sytrd#
Reduces a real symmetric matrix to tridiagonal form.
Description
sytrd
supports the following precisions.
T
float
double
The routine reduces a real symmetric matrix \(A\) to symmetric tridiagonal form \(T\) by an orthogonal similarity transformation: \(A = QTQ^T\). The orthogonal matrix \(Q\) is not formed explicitly but is represented as a product of \(n-1\) elementary reflectors. Routines are provided for working with \(Q\) in this representation .
sytrd (Buffer Version)#
Syntax
namespace oneapi::mkl::lapack {
void sytrd(sycl::queue &queue, oneapi::mkl::uplo upper_lower, std::int64_t n, sycl::buffer<T,1> &a, std::int64_t lda, sycl::buffer<T,1> &d, sycl::buffer<T,1> &e, sycl::buffer<T,1> &tau, sycl::buffer<T,1> &scratchpad, std::int64_t scratchpad_size)
}
Input Parameters
- queue
The queue where the routine should be executed.
- upper_lower
Must be
uplo::upper
oruplo::lower
.If
upper_lower = uplo::upper
,a
stores the upper triangular part of \(A\).If
upper_lower = uplo::lower
,a
stores the lower triangular part of \(A\).- n
The order of the matrices \(A\) \((0 \le n)\).
- a
The buffer
a
, size(lda,*)
. Contains the upper or lower triangle of the symmetric matrix \(A\), as specified byupper_lower
.The second dimension of
a
must be at least \(\max(1,n)\).- lda
The leading dimension of
a
; 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 sytrd_scratchpad_size function.
Output Parameters
- a
On exit,
if
upper_lower = uplo::upper
, the diagonal and first superdiagonal of \(A\) are overwritten by the corresponding elements of the tridiagonal matrix \(T\), and the elements above the first superdiagonal, with the buffertau
, represent the orthogonal matrix \(Q\) as a product of elementary reflectors;if
upper_lower = uplo::lower
, the diagonal and first subdiagonal of \(A\) are overwritten by the corresponding elements of the tridiagonal matrix \(T\), and the elements below the first subdiagonal, with the buffertau
, represent the orthogonal matrix \(Q\) as a product of elementary reflectors.- d
Buffer containing the diagonal elements of the matrix \(T\). The dimension of
d
must be at least \(\max(1, n)\).- e
Buffer containing the off diagonal elements of the matrix \(T\). The dimension of
e
must be at least \(\max(1, n-1)\).- tau
Buffer, size at least \(\max(1, n)\). Stores \((n-1)\) scalars that define elementary reflectors in decomposition of the unitary matrix \(Q\) in a product of \(n-1\) elementary reflectors. \(\tau(n)\) is used as workspace.
- scratchpad
Buffer holding scratchpad memory to be used by routine for storing intermediate results.
sytrd (USM Version)#
Syntax
namespace oneapi::mkl::lapack {
sycl::event sytrd(sycl::queue &queue, oneapi::mkl::uplo upper_lower, std::int64_t n, T *a, std::int64_t lda, T *d, T *e, T *tau, T *scratchpad, std::int64_t scratchpad_size, const std::vector<sycl::event> &events = {})
}
Input Parameters
- queue
The queue where the routine should be executed.
- upper_lower
Must be
uplo::upper
oruplo::lower
.If
upper_lower = uplo::upper
,a
stores the upper triangular part of \(A\).If
upper_lower = uplo::lower
,a
stores the lower triangular part of \(A\).- n
The order of the matrices \(A\) \((0 \le n)\).
- a
The pointer to matrix \(A\), size
(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 \(\max(1,n)\).- 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 sytrd_scratchpad_size function.- events
List of events to wait for before starting computation. Defaults to empty list.
Output Parameters
- a
On exit,
if
upper_lower = uplo::upper
, the diagonal and first superdiagonal of \(A\) are overwritten by the corresponding elements of the tridiagonal matrix \(T\), and the elements above the first superdiagonal, with the arraytau
, represent the orthogonal matrix \(Q\) as a product of elementary reflectors;if
upper_lower = uplo::lower
, the diagonal and first subdiagonal of \(A\) are overwritten by the corresponding elements of the tridiagonal matrix \(T\), and the elements below the first subdiagonal, with the arraytau
, represent the orthogonal matrix \(Q\) as a product of elementary reflectors.- d
Pointer to diagonal elements of the matrix \(T\). The dimension of
d
must be at least \(\max(1, n)\).- e
Pointer to off diagonal elements of the matrix \(T\). The dimension of
e
must be at least \(\max(1, n-1)\).- tau
Pointer to array of size at least \(\max(1, n)\). Stores \((n-1)\) scalars that define elementary reflectors in decomposition of the unitary matrix \(Q\) in a product of \(n-1\) elementary reflectors. \(\tau(n)\) is used as workspace.
- 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