geesx

sgeesx

Functions

void sgeesx(
    const char*      jobvs,
    const char*      sort,
          sselect2_t select,
    const char*      sense,
    const INT        n,
          f32*       A,
    const INT        lda,
          INT*       sdim,
          f32*       wr,
          f32*       wi,
          f32*       VS,
    const INT        ldvs,
          f32*       rconde,
          f32*       rcondv,
          f32*       work,
    const INT        lwork,
          INT*       iwork,
    const INT        liwork,
          INT*       bwork,
          INT*       info
);

void sgeesx(const char *jobvs, const char *sort, sselect2_t select, const char *sense, const INT n, f32 *A, const INT lda, INT *sdim, f32 *wr, f32 *wi, f32 *VS, const INT ldvs, f32 *rconde, f32 *rcondv, f32 *work, const INT lwork, INT *iwork, const INT liwork, INT *bwork, INT *info)

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.

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

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.

in

sense

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’.

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

rconde

If sense = ‘E’ or ‘B’, contains the reciprocal condition number for the average of the selected eigenvalues.

out

rcondv

If sense = ‘V’ or ‘B’, contains the reciprocal condition number for the selected right invariant subspace.

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

iwork

Integer array, dimension (max(1, liwork)). On exit, if info = 0, iwork[0] returns optimal liwork.

in

liwork

The dimension of iwork. liwork >= 1; if sense = ‘V’ or ‘B’, liwork >= sdim*(n-sdim). If liwork = -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;

• = 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.

dgeesx

Functions

void dgeesx(
    const char*      jobvs,
    const char*      sort,
          dselect2_t select,
    const char*      sense,
    const INT        n,
          f64*       A,
    const INT        lda,
          INT*       sdim,
          f64*       wr,
          f64*       wi,
          f64*       VS,
    const INT        ldvs,
          f64*       rconde,
          f64*       rcondv,
          f64*       work,
    const INT        lwork,
          INT*       iwork,
    const INT        liwork,
          INT*       bwork,
          INT*       info
);

void dgeesx(const char *jobvs, const char *sort, dselect2_t select, const char *sense, const INT n, f64 *A, const INT lda, INT *sdim, f64 *wr, f64 *wi, f64 *VS, const INT ldvs, f64 *rconde, f64 *rcondv, f64 *work, const INT lwork, INT *iwork, const INT liwork, INT *bwork, INT *info)

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.

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

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.

in

sense

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’.

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

rconde

If sense = ‘E’ or ‘B’, contains the reciprocal condition number for the average of the selected eigenvalues.

out

rcondv

If sense = ‘V’ or ‘B’, contains the reciprocal condition number for the selected right invariant subspace.

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

iwork

Integer array, dimension (max(1, liwork)). On exit, if info = 0, iwork[0] returns optimal liwork.

in

liwork

The dimension of iwork. liwork >= 1; if sense = ‘V’ or ‘B’, liwork >= sdim*(n-sdim). If liwork = -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;

• = 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.

cgeesx

Functions

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

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

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.

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. An eigenvalue W(j) is selected if select(W(j)) is true.

in

sense

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’.

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

rconde

If sense = ‘E’ or ‘B’, contains the reciprocal condition number for the average of the selected eigenvalues.

out

rcondv

If sense = ‘V’ or ‘B’, contains the reciprocal condition number for the selected right invariant subspace.

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). 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. 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; if jobvs = ‘V’, VS contains the transformation which reduces A to its partially converged Schur form.

• = n+1: 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.

zgeesx

Functions

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

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

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.

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. An eigenvalue W(j) is selected if select(W(j)) is true.

in

sense

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’.

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

rconde

If sense = ‘E’ or ‘B’, contains the reciprocal condition number for the average of the selected eigenvalues.

out

rcondv

If sense = ‘V’ or ‘B’, contains the reciprocal condition number for the selected right invariant subspace.

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). 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. 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; if jobvs = ‘V’, VS contains the transformation which reduces A to its partially converged Schur form.

• = n+1: 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.