Computes the generalized eigenvalues, and the left and/or right generalized eigenvectors for a pair of nonsymmetric matrices.
FORTRAN 77:
call sggev(jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)
call dggev(jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)
call cggev(jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)
call zggev(jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)
Fortran 95:
call ggev(a, b, alphar, alphai, beta [,vl] [,vr] [,info])
call ggev(a, b, alpha, beta [, vl] [,vr] [,info])
C:
lapack_int LAPACKE_sggev( int matrix_order, char jobvl, char jobvr, lapack_int n, float* a, lapack_int lda, float* b, lapack_int ldb, float* alphar, float* alphai, float* beta, float* vl, lapack_int ldvl, float* vr, lapack_int ldvr );
lapack_int LAPACKE_dggev( int matrix_order, char jobvl, char jobvr, lapack_int n, double* a, lapack_int lda, double* b, lapack_int ldb, double* alphar, double* alphai, double* beta, double* vl, lapack_int ldvl, double* vr, lapack_int ldvr );
lapack_int LAPACKE_cggev( int matrix_order, char jobvl, char jobvr, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb, lapack_complex_float* alpha, lapack_complex_float* beta, lapack_complex_float* vl, lapack_int ldvl, lapack_complex_float* vr, lapack_int ldvr );
lapack_int LAPACKE_zggev( int matrix_order, char jobvl, char jobvr, lapack_int n, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb, lapack_complex_double* alpha, lapack_complex_double* beta, lapack_complex_double* vl, lapack_int ldvl, lapack_complex_double* vr, lapack_int ldvr );
The FORTRAN 77 interfaces are specified in the mkl_lapack.fi and mkl_lapack.h include files, the Fortran 95 interfaces are specified in the lapack.f90 include file, and the C interfaces are specified in the mkl_lapacke.h include file.
The ?ggev routine computes the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors for a pair of n-by-n real/complex nonsymmetric matrices (A,B).
A generalized eigenvalue for a pair of matrices (A,B) is a scalar λ or a ratio alpha / beta = λ, such that A - λ*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.
The right generalized eigenvector v(j) corresponding to the generalized eigenvalue λ(j) of (A,B) satisfies
A*v(j) = λ(j)*B*v(j).
The left generalized eigenvector u(j) corresponding to the generalized eigenvalue λ(j) of (A,B) satisfies
u(j)H*A = λ(j)*u(j)H*B
where u(j)H denotes the conjugate transpose of u(j).
The ?ggev routine replaces the deprecated ?gegv routine.
The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.
CHARACTER*1. Must be 'N' or 'V'.
If jobvl = 'N', the left generalized eigenvectors are not computed;
If jobvl = 'V', the left generalized eigenvectors are computed.
CHARACTER*1. Must be 'N' or 'V'.
If jobvr = 'N', the right generalized eigenvectors are not computed;
If jobvr = 'V', the right generalized eigenvectors are computed.
INTEGER. The order of the matrices A, B, vl, and vr (n ≥ 0).
REAL for sggev
DOUBLE PRECISION for dggev
COMPLEX for cggev
DOUBLE COMPLEX for zggev.
Arrays:
a(lda,*) is an array containing the n-by-n matrix A (first of the pair of matrices).
The second dimension of a must be at least max(1, n).
b(ldb,*) is an array containing the n-by-n matrix B (second of the pair of matrices).
The second dimension of b must be at least max(1, n).
work is a workspace array, its dimension max(1, lwork).
INTEGER. The leading dimension of the array a. Must be at least max(1, n).
INTEGER. The leading dimension of the array b. Must be at least max(1, n).
INTEGER. The leading dimensions of the output matrices vl and vr, respectively.
Constraints:
ldvl ≥ 1. If jobvl = 'V', ldvl ≥ max(1, n).
ldvr ≥ 1. If jobvr = 'V', ldvr ≥ max(1, n).
INTEGER.
The dimension of the array work.
lwork ≥ max(1, 8n+16) for real flavors;
lwork ≥ max(1, 2n) for complex flavors.
For good performance, lwork must generally be larger.
If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.
REAL for cggev
DOUBLE PRECISION for zggev
Workspace array, DIMENSION at least max(1, 8n).
This array is used in complex flavors only.
On exit, these arrays have been overwritten.
REAL for sggev;
DOUBLE PRECISION for dggev.
Arrays, DIMENSION at least max(1, n) each. Contain values that form generalized eigenvalues in real flavors.
See beta.
COMPLEX for cggev;
DOUBLE COMPLEX for zggev.
Array, DIMENSION at least max(1, n). Contain values that form generalized eigenvalues in complex flavors. See beta.
REAL for sggev
DOUBLE PRECISION for dggev
COMPLEX for cggev
DOUBLE COMPLEX for zggev.
Array, DIMENSION at least max(1, n).
For real flavors:
On exit, (alphar(j)+ alphai(j)*i)/beta(j), j=1,..., n, are the generalized eigenvalues.
If alphai(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with alphai(j+1) negative.
For complex flavors:
On exit, alpha(j)/beta(j), j=1,..., n, are the generalized eigenvalues.
See also Application Notes below.
REAL for sggev
DOUBLE PRECISION for dggev
COMPLEX for cggev
DOUBLE COMPLEX for zggev.
Arrays:
vl(ldvl,*); the second dimension of vl must be at least max(1, n).
If jobvl = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of vl, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(Re) + abs(Im) = 1.
If jobvl = 'N', vl is not referenced.
For real flavors:
If the j-th eigenvalue is real, then u(j) = vl(:,j), the j-th column of vl.
If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then u(j) = vl(:,j) + i*vl(:,j+1) and u(j+1) = vl(:,j) - i*vl(:,j+1), where i = sqrt(-1).
For complex flavors:
u(j) = vl(:,j), the j-th column of vl.
vr(ldvr,*); the second dimension of vr must be at least max(1, n).
If jobvr = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of vr, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(Re) + abs(Im) = 1.
If jobvr = 'N', vr is not referenced.
For real flavors:
If the j-th eigenvalue is real, then v(j) = vr(:,j), the j-th column of vr.
If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then v(j) = vr(:,j) + i*vr(:,j+1) and v(j+1) = vr(:,j) - i*vr(:,j+1).
For complex flavors:
v(j) = vr(:,j), the j-th column of vr.
On exit, if info = 0, then work(1) returns the required minimal size of lwork.
INTEGER.
If info = 0, the execution is successful.
If info = -i, the i-th parameter had an illegal value.
If info = i, and
i ≤ n: the QZ iteration failed. No eigenvectors have been calculated, but alphar(j), alphai(j) (for real flavors), or alpha(j) (for complex flavors), and beta(j), j=info+1,..., n should be correct.
i > n: errors that usually indicate LAPACK problems:
i = n+1: other than QZ iteration failed in ?hgeqz;
i = n+2: error return from ?tgevc.
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.
Specific details for the routine ggev interface are the following:
Holds the matrix A of size (n, n).
Holds the matrix B of size (n, n).
Holds the vector of length n. Used in real flavors only.
Holds the vector of length n. Used in real flavors only.
Holds the vector of length n. Used in complex flavors only.
Holds the vector of length n.
Holds the matrix VL of size (n, n).
Holds the matrix VR of size (n, n).
Restored based on the presence of the argument vl as follows:
jobvl = 'V', if vl is present,
jobvl = 'N', if vl is omitted.
Restored based on the presence of the argument vr as follows:
jobvr = 'V', if vr is present,
jobvr = 'N', if vr is omitted.
If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1.
If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (work(1)) for subsequent runs.
If you set lwork = -1, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.
Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.
The quotients alphar(j)/beta(j) and alphai(j)/beta(j) may easily over- or underflow, and beta(j) may even be zero. Thus, you should avoid simply computing the ratio. However, alphar and alphai (for real flavors) or alpha (for complex flavors) will be always less than and usually comparable with norm(A) in magnitude, and beta always less than and usually comparable with norm(B).
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