hetrd

Reduces a complex Hermitian matrix to tridiagonal form.

Description

hetrd supports the following precisions.

Routine name	T
chetrd	std::complex<float>
zhetrd	std::complex<double>

The routine reduces a complex Hermitian matrix \(A\) to symmetric tridiagonal form \(T\) by a unitary similarity transformation: \(A = QTQ^H\). The unitary matrix \(Q\) is not formed explicitly but is represented as a product of \(n-1\) elementary reflectors. Routines are provided to work with \(Q\) in this representation.

hetrd (Buffer Version)

Syntax

namespace oneapi::mkl::lapack {
  void hetrd(sycl::queue &queue, oneapi::mkl::uplo upper_lower, std::int64_t n, sycl::buffer<T,1> &a, std::int64_t lda, sycl::buffer<realT,1> &d, sycl::buffer<realT,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 or uplo::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

Buffer, size (lda,*). The buffer a contains either 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 \(\max(1, n)\)

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 hetrd_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 buffer tau, 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 buffer tau, 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 \(\max(1, n-1)\). Stores \((n-1)\) scalars that define elementary reflectors in decomposition of the unitary matrix \(Q\) in a product of \(n-1\) elementary reflectors.

scratchpad

Buffer holding scratchpad memory to be used by routine for storing intermediate results.

hetrd (USM Version)

Syntax

namespace oneapi::mkl::lapack {
  sycl::event hetrd(sycl::queue &queue, oneapi::mkl::uplo upper_lower, std::int64_t n, T *a, std::int64_t lda, RealT *d, RealT *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 or uplo::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 either 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 \(\max(1, n)\)

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 hetrd_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 array tau, 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 array tau, 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 \(\max(1, n-1)\). Stores \((n-1)\) scalars that define elementary reflectors in decomposition of the unitary matrix \(Q\) in a product of \(n-1\) elementary reflectors.

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