.TH "geesx" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
geesx \- geesx: Schur form, expert
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBcgeesx\fP (jobvs, sort, select, sense, n, a, lda, sdim, w, vs, ldvs, rconde, rcondv, work, lwork, rwork, bwork, info)"
.br
.RI "\fB CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP "
.ti -1c
.RI "subroutine \fBdgeesx\fP (jobvs, sort, select, sense, n, a, lda, sdim, wr, wi, vs, ldvs, rconde, rcondv, work, lwork, iwork, liwork, bwork, info)"
.br
.RI "\fB DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP "
.ti -1c
.RI "subroutine \fBsgeesx\fP (jobvs, sort, select, sense, n, a, lda, sdim, wr, wi, vs, ldvs, rconde, rcondv, work, lwork, iwork, liwork, bwork, info)"
.br
.RI "\fB SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP "
.ti -1c
.RI "subroutine \fBzgeesx\fP (jobvs, sort, select, sense, n, a, lda, sdim, w, vs, ldvs, rconde, rcondv, work, lwork, rwork, bwork, info)"
.br
.RI "\fB ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine cgeesx (character jobvs, character sort, external select, character sense, integer n, complex, dimension( lda, * ) a, integer lda, integer sdim, complex, dimension( * ) w, complex, dimension( ldvs, * ) vs, integer ldvs, real rconde, real rcondv, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)"

.PP
\fB CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBVS\fP 
.PP
.nf
          JOBVS is CHARACTER*1
          = 'N': Schur vectors are not computed;
          = 'V': Schur vectors are computed\&.
.fi
.PP
.br
\fISORT\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
          SELECT is a 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\&.
.fi
.PP
.br
\fISENSE\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&. N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX array, dimension (LDA, N)
          On entry, the N-by-N matrix A\&.
          On exit, A is overwritten by its Schur form T\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fISDIM\fP 
.PP
.nf
          SDIM is INTEGER
          If SORT = 'N', SDIM = 0\&.
          If SORT = 'S', SDIM = number of eigenvalues for which
                         SELECT is true\&.
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is 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\&.
.fi
.PP
.br
\fIVS\fP 
.PP
.nf
          VS is COMPLEX array, dimension (LDVS,N)
          If JOBVS = 'V', VS contains the unitary matrix Z of Schur
          vectors\&.
          If JOBVS = 'N', VS is not referenced\&.
.fi
.PP
.br
\fILDVS\fP 
.PP
.nf
          LDVS is INTEGER
          The leading dimension of the array VS\&.  LDVS >= 1, and if
          JOBVS = 'V', LDVS >= N\&.
.fi
.PP
.br
\fIRCONDE\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fIRCONDV\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is 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\&. Note also
          that an error is only returned if LWORK < max(1,2*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 bound on the optimal size of the
          array WORK, returns this value as the first entry of the WORK
          array, and no error message related to LWORK is issued by
          XERBLA\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is REAL array, dimension (N)
.fi
.PP
.br
\fIBWORK\fP 
.PP
.nf
          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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 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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SS "subroutine dgeesx (character jobvs, character sort, external select, character sense, integer n, double precision, dimension( lda, * ) a, integer lda, integer sdim, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldvs, * ) vs, integer ldvs, double precision rconde, double precision rcondv, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)"

.PP
\fB DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DGEESX 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)\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBVS\fP 
.PP
.nf
          JOBVS is CHARACTER*1
          = 'N': Schur vectors are not computed;
          = 'V': Schur vectors are computed\&.
.fi
.PP
.br
\fISORT\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
          SELECT is a LOGICAL FUNCTION of two DOUBLE PRECISION 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)\&.
.fi
.PP
.br
\fISENSE\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&. N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the N-by-N matrix A\&.
          On exit, A is overwritten by its real Schur form T\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fISDIM\fP 
.PP
.nf
          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\&.)
.fi
.PP
.br
\fIWR\fP 
.PP
.nf
          WR is DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fIWI\fP 
.PP
.nf
          WI is DOUBLE PRECISION 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\&.
.fi
.PP
.br
\fIVS\fP 
.PP
.nf
          VS is DOUBLE PRECISION array, dimension (LDVS,N)
          If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
          vectors\&.
          If JOBVS = 'N', VS is not referenced\&.
.fi
.PP
.br
\fILDVS\fP 
.PP
.nf
          LDVS is INTEGER
          The leading dimension of the array VS\&.  LDVS >= 1, and if
          JOBVS = 'V', LDVS >= N\&.
.fi
.PP
.br
\fIRCONDE\fP 
.PP
.nf
          RCONDE is DOUBLE PRECISION
          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'\&.
.fi
.PP
.br
\fIRCONDV\fP 
.PP
.nf
          RCONDV is DOUBLE PRECISION
          If SENSE = 'V' or 'B', RCONDV contains the reciprocal
          condition number for the selected right invariant subspace\&.
          Not referenced if SENSE = 'N' or 'E'\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK\&.
.fi
.PP
.br
\fILIWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIBWORK\fP 
.PP
.nf
          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SS "subroutine sgeesx (character jobvs, character sort, external select, character sense, integer n, real, dimension( lda, * ) a, integer lda, integer sdim, real, dimension( * ) wr, real, dimension( * ) wi, real, dimension( ldvs, * ) vs, integer ldvs, real rconde, real rcondv, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)"

.PP
\fB SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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)\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBVS\fP 
.PP
.nf
          JOBVS is CHARACTER*1
          = 'N': Schur vectors are not computed;
          = 'V': Schur vectors are computed\&.
.fi
.PP
.br
\fISORT\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fISENSE\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&. N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fISDIM\fP 
.PP
.nf
          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\&.)
.fi
.PP
.br
\fIWR\fP 
.PP
.nf
          WR is REAL array, dimension (N)
.fi
.PP
.br
\fIWI\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIVS\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDVS\fP 
.PP
.nf
          LDVS is INTEGER
          The leading dimension of the array VS\&.  LDVS >= 1, and if
          JOBVS = 'V', LDVS >= N\&.
.fi
.PP
.br
\fIRCONDE\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fIRCONDV\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK\&.
.fi
.PP
.br
\fILIWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIBWORK\fP 
.PP
.nf
          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SS "subroutine zgeesx (character jobvs, character sort, external select, character sense, integer n, complex*16, dimension( lda, * ) a, integer lda, integer sdim, complex*16, dimension( * ) w, complex*16, dimension( ldvs, * ) vs, integer ldvs, double precision rconde, double precision rcondv, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)"

.PP
\fB ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZGEESX 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBVS\fP 
.PP
.nf
          JOBVS is CHARACTER*1
          = 'N': Schur vectors are not computed;
          = 'V': Schur vectors are computed\&.
.fi
.PP
.br
\fISORT\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
          SELECT is a 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\&.
          An eigenvalue W(j) is selected if SELECT(W(j)) is true\&.
.fi
.PP
.br
\fISENSE\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&. N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX*16 array, dimension (LDA, N)
          On entry, the N-by-N matrix A\&.
          On exit, A is overwritten by its Schur form T\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fISDIM\fP 
.PP
.nf
          SDIM is INTEGER
          If SORT = 'N', SDIM = 0\&.
          If SORT = 'S', SDIM = number of eigenvalues for which
                         SELECT is true\&.
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is 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\&.
.fi
.PP
.br
\fIVS\fP 
.PP
.nf
          VS is 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\&.
.fi
.PP
.br
\fILDVS\fP 
.PP
.nf
          LDVS is INTEGER
          The leading dimension of the array VS\&.  LDVS >= 1, and if
          JOBVS = 'V', LDVS >= N\&.
.fi
.PP
.br
\fIRCONDE\fP 
.PP
.nf
          RCONDE is DOUBLE PRECISION
          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'\&.
.fi
.PP
.br
\fIRCONDV\fP 
.PP
.nf
          RCONDV is DOUBLE PRECISION
          If SENSE = 'V' or 'B', RCONDV contains the reciprocal
          condition number for the selected right invariant subspace\&.
          Not referenced if SENSE = 'N' or 'E'\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is 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\&. Note also
          that an error is only returned if LWORK < max(1,2*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 bound on the optimal size of the
          array WORK, returns this value as the first entry of the WORK
          array, and no error message related to LWORK is issued by
          XERBLA\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fIBWORK\fP 
.PP
.nf
          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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 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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SH "Author"
.PP 
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