gges#

sgges

Functions

void sgges(
    const char*               jobvsl,
    const char*               jobvsr,
    const char*               sort,
          sselect3_t          selctg,
    const INT                 n,
          f32*       restrict A,
    const INT                 lda,
          f32*       restrict B,
    const INT                 ldb,
          INT*                sdim,
          f32*       restrict alphar,
          f32*       restrict alphai,
          f32*       restrict beta,
          f32*       restrict VSL,
    const INT                 ldvsl,
          f32*       restrict VSR,
    const INT                 ldvsr,
          f32*       restrict work,
    const INT                 lwork,
          INT*       restrict bwork,
          INT*                info
);

void sgges(const char *jobvsl, const char *jobvsr, const char *sort, sselect3_t selctg, const INT n, f32 *restrict A, const INT lda, f32 *restrict B, const INT ldb, INT *sdim, f32 *restrict alphar, f32 *restrict alphai, f32 *restrict beta, f32 *restrict VSL, const INT ldvsl, f32 *restrict VSR, const INT ldvsr, f32 *restrict work, const INT lwork, INT *restrict bwork, INT *info)#

SGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized real Schur form (S,T), optionally, the left and/or right matrices of Schur vectors (VSL and VSR).

This gives the generalized Schur factorization

     (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T.

Parameters

in

jobvsl

= ‘N’: do not compute the left Schur vectors; = ‘V’: compute the left Schur vectors.

in

jobvsr

= ‘N’: do not compute the right Schur vectors; = ‘V’: compute the right Schur vectors.

in

sort

= ‘N’: Eigenvalues are not ordered; = ‘S’: Eigenvalues are ordered (see selctg).

in

selctg

Selection function. If sort = ‘S’, selctg is used to select eigenvalues to sort to the top left of the Schur form. If sort = ‘N’, selctg is not referenced.

in

n

The order of the matrices A, B, VSL, and VSR. n >= 0.

inout

A

On entry, the first of the pair of matrices. On exit, A has been overwritten by its Schur form S.

in

lda

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

inout

B

On entry, the second of the pair of matrices. On exit, B has been overwritten by its Schur form T.

in

ldb

The leading dimension of B. ldb >= max(1,n).

out

sdim

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

out

alphar

Real parts of generalized eigenvalues.

out

alphai

Imaginary parts of generalized eigenvalues.

out

beta

Beta values of generalized eigenvalues.

out

VSL

If jobvsl = ‘V’, the left Schur vectors.

in

ldvsl

The leading dimension of VSL.

out

VSR

If jobvsr = ‘V’, the right Schur vectors.

in

ldvsr

The leading dimension of VSR.

out

work

Workspace array, dimension (max(1,lwork)).

in

lwork

The dimension of work.

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: errors from QZ iteration or reordering

dgges

Functions

void dgges(
    const char*               jobvsl,
    const char*               jobvsr,
    const char*               sort,
          dselect3_t          selctg,
    const INT                 n,
          f64*       restrict A,
    const INT                 lda,
          f64*       restrict B,
    const INT                 ldb,
          INT*                sdim,
          f64*       restrict alphar,
          f64*       restrict alphai,
          f64*       restrict beta,
          f64*       restrict VSL,
    const INT                 ldvsl,
          f64*       restrict VSR,
    const INT                 ldvsr,
          f64*       restrict work,
    const INT                 lwork,
          INT*       restrict bwork,
          INT*                info
);

void dgges(const char *jobvsl, const char *jobvsr, const char *sort, dselect3_t selctg, const INT n, f64 *restrict A, const INT lda, f64 *restrict B, const INT ldb, INT *sdim, f64 *restrict alphar, f64 *restrict alphai, f64 *restrict beta, f64 *restrict VSL, const INT ldvsl, f64 *restrict VSR, const INT ldvsr, f64 *restrict work, const INT lwork, INT *restrict bwork, INT *info)#

DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized real Schur form (S,T), optionally, the left and/or right matrices of Schur vectors (VSL and VSR).

This gives the generalized Schur factorization

     (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T.

Parameters

in

jobvsl

= ‘N’: do not compute the left Schur vectors; = ‘V’: compute the left Schur vectors.

in

jobvsr

= ‘N’: do not compute the right Schur vectors; = ‘V’: compute the right Schur vectors.

in

sort

= ‘N’: Eigenvalues are not ordered; = ‘S’: Eigenvalues are ordered (see selctg).

in

selctg

Selection function. If sort = ‘S’, selctg is used to select eigenvalues to sort to the top left of the Schur form. If sort = ‘N’, selctg is not referenced.

in

n

The order of the matrices A, B, VSL, and VSR. n >= 0.

inout

A

On entry, the first of the pair of matrices. On exit, A has been overwritten by its Schur form S.

in

lda

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

inout

B

On entry, the second of the pair of matrices. On exit, B has been overwritten by its Schur form T.

in

ldb

The leading dimension of B. ldb >= max(1,n).

out

sdim

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

out

alphar

Real parts of generalized eigenvalues.

out

alphai

Imaginary parts of generalized eigenvalues.

out

beta

Beta values of generalized eigenvalues.

out

VSL

If jobvsl = ‘V’, the left Schur vectors.

in

ldvsl

The leading dimension of VSL.

out

VSR

If jobvsr = ‘V’, the right Schur vectors.

in

ldvsr

The leading dimension of VSR.

out

work

Workspace array, dimension (max(1,lwork)).

in

lwork

The dimension of work.

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: errors from QZ iteration or reordering

cgges

Functions

void cgges(
    const char*      jobvsl,
    const char*      jobvsr,
    const char*      sort,
          cselect2_t selctg,
    const INT        n,
          c64*       A,
    const INT        lda,
          c64*       B,
    const INT        ldb,
          INT*       sdim,
          c64*       alpha,
          c64*       beta,
          c64*       VSL,
    const INT        ldvsl,
          c64*       VSR,
    const INT        ldvsr,
          c64*       work,
    const INT        lwork,
          f32*       rwork,
          INT*       bwork,
          INT*       info
);

void cgges(const char *jobvsl, const char *jobvsr, const char *sort, cselect2_t selctg, const INT n, c64 *A, const INT lda, c64 *B, const INT ldb, INT *sdim, c64 *alpha, c64 *beta, c64 *VSL, const INT ldvsl, c64 *VSR, const INT ldvsr, c64 *work, const INT lwork, f32 *rwork, INT *bwork, INT *info)#

CGGES computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized complex Schur form (S, T), and optionally left and/or right Schur vectors (VSL and VSR).

This gives the generalized Schur factorization

    (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )

where (VSR)**H is the conjugate-transpose of VSR.

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T. The leading columns of VSL and VSR then form an unitary basis for the corresponding left and right eigenspaces (deflating subspaces).

(If only the generalized eigenvalues are needed, use the driver CGGEV instead, which is faster.)

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero.

A pair of matrices (S,T) is in generalized complex Schur form if S and T are upper triangular and, in addition, the diagonal elements of T are non-negative real numbers.

Parameters

in

jobvsl

= ‘N’: do not compute the left Schur vectors; = ‘V’: compute the left Schur vectors.

in

jobvsr

= ‘N’: do not compute the right Schur vectors; = ‘V’: compute the right Schur vectors.

in

sort

= ‘N’: Eigenvalues are not ordered; = ‘S’: Eigenvalues are ordered (see selctg).

in

selctg

Selection function of two complex arguments. If sort = ‘S’, selctg is used to select eigenvalues to sort to the top left of the Schur form. An eigenvalue ALPHA(j)/BETA(j) is selected if SELCTG(ALPHA(j),BETA(j)) is true. If sort = ‘N’, selctg is not referenced.

in

n

The order of the matrices A, B, VSL, and VSR. n >= 0.

inout

A

On entry, the first of the pair of matrices. On exit, A has been overwritten by its generalized Schur form S. Dimension (lda, n).

in

lda

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

inout

B

On entry, the second of the pair of matrices. On exit, B has been overwritten by its generalized Schur form T. Dimension (ldb, n).

in

ldb

The leading dimension of B. ldb >= max(1,n).

out

sdim

If sort = ‘N’, sdim = 0. If sort = ‘S’, sdim = number of eigenvalues (after sorting) for which selctg is true.

out

alpha

Complex array, dimension (n).

out

beta

Complex array, dimension (n). On exit, ALPHA(j)/BETA(j), j=1,…,N, will be the generalized eigenvalues. BETA(j) will be non-negative real.

out

VSL

If jobvsl = ‘V’, the left Schur vectors. Dimension (ldvsl, n).

in

ldvsl

The leading dimension of VSL. ldvsl >= 1, and if jobvsl = ‘V’, ldvsl >= n.

out

VSR

If jobvsr = ‘V’, the right Schur vectors. Dimension (ldvsr, n).

in

ldvsr

The leading dimension of VSR. ldvsr >= 1, and if jobvsr = ‘V’, ldvsr >= n.

out

work

Complex workspace array, dimension (max(1,lwork)). On exit, if info = 0, work[0] returns the 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 (8*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
= 1,…,n: the QZ iteration failed. (A,B) are not in Schur form, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,…,N.
= n+1: other than QZ iteration failed in CHGEQZ
= n+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy SELCTG=.TRUE. This could also be caused due to scaling.
= n+3: reordering failed in CTGSEN.

zgges

Functions

void zgges(
    const char*      jobvsl,
    const char*      jobvsr,
    const char*      sort,
          zselect2_t selctg,
    const INT        n,
          c128*      A,
    const INT        lda,
          c128*      B,
    const INT        ldb,
          INT*       sdim,
          c128*      alpha,
          c128*      beta,
          c128*      VSL,
    const INT        ldvsl,
          c128*      VSR,
    const INT        ldvsr,
          c128*      work,
    const INT        lwork,
          f64*       rwork,
          INT*       bwork,
          INT*       info
);

void zgges(const char *jobvsl, const char *jobvsr, const char *sort, zselect2_t selctg, const INT n, c128 *A, const INT lda, c128 *B, const INT ldb, INT *sdim, c128 *alpha, c128 *beta, c128 *VSL, const INT ldvsl, c128 *VSR, const INT ldvsr, c128 *work, const INT lwork, f64 *rwork, INT *bwork, INT *info)#

ZGGES computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized complex Schur form (S, T), and optionally left and/or right Schur vectors (VSL and VSR).

This gives the generalized Schur factorization

    (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )

where (VSR)**H is the conjugate-transpose of VSR.

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T. The leading columns of VSL and VSR then form an unitary basis for the corresponding left and right eigenspaces (deflating subspaces).

(If only the generalized eigenvalues are needed, use the driver ZGGEV instead, which is faster.)

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero.

A pair of matrices (S,T) is in generalized complex Schur form if S and T are upper triangular and, in addition, the diagonal elements of T are non-negative real numbers.

Parameters

in

jobvsl

= ‘N’: do not compute the left Schur vectors; = ‘V’: compute the left Schur vectors.

in

jobvsr

= ‘N’: do not compute the right Schur vectors; = ‘V’: compute the right Schur vectors.

in

sort

= ‘N’: Eigenvalues are not ordered; = ‘S’: Eigenvalues are ordered (see selctg).

in

selctg

Selection function of two complex arguments. If sort = ‘S’, selctg is used to select eigenvalues to sort to the top left of the Schur form. An eigenvalue ALPHA(j)/BETA(j) is selected if SELCTG(ALPHA(j),BETA(j)) is true. If sort = ‘N’, selctg is not referenced.

in

n

The order of the matrices A, B, VSL, and VSR. n >= 0.

inout

A

On entry, the first of the pair of matrices. On exit, A has been overwritten by its generalized Schur form S. Dimension (lda, n).

in

lda

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

inout

B

On entry, the second of the pair of matrices. On exit, B has been overwritten by its generalized Schur form T. Dimension (ldb, n).

in

ldb

The leading dimension of B. ldb >= max(1,n).

out

sdim

If sort = ‘N’, sdim = 0. If sort = ‘S’, sdim = number of eigenvalues (after sorting) for which selctg is true.

out

alpha

Complex array, dimension (n).

out

beta

Complex array, dimension (n). On exit, ALPHA(j)/BETA(j), j=1,…,N, will be the generalized eigenvalues. BETA(j) will be non-negative real.

out

VSL

If jobvsl = ‘V’, the left Schur vectors. Dimension (ldvsl, n).

in

ldvsl

The leading dimension of VSL. ldvsl >= 1, and if jobvsl = ‘V’, ldvsl >= n.

out

VSR

If jobvsr = ‘V’, the right Schur vectors. Dimension (ldvsr, n).

in

ldvsr

The leading dimension of VSR. ldvsr >= 1, and if jobvsr = ‘V’, ldvsr >= n.

out

work

Complex workspace array, dimension (max(1,lwork)). On exit, if info = 0, work[0] returns the 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 (8*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
= 1,…,n: the QZ iteration failed. (A,B) are not in Schur form, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,…,N.
= n+1: other than QZ iteration failed in ZHGEQZ
= n+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy SELCTG=.TRUE. This could also be caused due to scaling.
= n+3: reordering failed in ZTGSEN.

gges#

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