---
name: "gees"
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/gees"
---

# gees(3)

gees: Schur form

## NAME

gees - gees: Schur form

## SYNOPSIS

### Functions

subroutine **cgees** (jobvs, sort, select, n, a, lda, sdim, w, vs, ldvs, work, lwork, rwork, bwork, info)\
  **CGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**  subroutine **dgees** (jobvs, sort, select, n, a, lda, sdim, wr, wi, vs, ldvs, work, lwork, bwork, info)\
  **DGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**  subroutine **sgees** (jobvs, sort, select, n, a, lda, sdim, wr, wi, vs, ldvs, work, lwork, bwork, info)\
  **SGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**  subroutine **zgees** (jobvs, sort, select, n, a, lda, sdim, w, vs, ldvs, work, lwork, rwork, bwork, info)\
  **ZGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**

## Detailed Description

## Function Documentation

### subroutine cgees (character jobvs, character sort, external select, integer n, complex, dimension( lda, \* ) a, integer lda, integer sdim, complex, dimension( \* ) w, complex, dimension( ldvs, \* ) vs, integer ldvs, complex, dimension( \* ) work, integer lwork, real, dimension( \* ) rwork, logical, dimension( \* ) bwork, integer info)

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

**Purpose:**

```
!>
!> CGEES 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.
!>
```

**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 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.
!>          The eigenvalue W(j) is selected if SELECT(W(j)) is true.
!>
```

*N*

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

*A*

```
!>          A is COMPLEX array, dimension (LDA,N)
!>          On entry, the N-by-N matrix A.
!>          On exit, A has been overwritten by its 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 for which
!>                         SELECT is true.
!>
```

*W*

```
!>          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.
!>
```

*VS*

```
!>          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.
!>
```

*LDVS*

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

*WORK*

```
!>          WORK is COMPLEX 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(1,2*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*

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

*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 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.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **195** of file **cgees.f**.

### subroutine dgees (character jobvs, character sort, external select, 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, dimension( \* ) work, integer lwork, logical, dimension( \* ) bwork, integer info)

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

**Purpose:**

```
!>
!> DGEES 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.
!> The leading columns of Z then form an orthonormal basis for the
!> invariant subspace corresponding to the selected eigenvalues.
!>
!> A 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 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 complex
!>          eigenvalues are selected.
!>          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 is set to N+2 (see INFO below).
!>
```

*N*

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

*A*

```
!>          A is DOUBLE PRECISION array, dimension (LDA,N)
!>          On entry, the N-by-N matrix A.
!>          On exit, A has been 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 DOUBLE PRECISION array, dimension (N)
!>
```

*WI*

```
!>          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 will appear
!>          consecutively with the eigenvalue having the positive
!>          imaginary part first.
!>
```

*VS*

```
!>          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.
!>
```

*LDVS*

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

*WORK*

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

*LWORK*

```
!>          LWORK is INTEGER
!>          The dimension of the array WORK.  LWORK >= max(1,3*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.
!>
```

*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 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.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **214** of file **dgees.f**.

### subroutine sgees (character jobvs, character sort, external select, integer n, real, dimension( lda, \* ) a, integer lda, integer sdim, real, dimension( \* ) wr, real, dimension( \* ) wi, real, dimension( ldvs, \* ) vs, integer ldvs, real, dimension( \* ) work, integer lwork, logical, dimension( \* ) bwork, integer info)

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

**Purpose:**

```
!>
!> SGEES 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.
!> The leading columns of Z then form an orthonormal basis for the
!> invariant subspace corresponding to the selected eigenvalues.
!>
!> A 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 complex
!>          eigenvalues are selected.
!>          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 is set to N+2 (see INFO below).
!>
```

*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 has been 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 will 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; if
!>          JOBVS = 'V', LDVS >= N.
!>
```

*WORK*

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

*LWORK*

```
!>          LWORK is INTEGER
!>          The dimension of the array WORK.  LWORK >= max(1,3*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.
!>
```

*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 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.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **214** of file **sgees.f**.

### subroutine zgees (character jobvs, character sort, external select, integer n, complex\*16, dimension( lda, \* ) a, integer lda, integer sdim, complex\*16, dimension( \* ) w, complex\*16, dimension( ldvs, \* ) vs, integer ldvs, complex\*16, dimension( \* ) work, integer lwork, double precision, dimension( \* ) rwork, logical, dimension( \* ) bwork, integer info)

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

**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.
!>
```

**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 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*

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

*A*

```
!>          A is 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*

```
!>          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 for which
!>                         SELECT is true.
!>
```

*W*

```
!>          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.
!>
```

*VS*

```
!>          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.
!>
```

*LDVS*

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

*WORK*

```
!>          WORK is COMPLEX*16 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(1,2*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*

```
!>          RWORK is DOUBLE PRECISION array, dimension (N)
!>
```

*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 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.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **195** of file **zgees.f**.

## Author

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