gees

sgees

Functions

void sgees(
    const char*      jobvs,
    const char*      sort,
          sselect2_t select,
    const INT        n,
          f32*       A,
    const INT        lda,
          INT*       sdim,
          f32*       wr,
          f32*       wi,
          f32*       VS,
    const INT        ldvs,
          f32*       work,
    const INT        lwork,
          INT*       bwork,
          INT*       info
);

void sgees(const char *jobvs, const char *sort, sselect2_t select, const INT n, f32 *A, const INT lda, INT *sdim, f32 *wr, f32 *wi, f32 *VS, const INT ldvs, f32 *work, const INT lwork, INT *bwork, INT *info)

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

in

jobvs

= ‘N’: Schur vectors are not computed; = ‘V’: Schur vectors are computed.

in

sort

Specifies whether or not to order the eigenvalues: = ‘N’: Eigenvalues are not ordered; = ‘S’: Eigenvalues are ordered (see select).

in

select

Eigenvalue selection callback. 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 and may be NULL. The callback should return nonzero if the eigenvalue wr + i*wi should be selected.

in

n

The order of the matrix A. n >= 0.

inout

A

On entry, the N-by-N matrix A. On exit, A has been overwritten by its real Schur form T. Dimension (lda, n).

in

lda

The leading dimension of A. lda >= max(1, n).

out

sdim

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.)

out

wr

Array, dimension (n). Real parts of eigenvalues.

out

wi

Array, dimension (n). Imaginary parts of eigenvalues.

out

VS

If jobvs = ‘V’, VS contains the orthogonal matrix Z of Schur vectors. Dimension (ldvs, n). If jobvs = ‘N’, VS is not referenced.

in

ldvs

The leading dimension of VS. ldvs >= 1; if jobvs = ‘V’, ldvs >= n.

out

work

Workspace array, dimension (max(1, lwork)). On exit, if info = 0, work[0] returns optimal lwork.

in

lwork

The dimension of work. lwork >= max(1, 3*n). If lwork = -1, a workspace query is assumed.

out

bwork

Integer array, dimension (n). Not referenced if sort = ‘N’.

out

info

• = 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 eigenvalues; elements 0:ilo-1 and i:n-1 of wr and wi contain those eigenvalues which have converged;

• = n+1: eigenvalues could not be reordered because some eigenvalues were too close to separate;

• = n+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy select=true.

dgees

Functions

void dgees(
    const char*      jobvs,
    const char*      sort,
          dselect2_t select,
    const INT        n,
          f64*       A,
    const INT        lda,
          INT*       sdim,
          f64*       wr,
          f64*       wi,
          f64*       VS,
    const INT        ldvs,
          f64*       work,
    const INT        lwork,
          INT*       bwork,
          INT*       info
);

void dgees(const char *jobvs, const char *sort, dselect2_t select, const INT n, f64 *A, const INT lda, INT *sdim, f64 *wr, f64 *wi, f64 *VS, const INT ldvs, f64 *work, const INT lwork, INT *bwork, INT *info)

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

in

jobvs

= ‘N’: Schur vectors are not computed; = ‘V’: Schur vectors are computed.

in

sort

Specifies whether or not to order the eigenvalues: = ‘N’: Eigenvalues are not ordered; = ‘S’: Eigenvalues are ordered (see select).

in

select

Eigenvalue selection callback. 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 and may be NULL. The callback should return nonzero if the eigenvalue wr + i*wi should be selected.

in

n

The order of the matrix A. n >= 0.

inout

A

On entry, the N-by-N matrix A. On exit, A has been overwritten by its real Schur form T. Dimension (lda, n).

in

lda

The leading dimension of A. lda >= max(1, n).

out

sdim

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.)

out

wr

Array, dimension (n). Real parts of eigenvalues.

out

wi

Array, dimension (n). Imaginary parts of eigenvalues.

out

VS

If jobvs = ‘V’, VS contains the orthogonal matrix Z of Schur vectors. Dimension (ldvs, n). If jobvs = ‘N’, VS is not referenced.

in

ldvs

The leading dimension of VS. ldvs >= 1; if jobvs = ‘V’, ldvs >= n.

out

work

Workspace array, dimension (max(1, lwork)). On exit, if info = 0, work[0] returns optimal lwork.

in

lwork

The dimension of work. lwork >= max(1, 3*n). If lwork = -1, a workspace query is assumed.

out

bwork

Integer array, dimension (n). Not referenced if sort = ‘N’.

out

info

• = 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 eigenvalues; elements 0:ilo-1 and i:n-1 of wr and wi contain those eigenvalues which have converged;

• = n+1: eigenvalues could not be reordered because some eigenvalues were too close to separate;

• = n+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy select=true.

cgees

Functions

void cgees(
    const char*      jobvs,
    const char*      sort,
          cselect1_t select,
    const INT        n,
          c64*       A,
    const INT        lda,
          INT*       sdim,
          c64*       W,
          c64*       VS,
    const INT        ldvs,
          c64*       work,
    const INT        lwork,
          f32*       rwork,
          INT*       bwork,
          INT*       info
);

void cgees(const char *jobvs, const char *sort, cselect1_t select, const INT n, c64 *A, const INT lda, INT *sdim, c64 *W, c64 *VS, const INT ldvs, c64 *work, const INT lwork, f32 *rwork, INT *bwork, INT *info)

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

in

jobvs

= ‘N’: Schur vectors are not computed; = ‘V’: Schur vectors are computed.

in

sort

Specifies whether or not to order the eigenvalues: = ‘N’: Eigenvalues are not ordered; = ‘S’: Eigenvalues are ordered (see select).

in

select

Eigenvalue selection callback. 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 and may be NULL. The callback should return nonzero if the eigenvalue W(j) should be selected.

in

n

The order of the matrix A. n >= 0.

inout

A

On entry, the N-by-N matrix A. On exit, A has been overwritten by its Schur form T. Dimension (lda, n).

in

lda

The leading dimension of A. lda >= max(1, n).

out

sdim

If sort = ‘N’, sdim = 0. If sort = ‘S’, sdim = number of eigenvalues for which select is true.

out

W

Complex array, dimension (n). Contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.

out

VS

If jobvs = ‘V’, VS contains the unitary matrix Z of Schur vectors. Dimension (ldvs, n). If jobvs = ‘N’, VS is not referenced.

in

ldvs

The leading dimension of VS. ldvs >= 1; if jobvs = ‘V’, ldvs >= n.

out

work

Complex workspace array, dimension (max(1, lwork)). On exit, if info = 0, work[0] returns optimal lwork.

in

lwork

The dimension of work. lwork >= max(1, 2*n). If lwork = -1, a workspace query is assumed.

out

rwork

Single precision array, dimension (n).

out

bwork

Integer array, dimension (n). Not referenced if sort = ‘N’.

out

info

• = 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 eigenvalues; elements 0:ilo-1 and i:n-1 of W contain those eigenvalues which have converged;

• = n+1: eigenvalues could not be reordered because some eigenvalues were too close to separate;

• = n+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy select=true.

zgees

Functions

void zgees(
    const char*      jobvs,
    const char*      sort,
          zselect1_t select,
    const INT        n,
          c128*      A,
    const INT        lda,
          INT*       sdim,
          c128*      W,
          c128*      VS,
    const INT        ldvs,
          c128*      work,
    const INT        lwork,
          f64*       rwork,
          INT*       bwork,
          INT*       info
);

void zgees(const char *jobvs, const char *sort, zselect1_t select, const INT n, c128 *A, const INT lda, INT *sdim, c128 *W, c128 *VS, const INT ldvs, c128 *work, const INT lwork, f64 *rwork, INT *bwork, INT *info)

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

in

jobvs

= ‘N’: Schur vectors are not computed; = ‘V’: Schur vectors are computed.

in

sort

Specifies whether or not to order the eigenvalues: = ‘N’: Eigenvalues are not ordered; = ‘S’: Eigenvalues are ordered (see select).

in

select

Eigenvalue selection callback. 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 and may be NULL. The callback should return nonzero if the eigenvalue W(j) should be selected.

in

n

The order of the matrix A. n >= 0.

inout

A

On entry, the N-by-N matrix A. On exit, A has been overwritten by its Schur form T. Dimension (lda, n).

in

lda

The leading dimension of A. lda >= max(1, n).

out

sdim

If sort = ‘N’, sdim = 0. If sort = ‘S’, sdim = number of eigenvalues for which select is true.

out

W

Complex array, dimension (n). Contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.

out

VS

If jobvs = ‘V’, VS contains the unitary matrix Z of Schur vectors. Dimension (ldvs, n). If jobvs = ‘N’, VS is not referenced.

in

ldvs

The leading dimension of VS. ldvs >= 1; if jobvs = ‘V’, ldvs >= n.

out

work

Complex workspace array, dimension (max(1, lwork)). On exit, if info = 0, work[0] returns optimal lwork.

in

lwork

The dimension of work. lwork >= max(1, 2*n). If lwork = -1, a workspace query is assumed.

out

rwork

Double precision array, dimension (n).

out

bwork

Integer array, dimension (n). Not referenced if sort = ‘N’.

out

info

• = 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 eigenvalues; elements 0:ilo-1 and i:n-1 of W contain those eigenvalues which have converged;

• = n+1: eigenvalues could not be reordered because some eigenvalues were too close to separate;

• = n+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy select=true.