hegvd#
Computes all eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem using a divide and conquer method.
Description
hegvd
supports the following precisions.
T
std::complex<float>
std::complex<double>
The routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian positive-definite eigenproblem, of the form
\(Ax = \lambda Bx, ABx = \lambda x\), or \(BAx =\lambda x\).
Here \(A\) and \(B\) are assumed to be Hermitian and \(B\) is also positive definite.
It uses a divide and conquer algorithm.
hegvd (Buffer Version)#
Syntax
namespace oneapi::mkl::lapack {
void hegvd(sycl::queue &queue, std::int64_t itype, oneapi::mkl::job jobz, oneapi::mkl::uplo upper_lower, std::int64_t n, sycl::buffer<T,1> &a, std::int64_t lda, sycl::buffer<T,1> &b, std::int64_t ldb, sycl::buffer<realT,1> &w, sycl::buffer<T,1> &scratchpad, std::int64_t scratchpad_size)
}
Input Parameters
- queue
The queue where the routine should be executed.
- itype
Must be 1 or 2 or 3. Specifies the problem type to be solved:
if \(\text{itype} = 1\), the problem type is \(Ax = \lambda Bx;\)
if \(\text{itype} = 2\), the problem type is \(ABx = \lambda x;\)
if \(\text{itype} = 3\), the problem type is \(BAx = \lambda x\).
- jobz
Must be
job::novec
orjob::vec
.If
jobz = job::novec
, then only eigenvalues are computed.If
jobz = job::vec
, then eigenvalues and eigenvectors are computed.- upper_lower
Must be
uplo::upper
oruplo::lower
.If
upper_lower = uplo::upper
,a
andb
store the upper triangular part of \(A\) and \(B\).If
upper_lower = uplo::lower
,a
andb
stores the lower triangular part of \(A\) and \(B\).- n
The order of the matrices \(A\) and \(B\) (\(0 \le n\)).
- a
Buffer, size
a(lda,*)
contains the upper or lower triangle of the Hermitian matrix \(A\), as specified by upper_lower.The second dimension of
a
must be at least \(\max(1, n)\).- lda
The leading dimension of
a
; at least \(\max(1,n)\).- b
Buffer, size
b(ldb,*)
contains the upper or lower triangle of the Hermitian matrix \(B\), as specified by upper_lower.The second dimension of
b
must be at least \(\max(1, n)\).- ldb
The leading dimension of
b
; at least \(\max(1,n)\).- scratchpad_size
Size of scratchpad memory as a number of floating point elements of type
T
. Size should not be less than the value returned by hegvd_scratchpad_size function.
Output Parameters
- a
On exit, if
jobz = job::vec
, then if \(\text{info} = 0\),a
contains the matrix \(Z\) of eigenvectors. The eigenvectors are normalized as follows:if \(\text{itype} = 1\) or \(\text{itype} = 2\), \(Z^{H}BZ = I\);
if \(\text{itype} = 3\), \(Z^{H}B^{-1}Z = I\);
If
jobz = job::novec
, then on exit the upper triangle (ifupper_lower = uplo::upper
) or the lower triangle (ifupper_lower = uplo::lower
) of \(A\), including the diagonal, is destroyed.- b
On exit, if \(\text{info} \le n\), the part of
b
containing the matrix is overwritten by the triangular factor \(U\) or \(L\) from the Cholesky factorization \(B = U^{H}U\)or \(B = LL^{H}\).- w
Buffer, size at least \(n\). If \(\text{info} = 0\), contains the eigenvalues of the matrix \(A\) in ascending order.
- scratchpad
Buffer holding scratchpad memory to be used by routine for storing intermediate results.
hegvd (USM Version)#
Syntax
namespace oneapi::mkl::lapack {
sycl::event hegvd(sycl::queue &queue, std::int64_t itype, oneapi::mkl::job jobz, oneapi::mkl::uplo upper_lower, std::int64_t n, T *a, std::int64_t lda, T *b, std::int64_t ldb, RealT *w, T *scratchpad, std::int64_t scratchpad_size, const std::vector<sycl::event> &events = {})
}
Input Parameters
- queue
The queue where the routine should be executed.
- itype
Must be 1 or 2 or 3. Specifies the problem type to be solved:
if \(\text{itype} = 1\), the problem type is \(Ax = \lambda Bx;\)
if \(\text{itype} = 2\), the problem type is \(ABx = \lambda x;\)
if \(\text{itype} = 3\), the problem type is \(BAx = \lambda x\).
- jobz
Must be
job::novec
orjob::vec
.If
jobz = job::novec
, then only eigenvalues are computed.If
jobz = job::vec
, then eigenvalues and eigenvectors are computed.- upper_lower
Must be
uplo::upper
oruplo::lower
.If
upper_lower = uplo::upper
,a
andb
store the upper triangular part of \(A\) and \(B\).If
upper_lower = uplo::lower
,a
andb
stores the lower triangular part of \(A\) and \(B\).- n
The order of the matrices \(A\) and \(B\) (\(0 \le n\)).
- a
Pointer to array of size
a(lda,*)
containing the upper or lower triangle of the Hermitian matrix \(A\), as specified by upper_lower. The second dimension ofa
must be at least \(\max(1, n)\).- lda
The leading dimension of
a
; at least \(\max(1,n)\).- b
Pointer to array of size
b(ldb,*)
containing the upper or lower triangle of the Hermitian matrix \(B\), as specified by upper_lower. The second dimension ofb
must be at least \(\max(1, n)\).- ldb
The leading dimension of
b
; at least \(\max(1,n)\).- scratchpad_size
Size of scratchpad memory as a number of floating point elements of type
T
. Size should not be less than the value returned by hegvd_scratchpad_size function.- events
List of events to wait for before starting computation. Defaults to empty list.
Output Parameters
- a
On exit, if
jobz = job::vec
, then if \(\text{info} = 0\),a
contains the matrix \(Z\) of eigenvectors. The eigenvectors are normalized as follows:if \(\text{itype} = 1`\) or \(\text{itype} = 2\), \(Z^{H}BZ = I\);
if \(\text{itype} = 3\), \(Z^{H} B^{-1} Z = I\);
If
jobz = job::novec
, then on exit the upper triangle (ifupper_lower = uplo::upper
) or the lower triangle (ifupper_lower = uplo::lower
) of \(A\), including the diagonal, is destroyed.- b
On exit, if \(\text{info} \le n\), the part of
b
containing the matrix is overwritten by the triangular factor \(U\) or \(L\) from the Cholesky factorization \(B = U^{H}U\)or \(B\) = \(LL^{H}\).- w
Pointer to array of size at least n. If \(\text{info} = 0\), contains the eigenvalues of the matrix \(A\) in ascending order.
- scratchpad
Pointer to scratchpad memory to be used by routine for storing intermediate results.
Return Values
Output event to wait on to ensure computation is complete.
Parent topic: LAPACK Singular Value and Eigenvalue Problem Routines