StarNEig Library  v0.1.1
A task-based library for solving dense nonsymmetric eigenvalue problems
Shared Memory / Generalized EVP

Functions for solving non-symmetric generalized eigenvalue problems on shared memory systems. More...

Computational functions

starneig_error_t starneig_GEP_SM_HessenbergTriangular (int n, double A[], int ldA, double B[], int ldB, double Q[], int ldQ, double Z[], int ldZ)
 Computes a Hessenberg-triangular decomposition of a general matrix pencil. More...
 
starneig_error_t starneig_GEP_SM_Schur (int n, double H[], int ldH, double T[], int ldT, double Q[], int ldQ, double Z[], int ldZ, double real[], double imag[], double beta[])
 Computes a generalized Schur decomposition given a Hessenberg-triangular decomposition. More...
 
starneig_error_t starneig_GEP_SM_ReorderSchur (int n, int selected[], double S[], int ldS, double T[], int ldT, double Q[], int ldQ, double Z[], int ldZ, double real[], double imag[], double beta[])
 Reorders selected generalized eigenvalues to the top left corner of a generalized Schur decomposition. More...
 
starneig_error_t starneig_GEP_SM_Reduce (int n, double A[], int ldA, double B[], int ldB, double Q[], int ldQ, double Z[], int ldZ, double real[], double imag[], double beta[], int(*predicate)(double real, double imag, double beta, void *arg), void *arg, int selected[], int *num_selected)
 Computes a (reordered) generalized Schur decomposition given a general matrix pencil. More...
 
starneig_error_t starneig_GEP_SM_Eigenvectors (int n, int selected[], double S[], int ldS, double T[], int ldT, double Z[], int ldZ, double X[], int ldX)
 Computes a generalized eigenvector for each selected generalized eigenvalue. More...
 

Helper functions

starneig_error_t starneig_GEP_SM_Select (int n, double S[], int ldS, double T[], int ldT, int(*predicate)(double real, double imag, double beta, void *arg), void *arg, int selected[], int *num_selected)
 Generates a selection array for a Schur-triangular matrix pencil using a user-supplied predicate function. More...
 

Expert computational functions

starneig_error_t starneig_GEP_SM_Schur_expert (struct starneig_schur_conf *conf, int n, double H[], int ldH, double T[], int ldT, double Q[], int ldQ, double Z[], int ldZ, double real[], double imag[], double beta[])
 Computes a generalized Schur decomposition given a Hessenberg-triangular decomposition. More...
 
starneig_error_t starneig_GEP_SM_ReorderSchur_expert (struct starneig_reorder_conf *conf, int n, int selected[], double S[], int ldS, double T[], int ldT, double Q[], int ldQ, double Z[], int ldZ, double real[], double imag[], double beta[])
 Reorders selected eigenvalues to the top left corner of a generalized Schur decomposition. More...
 
starneig_error_t starneig_GEP_SM_Eigenvectors_expert (struct starneig_eigenvectors_conf *conf, int n, int selected[], double S[], int ldS, double T[], int ldT, double Z[], int ldZ, double X[], int ldX)
 Computes a generalized eigenvector for each selected generalized eigenvalue. More...
 

Detailed Description

Functions for solving non-symmetric generalized eigenvalue problems on shared memory systems.

Function Documentation

◆ starneig_GEP_SM_HessenbergTriangular()

starneig_error_t starneig_GEP_SM_HessenbergTriangular ( int  n,
double  A[],
int  ldA,
double  B[],
int  ldB,
double  Q[],
int  ldQ,
double  Z[],
int  ldZ 
)

Computes a Hessenberg-triangular decomposition of a general matrix pencil.

Remarks
This function is a wrapper for several LAPACK subroutines.
Parameters
[in]nThe order of $A$, $B$, $Q$ and $Z$.
[in,out]AOn entry, the general matrix $A$. On exit, the upper Hessenberg matrix $H$.
[in]ldAThe leading dimension of $A$.
[in,out]BOn entry, the general matrix $B$. On exit, the upper triangular matrix $T$.
[in]ldBThe leading dimension of $B$.
[in,out]QOn entry, the orthogonal matrix $Q$. On exit, the product matrix $Q * U_1$.
[in]ldQThe leading dimension of $Q$.
[in,out]ZOn entry, the orthogonal matrix $Z$. On exit, the product matrix $Z * U_2$.
[in]ldZThe leading dimension of $Z$.
Returns
STARNEIG_SUCCESS (0) on success. Negative integer -i when i'th argument is invalid. Positive error code otherwise.
Examples:
gep_sm_full_chain.c.

◆ starneig_GEP_SM_Schur()

starneig_error_t starneig_GEP_SM_Schur ( int  n,
double  H[],
int  ldH,
double  T[],
int  ldT,
double  Q[],
int  ldQ,
double  Z[],
int  ldZ,
double  real[],
double  imag[],
double  beta[] 
)

Computes a generalized Schur decomposition given a Hessenberg-triangular decomposition.

Parameters
[in]nThe order of $H$, $T$, $Q$ and $Z$.
[in,out]HOn entry, the upper Hessenberg matrix $H$. On exit, the Schur matrix $S$.
[in]ldHThe leading dimension of $H$.
[in,out]TOn entry, the upper triangular matrix $T$. On exit, the upper triangular matrix $\hat{T}$.
[in]ldTThe leading dimension of $T$.
[in,out]QOn entry, the orthogonal matrix $Q$. On exit, the product matrix $Q * U_1$.
[in]ldQThe leading dimension of $Q$.
[in,out]ZOn entry, the orthogonal matrix $Z$. On exit, the product matrix $Z * U_2$.
[in]ldZThe leading dimension of $Z$.
[out]realAn array of the same size as $H$ containing the real parts of the $\alpha$ values of the computed generalized eigenvalues.
[out]imagAn array of the same size as $H$ containing the imaginary parts of the $\alpha$ values of the computed generalized eigenvalues.
[out]betaAn array of the same size as $H$ containing the $\beta$ values of computed generalized eigenvalues.
Returns
STARNEIG_SUCCESS (0) on success. Negative integer -i when i'th argument is invalid. Positive error code otherwise. STARNEIG_DID_NOT_CONVERGE if the QZ algorithm failed to converge.
Examples:
gep_sm_full_chain.c.

◆ starneig_GEP_SM_ReorderSchur()

starneig_error_t starneig_GEP_SM_ReorderSchur ( int  n,
int  selected[],
double  S[],
int  ldS,
double  T[],
int  ldT,
double  Q[],
int  ldQ,
double  Z[],
int  ldZ,
double  real[],
double  imag[],
double  beta[] 
)

Reorders selected generalized eigenvalues to the top left corner of a generalized Schur decomposition.

Parameters
[in]nThe order of $H$, $T$, $Q$ and $Z$.
[in,out]selectedThe selection array. On entry, the initial positions of the selected generalized eigenvalues. On exit, the final positions of all correctly placed selected generalized eigenvalues. In case of failure, the number of 1's in the output may be less than the number of 1's in the input.
[in,out]SOn entry, the Schur matrix $S$. On exit, the updated Schur matrix $\hat{S}$.
[in]ldSThe leading dimension of $S$.
[in,out]TOn entry, the upper triangular $T$. On exit, the updates upper triangular matrix $\hat{T}$.
[in]ldTThe leading dimension of $T$.
[in,out]QOn entry, the orthogonal matrix $Q$. On exit, the product matrix $Q * U_1$.
[in]ldQThe leading dimension of $Q$.
[in,out]ZOn entry, the orthogonal matrix $Z$. On exit, the product matrix $Z * U_2$.
[in]ldZThe leading dimension of $Z$.
[out]realAn array of the same size as $S$ containing the real parts of the $\alpha$ values of the computed generalized eigenvalues.
[out]imagAn array of the same size as $S$ containing the imaginary parts of the $\alpha$ values of the computed generalized eigenvalues.
[out]betaAn array of the same size as $S$ containing the $\beta$ values of computed generalized eigenvalues.
Returns
STARNEIG_SUCCESS (0) on success. Negative integer -i when i'th argument is invalid. Positive error code otherwise. STARNEIG_PARTIAL_REORDERING if the generalized Schur form is not fully reordered.
See also
starneig_GEP_SM_Select
Examples:
gep_sm_full_chain.c.

◆ starneig_GEP_SM_Reduce()

starneig_error_t starneig_GEP_SM_Reduce ( int  n,
double  A[],
int  ldA,
double  B[],
int  ldB,
double  Q[],
int  ldQ,
double  Z[],
int  ldZ,
double  real[],
double  imag[],
double  beta[],
int(*)(double real, double imag, double beta, void *arg)  predicate,
void *  arg,
int  selected[],
int *  num_selected 
)

Computes a (reordered) generalized Schur decomposition given a general matrix pencil.

Parameters
[in]nThe order of $A$, $B$, $Q$ and $Z$.
[in,out]AOn entry, the general matrix $A$. On exit, the Schur matrix $S$.
[in]ldAThe leading dimension of $A$.
[in,out]BOn entry, the general matrix $B$. On exit, the upper triangular matrix $T$.
[in]ldBThe leading dimension of $B$.
[in,out]QOn entry, the orthogonal matrix $Q$. On exit, the product matrix $Q * U_1$.
[in]ldQThe leading dimension of $Q$.
[in,out]ZOn entry, the orthogonal matrix $Z$. On exit, the product matrix $Z * U_2$.
[in]ldZThe leading dimension of $Z$.
[out]realAn array of the same size as $A$ containing the real parts of the $\alpha$ values of the computed generalized eigenvalues.
[out]imagAn array of the same size as $A$ containing the imaginary parts of the $\alpha$ values of the computed generalized eigenvalues.
[out]betaAn array of the same size as $A$ containing the $\beta$ values of computed generalized eigenvalues.
[in]predicateA function that takes a (complex) generalized eigenvalue as input and returns non-zero if it should be selected. For complex conjugate pairs of generalized eigenvalues, the predicate is called only for the generalized eigenvalue with positive imaginary part and the corresponding $2 \times 2$ block is either selected or deselected. The reordering step is skipped if the argument is a NULL pointer.
[in]argAn optional argument for the predicate function.
[out]selectedThe final positions of all correctly placed selected generalized eigenvalues.
[out]num_selectedThe number of selected generalized eigenvalues (a complex conjugate pair is counted as two selected generalized eigenvalues).
Returns
STARNEIG_SUCCESS (0) on success. Negative integer -i when i'th argument is invalid. Positive error code otherwise. STARNEIG_DID_NOT_CONVERGE if the QZ algorithm failed to converge. STARNEIG_PARTIAL_REORDERING if the generalized Schur form is not fully reordered.
Examples:
gep_sm_eigenvectors.c.

◆ starneig_GEP_SM_Eigenvectors()

starneig_error_t starneig_GEP_SM_Eigenvectors ( int  n,
int  selected[],
double  S[],
int  ldS,
double  T[],
int  ldT,
double  Z[],
int  ldZ,
double  X[],
int  ldX 
)

Computes a generalized eigenvector for each selected generalized eigenvalue.

Parameters
[in]nThe order of $S$ and $Q$ and the number of rows of $X$.
[in]selectedThe selection array specifying the locations of the selected generalized eigenvalues. The number of 1's in the array is the same as the number of columns in $X$.
[in]SThe Schur matrix $S$.
[in]ldSThe leading dimension of $S$.
[in]TThe upper triangular matrix $T$.
[in]ldTThe leading dimension of $T$.
[in]ZThe orthogonal matrix $Z$.
[in]ldZThe leading dimension of $Z$.
[out]XA matrix with $n$ rows and one column for each selected generalized eigenvalue. The columns represent the computed generalized eigenvectors as previously described.
[in]ldXThe leading dimension of $X$.
Returns
STARNEIG_SUCCESS (0) on success. Negative integer -i when i'th argument is invalid. Positive error code otherwise.
See also
starneig_GEP_SM_Select
Examples:
gep_sm_eigenvectors.c.

◆ starneig_GEP_SM_Select()

starneig_error_t starneig_GEP_SM_Select ( int  n,
double  S[],
int  ldS,
double  T[],
int  ldT,
int(*)(double real, double imag, double beta, void *arg)  predicate,
void *  arg,
int  selected[],
int *  num_selected 
)

Generates a selection array for a Schur-triangular matrix pencil using a user-supplied predicate function.

Parameters
[in]nThe order of $S$ and $T$.
[in]SThe Schur matrix $S$.
[in]ldSThe leading dimension of $S$.
[in]TThe upper triangular matrix $T$.
[in]ldTThe leading dimension of $T$.
[in]predicateA function that takes a (complex) generalized eigenvalue as input and returns non-zero if it should be selected. For complex conjugate pairs of generalized eigenvalues, the predicate is called only for the generallized eigenvalue with positive imaginary part and the corresponding $2 \times 2$ block is either selected or deselected.
[in]argAn optional argument for the predicate function.
[out]selectedThe selection array. Both elements of a selected complex conjugate pair are set to 1.
[out]num_selectedThe number of selected generalized eigenvalues (a complex conjugate pair is counted as two selected generalized eigenvalues).
Returns
STARNEIG_SUCCESS (0) on success. Negative integer -i when i'th argument is invalid. Positive error code otherwise.
Examples:
gep_sm_eigenvectors.c, and gep_sm_full_chain.c.

◆ starneig_GEP_SM_Schur_expert()

starneig_error_t starneig_GEP_SM_Schur_expert ( struct starneig_schur_conf *  conf,
int  n,
double  H[],
int  ldH,
double  T[],
int  ldT,
double  Q[],
int  ldQ,
double  Z[],
int  ldZ,
double  real[],
double  imag[],
double  beta[] 
)

Computes a generalized Schur decomposition given a Hessenberg-triangular decomposition.

Parameters
[in]confConfiguration structure.
[in]nThe order of $H$, $T$, $Q$ and $Z$.
[in,out]HOn entry, the upper Hessenberg matrix $H$. On exit, the Schur matrix $S$.
[in]ldHThe leading dimension of $H$.
[in,out]TOn entry, the upper triangular matrix $T$. On exit, the upper triangular matrix $\hat{T}$.
[in]ldTThe leading dimension of $T$.
[in,out]QOn entry, the orthogonal matrix $Q$. On exit, the product matrix $Q * U_1$.
[in]ldQThe leading dimension of $Q$.
[in,out]ZOn entry, the orthogonal matrix $Z$. On exit, the product matrix $Z * U_2$.
[in]ldZThe leading dimension of $Z$.
[out]realAn array of the same size as $H$ containing the real parts of the $\alpha$ values of the computed generalized eigenvalues.
[out]imagAn array of the same size as $H$ containing the imaginary parts of the $\alpha$ values of the computed generalized eigenvalues.
[out]betaAn array of the same size as $H$ containing the $\beta$ values of computed generalized eigenvalues.
Returns
STARNEIG_SUCCESS (0) on success. Negative integer -i when i'th argument is invalid. Positive error code otherwise.
See also
starneig_GEP_SM_Schur
starneig_schur_conf
starneig_schur_init_conf

◆ starneig_GEP_SM_ReorderSchur_expert()

starneig_error_t starneig_GEP_SM_ReorderSchur_expert ( struct starneig_reorder_conf *  conf,
int  n,
int  selected[],
double  S[],
int  ldS,
double  T[],
int  ldT,
double  Q[],
int  ldQ,
double  Z[],
int  ldZ,
double  real[],
double  imag[],
double  beta[] 
)

Reorders selected eigenvalues to the top left corner of a generalized Schur decomposition.

Parameters
[in]confConfiguration structure.
[in]nThe order of $H$, $T$, $Q$ and $Z$.
[in,out]selectedThe selection array.
[in,out]SOn entry, the Schur matrix $S$. On exit, the updated Schur matrix $\hat{S}$.
[in]ldSThe leading dimension of $S$.
[in,out]TOn entry, the upper triangular $T$. On exit, the updates upper triangular matrix $\hat{T}$.
[in]ldTThe leading dimension of $T$.
[in,out]QOn entry, the orthogonal matrix $Q$. On exit, the product matrix $Q * U_1$.
[in]ldQThe leading dimension of $Q$.
[in,out]ZOn entry, the orthogonal matrix $Z$. On exit, the product matrix $Z * U_2$.
[in]ldZThe leading dimension of $Z$.
[out]realAn array of the same size as $S$ containing the real parts of the $\alpha$ values of the computed generalized eigenvalues.
[out]imagAn array of the same size as $S$ containing the imaginary parts of the $\alpha$ values of the computed generalized eigenvalues.
[out]betaAn array of the same size as $S$ containing the $\beta$ values of computed generalized eigenvalues.
Returns
STARNEIG_SUCCESS (0) on success. Negative integer -i when i'th argument is invalid. Positive error code otherwise.
See also
starneig_GEP_SM_ReorderSchur
starneig_GEP_SM_Select
starneig_reorder_conf
starneig_reorder_init_conf

◆ starneig_GEP_SM_Eigenvectors_expert()

starneig_error_t starneig_GEP_SM_Eigenvectors_expert ( struct starneig_eigenvectors_conf *  conf,
int  n,
int  selected[],
double  S[],
int  ldS,
double  T[],
int  ldT,
double  Z[],
int  ldZ,
double  X[],
int  ldX 
)

Computes a generalized eigenvector for each selected generalized eigenvalue.

Parameters
[in]confConfiguration structure.
[in]nThe order of $S$ and $Q$ and the number of rows of $X$.
[in]selectedThe selection array specifying the locations of the selected generalized eigenvalues. The number of 1's in the array is the same as the number of columns in $X$.
[in]SThe Schur matrix $S$.
[in]ldSThe leading dimension of $S$.
[in]TThe upper triangular matrix $T$.
[in]ldTThe leading dimension of $T$.
[in]ZThe orthogonal matrix $Z$.
[in]ldZThe leading dimension of $Z$.
[out]XA matrix with $n$ rows and one column for each selected generalized eigenvalue. The columns represent the computed generalized eigenvectors as previously described.
[in]ldXThe leading dimension of $X$.
Returns
STARNEIG_SUCCESS (0) on success. Negative integer -i when i'th argument is invalid. Positive error code otherwise.
See also
starneig_GEP_SM_Select