGEESX * END