(S|D|C|Z)GGES¶

Single, double, single complex, and double complex GGES.

Description¶

Computes the generalized eigenvalues, Schur form, and (optionally) Schur vectors of a general matrix.

Given two N by N matrices A and B, computes the generalized Schur factorization:

$\pair{\fvar{A}}{\fvar{B}} = \pair{\fvar{VSL} \mult \fvar{S} \mult \herm{\fvar{VSR}}} {\fvar{VSL} \mult \fvar{T} \mult \herm{\fvar{VSR}}}$

The output matrix S will be upper block triangular with blocks of size 1x1 or 2x2. Blocks of size 2x2 will only occur in SGGES and DGGES and represent a complex conjugate pair of generalized eigenvalues. The output matrix T will be upper triangular with non-negative diagonal.

A generalized eigenvalue is a solution to the equation:

$A v = \lambda B v$

Generalized eigenvalues are represented as a ratio ALPHA / BETA with separate storage for ALPHA and BETA for increased numerical flexibility.

LAPACK Interface¶

void sgges(const char *JOBVSL, const char *JOBVSR, const char *SORT,
    void *SELCTG, const qml_long *N, float *A, const qml_long *LDA,
    float *B, const qml_long *LDB, qml_long *SDIM, float *ALPHAR,
    float *ALPHAI, float *BETA, float *VSL, const qml_long *LDVSL,
    float *VSR, const qml_long *LDVSR, float *WORK, const qml_long *LWORK,
    qml_int *BWORK, qml_long *INFO);

void dgges(const char *JOBVSL, const char *JOBVSR, const char *SORT,
    void *SELCTG, const qml_long *N, double *A, const qml_long *LDA,
    double *B, const qml_long *LDB, qml_long *SDIM, double *ALPHAR,
    double *ALPHAI, double *BETA, double *VSL, const qml_long *LDVSL,
    double *VSR, const qml_long *LDVSR, double *WORK, const qml_long *LWORK,
    qml_int *BWORK, qml_long *INFO);

void cgges(const char *JOBVSL, const char *JOBVSR, const char *SORT,
    void *SELCTG, const qml_long *N, qml_single_complex *A,
    const qml_long *LDA, qml_single_complex *B, const qml_long *LDB,
    qml_long *SDIM, qml_single_complex *ALPHA, qml_single_complex *BETA,
    qml_single_complex *VSL, const qml_long *LDVSL, qml_single_complex *VSR,
    const qml_long *LDVSR, qml_single_complex *WORK, const qml_long *LWORK,
    float *RWORK, qml_int *BWORK, qml_long *INFO);

void zgges(const char *JOBVSL, const char *JOBVSR, const char *SORT,
    void *SELCTG, const qml_long *N, qml_double_complex *A,
    const qml_long *LDA, qml_double_complex *B, const qml_long *LDB,
    qml_long *SDIM, qml_double_complex *ALPHA, qml_double_complex *BETA,
    qml_double_complex *VSL, const qml_long *LDVSL, qml_double_complex *VSR,
    const qml_long *LDVSR, qml_double_complex *WORK, const qml_long *LWORK,
    double *RWORK, qml_int *BWORK, qml_long *INFO);

Arguments¶

JOBVSL	Compute VSL if ‘V’, otherwise do not compute
JOBVSR	Compute VSR if ‘V’, otherwise do not compute
SORT	Sort eigenvalues, only ‘N’ supported
SELCTG	Function pointers, not supported (pass NULL)
N	Number of rows and columns of A, B, VSL, and VSR
A	Matrix of size N x N, overwritten with S on output
LDA	Leading dimension of A
B	Matrix of size N x N, overwritten with T on output
LDB	Leading dimension of B
SDIM	Not used
ALPHA	On exit contains generalized eigenvalues
ALPHAR	Real part of ALPHA for real versions
ALPHAI	Complex part of ALPHA for real version
VSL	On exit contains left Schur vectors if JOBVSL is ‘V’
LDVSL	Leading dimension of VSL
VSR	On exit contains right Schur vectors if JOBVSR is ‘V’
LDVSR	Leading dimension of VSR
WORK	Work space of size at least LWORK
LWORK	Size of work space, at least 2N but optimally a larger multiple of N (-1 to query)
RWORK	Work space of size 8N
BWORK	Not used
INFO	0 on success