---
name: "sgeesx.f"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/sgeesx.f"
---

# sgeesx.f(3)

## NAME

SRC/sgeesx.f

## SYNOPSIS

### Functions/Subroutines

subroutine **sgeesx** (jobvs, sort, select, sense, n, a, lda, sdim, wr, wi, vs, ldvs, rconde, rcondv, work, lwork, iwork, liwork, bwork, info)\
  **SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**

## Function/Subroutine Documentation

### 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)

 **SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**

**Purpose:**

```
!>
!> 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).
!>
```

**Parameters**

*JOBVS*

```
!>          JOBVS is CHARACTER*1
!>          = 'N': Schur vectors are not computed;
!>          = 'V': Schur vectors are computed.
!>
```

*SORT*

```
!>          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).
!>
```

*SELECT*

```
!>          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).
!>
```

*SENSE*

```
!>          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'.
!>
```

*N*

```
!>          N is INTEGER
!>          The order of the matrix A. N >= 0.
!>
```

*A*

```
!>          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.
!>
```

*LDA*

```
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!>
```

*SDIM*

```
!>          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.)
!>
```

*WR*

```
!>          WR is REAL array, dimension (N)
!>
```

*WI*

```
!>          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.
!>
```

*VS*

```
!>          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.
!>
```

*LDVS*

```
!>          LDVS is INTEGER
!>          The leading dimension of the array VS.  LDVS >= 1, and if
!>          JOBVS = 'V', LDVS >= N.
!>
```

*RCONDE*

```
!>          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'.
!>
```

*RCONDV*

```
!>          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'.
!>
```

*WORK*

```
!>          WORK is REAL array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>
```

*LWORK*

```
!>          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.
!>
```

*IWORK*

```
!>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
!>
```

*LIWORK*

```
!>          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.
!>
```

*BWORK*

```
!>          BWORK is LOGICAL array, dimension (N)
!>          Not referenced if SORT = 'N'.
!>
```

*INFO*

```
!>          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.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **278** of file **sgeesx.f**.

## Author

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