Developer Reference for Intel® oneAPI Math Kernel Library for C
?hpevd
Uses divide and conquer algorithm to compute all eigenvalues and, optionally, all eigenvectors of a complex Hermitian matrix held in packed storage.
Syntax
lapack_int LAPACKE_chpevd ( intmatrix_layout , charjobz , charuplo , lapack_intn , lapack_complex_float*ap , float*w , lapack_complex_float*z , lapack_intldz );
lapack_int LAPACKE_zhpevd ( intmatrix_layout , charjobz , charuplo , lapack_intn , lapack_complex_double*ap , double*w , lapack_complex_double*z , lapack_intldz );
Include Files
mkl.h
Description
chpevd zhpevd hpevd
The routine computes all the eigenvalues, and optionally all the eigenvectors, of a complex Hermitian matrix A (held in packed storage). In other words, it can compute the spectral factorization of A as: A = Z*Λ*Z^{H} .
Here Λ is a real diagonal matrix whose diagonal elements are the eigenvalues λ i , and Z is the (complex) unitary matrix whose columns are the eigenvectors zi . Thus,
A*z_{i} = λ_{i}*z_{i} for i = 1, 2, ..., n .
If the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan variant of the QL or QR algorithm.
Input Parameters
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
jobz
Must be ‘N’ or ‘V’ .
If jobz = 'N' , then only eigenvalues are computed. If jobz = 'V' , then eigenvalues and eigenvectors are computed.
uplo
Must be ‘U’ or ‘L’ .
If uplo = 'U' , ap stores the packed upper triangular part of A . If uplo = 'L' , ap stores the packed lower triangular part of A .
n
The order of the matrix A ( n≥ 0 ).
- ap , work
-
COMPLEX for chpevd DOUBLE COMPLEX for zhpevd Arrays: ap(*) contains the packed upper or lower triangle of Hermitian matrix A , as specified by uplo . The dimension of ap must be at least max(1, n *( n +1)/2) .work is a workspace array, its dimension max(1, lwork) .
ldz
The leading dimension of the output array z .
Constraints: if jobz = 'N' , then ldz≥ 1 ; if jobz = 'V' , then ldz≥ max(1, n) .
lwork
The dimension of the array work . Constraints: if n≤ 1 , then lwork≥ 1 ; if jobz = 'N' and n > 1 , then lwork≥n ; if jobz = 'V' and n > 1 , then lwork≥ 2*n . If lwork = -1 , then a workspace query is assumed; the routine only calculates the optimal size of the work , rwork and iwork arrays, returns these values as the first entries of the work , rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla . See Application Notes for details.
- rwork
-
REAL for chpevd DOUBLE PRECISION for zhpevd Workspace array, its dimension max(1, lrwork) .
lrwork
The dimension of the array rwork . Constraints: if n≤ 1 , then lrwork≥ 1 ; if jobz = 'N' and n > 1 , then lrwork≥n ; if jobz = 'V' and n > 1 , then lrwork≥ 2*n^{2} + 5*n + 1 . If lrwork = -1 , then a workspace query is assumed; the routine only calculates the optimal size of the work , rwork and iwork arrays, returns these values as the first entries of the work , rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla . See Application Notes for details.
iwork
Workspace array, its dimension max(1, liwork) .
liwork
The dimension of the array iwork . Constraints: if n ≤ 1 , then liwork ≥ 1 ; if jobz = 'N' and n > 1 , then liwork ≥ 1 ; if jobz = 'V' and n > 1 , then liwork ≥ 5*n+3 . If liwork = -1 , then a workspace query is assumed; the routine only calculates the optimal size of the work , rwork and iwork arrays, returns these values as the first entries of the work , rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla . See Application Notes for details.
Output Parameters
- w
-
REAL for chpevd DOUBLE PRECISION for zhpevd Array, size at least max(1, n ). If info = 0 , contains the eigenvalues of the matrix A in ascending order. See also info .
- z
-
COMPLEX for chpevd DOUBLE COMPLEX for zhpevd
Array, size 1 if jobz = 'N' and max(1, ldz * n ) if jobz = 'V' .
The second dimension of z must be: at least 1 if jobz = 'N' ; at least max(1, n ) if jobz = 'V' . If jobz = 'V' , then this array is overwritten by the unitary matrix Z which contains the eigenvectors of A . If jobz = 'N' , then z is not referenced.
- ap
-
On exit, this array is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of A.
- work(1)
-
On exit, if info = 0 , then work(1) returns the required minimal size of lwork .
- rwork(1)
-
On exit, if info = 0 , then rwork(1) returns the required minimal size of lrwork .
- iwork(1)
-
On exit, if info = 0 , then iwork(1) returns the required minimal size of liwork .
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 , then the algorithm failed to converge; i indicates the number of elements of an intermediate tridiagonal form which did not converge to zero.
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
The computed eigenvalues and eigenvectors are exact for a matrix \(A + E\) such that \(||E||_{2} = O(\varepsilon) ||A||_{2}\) , where \(\varepsilon\) is the machine precision.
The real analogue of this routine is spevd .
See also heevd for matrices held in full storage, and hbevd for banded matrices.