SUBROUTINE ZGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, W, VS, \$ LDVS, WORK, LWORK, RWORK, BWORK, INFO ) * * -- LAPACK driver routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER JOBVS, SORT INTEGER INFO, LDA, LDVS, LWORK, N, SDIM * .. * .. Array Arguments .. LOGICAL BWORK( * ) DOUBLE PRECISION RWORK( * ) COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * ) * .. * .. Function Arguments .. LOGICAL SELECT EXTERNAL SELECT * .. * * Purpose * ======= * * ZGEES 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. * The leading columns of Z then form an orthonormal basis for the * invariant subspace corresponding to the selected eigenvalues. * * 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 (external procedure) LOGICAL FUNCTION of one COMPLEX*16 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. * The eigenvalue W(j) is selected if SELECT(W(j)) is true. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) COMPLEX*16 array, dimension (LDA,N) * On entry, the N-by-N matrix A. * On exit, A has been 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*16 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*16 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; if * JOBVS = 'V', LDVS >= N. * * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,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). * 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. * * RWORK (workspace) DOUBLE PRECISION 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 matrix 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 .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) * .. * .. Local Scalars .. LOGICAL LQUERY, SCALEA, WANTST, WANTVS INTEGER HSWORK, I, IBAL, ICOND, IERR, IEVAL, IHI, ILO, \$ ITAU, IWRK, MAXWRK, MINWRK DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, S, SEP, SMLNUM * .. * .. Local Arrays .. DOUBLE PRECISION DUM( 1 ) * .. * .. External Subroutines .. EXTERNAL DLABAD, XERBLA, ZCOPY, ZGEBAK, ZGEBAL, ZGEHRD, \$ ZHSEQR, ZLACPY, ZLASCL, ZTRSEN, ZUNGHR * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, ZLANGE EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE * .. * .. Intrinsic Functions .. INTRINSIC MAX, SQRT * .. * .. Executable Statements .. * * Test the input arguments * INFO = 0 LQUERY = ( LWORK.EQ.-1 ) WANTVS = LSAME( JOBVS, 'V' ) WANTST = LSAME( SORT, 'S' ) 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( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.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 ZHSEQR, 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, 'ZGEHRD', ' ', N, 1, N, 0 ) MINWRK = 2*N * CALL ZHSEQR( 'S', JOBVS, N, 1, N, A, LDA, W, VS, LDVS, \$ WORK, -1, IEVAL ) HSWORK = WORK( 1 ) * IF( .NOT.WANTVS ) THEN MAXWRK = MAX( MAXWRK, HSWORK ) ELSE MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR', \$ ' ', N, 1, N, -1 ) ) MAXWRK = MAX( MAXWRK, HSWORK ) END IF 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( 'ZGEES ', -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 = 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 = ZLANGE( '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 ZLASCL( '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 ZGEBAL( '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 ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), \$ LWORK-IWRK+1, IERR ) * IF( WANTVS ) THEN * * Copy Householder vectors to VS * CALL ZLACPY( '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 ZUNGHR( 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 ZHSEQR( '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 ZLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, W, N, IERR ) DO 10 I = 1, N BWORK( I ) = SELECT( W( I ) ) 10 CONTINUE * * Reorder eigenvalues and transform Schur vectors * (CWorkspace: none) * (RWorkspace: none) * CALL ZTRSEN( 'N', JOBVS, BWORK, N, A, LDA, VS, LDVS, W, SDIM, \$ S, SEP, WORK( IWRK ), LWORK-IWRK+1, ICOND ) END IF * IF( WANTVS ) THEN * * Undo balancing * (CWorkspace: none) * (RWorkspace: need N) * CALL ZGEBAK( '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 ZLASCL( 'U', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR ) CALL ZCOPY( N, A, LDA+1, W, 1 ) END IF * WORK( 1 ) = MAXWRK RETURN * * End of ZGEES * END