Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?hpevd
Uses divide and conquer algorithm to compute all eigenvalues and, optionally, all eigenvectors of a complex Hermitian matrix held in packed storage.
Syntax
call chpevd ( job z, uplo , n , ap , w , z , ldz , work , lwork , rwork , lrwork , iwork , liwork , info )
call zhpevd ( jobz , uplo , n , ap , w , z , ldz , work , lwork , rwork , lrwork , iwork , liwork , info )
call hpevd ( ap , w [ , uplo ] [ , z ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
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
jobz
CHARACTER*1 . Must be ‘N’ or ‘V’ .
If jobz = 'N' , then only eigenvalues are computed. If jobz = 'V' , then eigenvalues and eigenvectors are computed.
uplo
CHARACTER*1 . 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
INTEGER . 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
INTEGER . The leading dimension of the output array z .
Constraints: if jobz = 'N' , then ldz≥ 1 ; if jobz = 'V' , then ldz≥ max(1, n) .
lwork
INTEGER .
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
INTEGER .
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
INTEGER . Workspace array, its dimension max(1, liwork) .
liwork
INTEGER .
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 ( ldz ,*) .
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 .
info
INTEGER .
If info = 0 , the execution is successful.
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.
If info = -i , the i- th parameter had an illegal value.
Return Values
No return value, info is an Output Parameter.
LAPACK 95 Interface Notes
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 LAPACK 95 Interface Conventions .
Specific details for the routine hpevd interface are the following:
ap
Holds the array A of size ( n*(n+1)/2 ).
w
Holds the vector with the number of elements n .
z
Holds the matrix Z of size ( n , n ).
uplo
Must be ‘U’ or ‘L’ . The default value is ‘U’ .
jobz
Restored based on the presence of the argument z as follows:
jobz = 'V' , if z is present,
jobz = 'N' , if z is omitted.
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.
If you are in doubt how much workspace to supply, use a generous value of lwork ( liwork or lrwork ) for the first run or set lwork = -1 ( liwork = -1 , lrwork = -1 ).
If you choose the first option and set any of admissible lwork ( liwork or lrwork ) 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 , iwork , rwork ) on exit. Use this value ( work(1) , iwork(1) , rwork(1) ) for subsequent runs.
If you set lwork = -1 ( liwork = -1 , lrwork = -1 ), the routine returns immediately and provides the recommended workspace in the first element of the corresponding array ( work , iwork , rwork ). This operation is called a workspace query.
Note that if you set lwork ( liwork , lrwork ) 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 real analogue of this routine is spevd .
See also heevd for matrices held in full storage, and hbevd for banded matrices.