sygvd#
Computes all eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem using a divide and conquer method.
Description
sygvd
supports the following precisions.
T
float
double
The routine computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
\(Ax = \lambda Bx\), \(ABx = \lambda x\), or \(BAx = \lambda x\) .
Here \(A\) and \(B\) are assumed to be symmetric and \(B\) is also positive definite.
It uses a divide and conquer algorithm.
sygvd (Buffer Version)#
Syntax
namespace oneapi::mkl::lapack {
void sygvd(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<T,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 = job::upper
,a
andb
store the upper triangular part of \(A\) and \(B\).If
upper_lower = job::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 symmetric matrix \(A\), as specified byupper_lower
. The second dimension ofa
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 symmetric matrix \(B\), as specified byupper_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 sygvd_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 \(2\) , \(Z^{T}BZ = I\);
if \(\text{itype} = 3\) , \(Z^{T}B^{-1}Z = I\);
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 \(B = U^{T}U\) or \(B = LL^{T}\).- 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.
sygvd (USM Version)#
Syntax
namespace oneapi::mkl::lapack {
sycl::event sygvd(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, T *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 = job::upper
,a
andb
store the upper triangular part of \(A\) and \(B\).If
upper_lower = job::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 symmetric matrix \(A\), as specified byupper_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,*)
contains the upper or lower triangle of the symmetric matrix \(B\), as specified byupper_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 sygvd_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 \(2\), \(Z^{T}BZ = I\);
if \(\text{itype} = 3\), \(Z^{T}B^{-1}Z = I\);
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 \(B\) = \(U^{T}U\) or \(B = LL^{T}\).- 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