Developer Reference for Intel® oneAPI Math Kernel Library for C
?geev
Computes the eigenvalues and left and right eigenvectors of a general matrix.
Syntax
lapack_int LAPACKE_sgeev ( intmatrix_layout , charjobvl , charjobvr , lapack_intn , float*a , lapack_intlda , float*wr , float*wi , float*vl , lapack_intldvl , float*vr , lapack_intldvr );
lapack_int LAPACKE_dgeev ( intmatrix_layout , charjobvl , charjobvr , lapack_intn , double*a , lapack_intlda , double*wr , double*wi , double*vl , lapack_intldvl , double*vr , lapack_intldvr );
lapack_int LAPACKE_cgeev ( intmatrix_layout , charjobvl , charjobvr , lapack_intn , lapack_complex_float*a , lapack_intlda , lapack_complex_float*w , lapack_complex_float*vl , lapack_intldvl , lapack_complex_float*vr , lapack_intldvr );
lapack_int LAPACKE_zgeev ( intmatrix_layout , charjobvl , charjobvr , lapack_intn , lapack_complex_double*a , lapack_intlda , lapack_complex_double*w , lapack_complex_double*vl , lapack_intldvl , lapack_complex_double*vr , lapack_intldvr );
Include Files
mkl.h
Description
sgeev dgeev cgeev zgeev geev
The routine computes for an n -by- n real/complex nonsymmetric matrix A , the eigenvalues and, optionally, the left and/or right eigenvectors. The right eigenvector v of A satisfies
A*v = λ*v
where λ is its eigenvalue.
The left eigenvector u of A satisfies
u^{H}*A = λ*u^{H}
where uH denotes the conjugate transpose of u . The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real.
Input Parameters
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
jobvl
Must be ‘N’ or ‘V’ .
If jobvl = 'N' , then left eigenvectors of A are not computed. If jobvl = 'V' , then left eigenvectors of A are computed.
jobvr
Must be ‘N’ or ‘V’ .
If jobvr = 'N' , then right eigenvectors of A are not computed. If jobvr = 'V' , then right eigenvectors of A are computed.
n
The order of the matrix A ( n≥ 0 ).
- a , work
-
REAL for sgeev DOUBLE PRECISION for dgeev COMPLEX for cgeev DOUBLE COMPLEX for zgeev . Arrays:
a (size at least max(1, lda * n )) is an array containing the n -by- n matrix A .
The second dimension of a must be at least max(1, n) . work is a workspace array, its dimension max(1, lwork) .
lda
The leading dimension of the array a . Must be at least max(1, n) .
ldvl , ldvr
The leading dimensions of the output arrays vl and vr , respectively.
Constraints: ldvl≥ 1 ; ldvr≥ 1 . If jobvl = 'V' , ldvl≥ max(1, n) ; If jobvr = 'V' , ldvr≥ max(1, n) .
lwork
The dimension of the array work . 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 .
- rwork
-
REAL for cgeev DOUBLE PRECISION for zgeev Workspace array, size at least max(1, 2n) . Used in complex flavors only.
Output Parameters
- a
-
On exit, this array is overwritten.
- wr , wi
-
REAL for sgeev DOUBLE PRECISION for dgeev Arrays, size at least max (1, n) each. Contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having positive imaginary part first.
- w
-
COMPLEX for cgeev DOUBLE COMPLEX for zgeev . Array, size at least max(1, n) . Contains the computed eigenvalues.
- vl , vr
-
REAL for sgeev DOUBLE PRECISION for dgeev COMPLEX for cgeev DOUBLE COMPLEX for zgeev . Arrays:
vl (size at least max(1, ldvl * n )) .
If jobvl = 'N' , vl is not referenced. For real flavors :
If the j -th eigenvalue is real, the i -th component of the j -th eigenvector uj is stored in vl[(i - 1) + (j - 1)*ldvl] for column major layout and in vl[(i - 1)*ldvl + (j - 1)] for row major layout. .
If the j -th and ( j +1)-st eigenvalues form a complex conjugate pair, then for i = sqrt(-1) , the k -th component of the j -th eigenvector uj is vl [( k - 1) + ( j - 1)* ldvl ] + i* vl [( k - 1) + j * ldvl ] for column major layout and as vl [( k - 1)* ldvl + ( j - 1)] + i* vl [( k -1)* ldvl + j ] for row major layout. Similarly, the k -th component of vector ( j +1) uj + 1 is vl [( k - 1) + ( j - 1)* ldvl ] - i* vl [( k - 1) + j * ldvl ] for column major layout and as vl [( k - 1)* ldvl + ( j - 1)] -i* vl [( k - 1)* ldvl + j ] for row major layout. .
For complex flavors : u_{j} = vl(:,j) , the j -th column of vl . The i -th component of the j -th eigenvector uj is stored in vl[(i - 1) + (j - 1)*ldvl] for column major layout and in vl[(i - 1)*ldvl+(j - 1)] for row major layout.
vr (size at least max(1, ldvr * n )) .
If jobvr = 'N' , vr is not referenced. For real flavors :
If the j -th eigenvalue is real, then the i -th component of j -th eigenvector vj is stored in vr[(i - 1) + (j - 1)*ldvr] for column major layout and in vr[(i - 1)*ldvr + (j - 1)] for row major layout. .
If the j -th and ( j +1)-st eigenvalues form a complex conjugate pair, then for i = sqrt(-1) , the k -th component of the j -th eigenvector vj is vr [( k - 1) + ( j - 1)* ldvr ] +i* vr [( k - 1) + j * ldvr ] for column major layout and as vr [( k - 1)* ldvr + ( j - 1)] + i* vr [( k - 1)* ldvr + j ] for row major layout. Similarly, the k -th component of vector j + 1) vj + 1 is vr [( k - 1) + ( j - 1)* ldvr ] - i* vr [( k - 1) + j * ldvr ] for column major layout and as vr [( k - 1)* ldvr + ( j - 1)] - i* vr [( k - 1)* ldvr + j ] for row major layout .
For complex flavors : v_{j} = vr(:,j) , the j -th column of vr . The i -th component of the j -th eigenvector vj is stored in vr[(i - 1) + (j - 1)*ldvr] for column major layout and in vr[(i - 1)*ldvr + (j - 1)] for row major layout.
- work(1)
-
On exit, if info = 0 , then work(1) returns the required minimal size of lwork .
Return Values
This function returns a value info .
If info=0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
If info = i , the QR algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements i +1: n of wr and wi (for real flavors) or w (for complex flavors) contain those eigenvalues which have converged.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
Application Notes
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 exits immediately with an error and does not provide any information on the recommended workspace.