sygvd
Contents
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(cl::sycl::queue &queue, std::int64_t itype, oneapi::mkl::job jobz, oneapi::mkl::uplo upper_lower, std::int64_t n, cl::sycl::buffer<T,1> &a, std::int64_t lda, cl::sycl::buffer<T,1> &b, std::int64_t ldb, cl::sycl::buffer<T,1> &w, cl::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.
Throws
This routine shall throw the following exceptions if the associated condition is detected. An implementation may throw additional implementation-specific exception(s) in case of error conditions not covered here.
oneapi::mkl::unsupported_device
oneapi::mkl::lapack::invalid_argument
oneapi::mkl::lapack::computation_error
Exception is thrown in case of problems during calculations. The
info
code of the problem can be obtained by info() method of exception object:If \(\text{info}=-i\), the \(i\)-th parameter had an illegal value.
For \(\text{info} \le n\):
If \(\text{info}=i\), and
jobz = oneapi::mkl::job::novec
, then the algorithm failed to converge; \(i\) indicates the number of off-diagonal elements of an intermediate tridiagonal form which did not converge to zero.If \(\text{info}=i\), and
jobz = oneapi::mkl::job::vec
, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns \(\text{info}/(n+1)\) through \(\text{mod}(\text{info},n+1)\).For \(\text{info}>n\):
If \(\text{info}=n+i\), for \(1 \le i \le n\), then the leading minor of order \(i\) of \(B\) is not positive-definite. The factorization of \(B\) could not be completed and no eigenvalues or eigenvectors were computed.
If
info
equals to value passed as scratchpad size, and detail() returns non zero, then passed scratchpad is of insufficient size, and required size should not be less than value return by detail() method of exception object.
sygvd (USM Version)#
Syntax
namespace oneapi::mkl::lapack {
cl::sycl::event sygvd(cl::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<cl::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.
Throws
This routine shall throw the following exceptions if the associated condition is detected. An implementation may throw additional implementation-specific exception(s) in case of error conditions not covered here.
oneapi::mkl::unsupported_device
oneapi::mkl::lapack::invalid_argument
oneapi::mkl::lapack::computation_error
Exception is thrown in case of problems during calculations. The
info
code of the problem can be obtained by info() method of exception object:If \(\text{info}=-i\), the \(i\)-th parameter had an illegal value.
For \(\text{info} \le n\):
If \(\text{info}=i\), and
jobz = oneapi::mkl::job::novec
, then the algorithm failed to converge; \(i\) indicates the number of off-diagonal elements of an intermediate tridiagonal form which did not converge to zero.If \(\text{info}=i\), and
jobz = oneapi::mkl::job::vec
, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns \(\text{info}/(n+1)\) through \(\text{mod}(\text{info},n+1)\).For \(\text{info}>n\):
If \(\text{info}=n+i\), for \(1 \le i \le n\), then the leading minor of order \(i\) of \(B\) is not positive-definite. The factorization of \(B\) could not be completed and no eigenvalues or eigenvectors were computed.
If
info
equals to value passed as scratchpad size, and detail() returns non zero, then passed scratchpad is of insufficient size, and required size should not be less than value return by detail() method of exception object.
Return Values
Output event to wait on to ensure computation is complete
Parent topic: LAPACK Singular Value and Eigenvalue Problem Routines