syevd

Computes all eigenvalues and, optionally, all eigenvectors of a real symmetric matrix using divide and conquer algorithm.

Description

syevd supports the following precisions.

T

float

double

The routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric matrix \(A\). In other words, it can compute the spectral factorization of \(A\) as: \(A = Z\lambda Z^T\).

Here \(\Lambda\) is a diagonal matrix whose diagonal elements are the eigenvalues \(\lambda_i\), and \(Z\) is the orthogonal matrix whose columns are the eigenvectors \(z_{i}\). Thus,

\(A z_i = \lambda_i z_i\) for \(i = 1, 2, ..., n\).

If the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan variant of the QL or QR algorithm.

syevd (Buffer Version)

Syntax

namespace oneapi::mkl::lapack {
  void syevd(cl::sycl::queue &queue, jobz 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> &w, cl::sycl::buffer<T,1> &scratchpad, std::int64_t scratchpad_size)
}

Input Parameters

queue

The queue where the routine should be executed.

jobz

Must be job::novec or job::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 or uplo::lower.

If upper_lower = job::upper, a stores the upper triangular part of \(A\).

If upper_lower = job::lower, a stores the lower triangular part of \(A\).

n

The order of the matrix \(A\) (\(0 \le n\)).

a

The buffer a, size (lda,*). The buffer a contains the matrix \(A\). The second dimension of a must be at least \(\max(1, n)\).

lda

The leading dimension of a. Must be 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 syevd_scratchpad_size function.

Output Parameters

a

If jobz = job::vec, then on exit this buffer is overwritten by the orthogonal matrix \(Z\) which contains the eigenvectors of \(A\).

w

Buffer, size at least \(n\). Contains the eigenvalues of the matrix \(A\) in ascending order.

scratchpad

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

syevd (USM Version)

Syntax

namespace oneapi::mkl::lapack {
  cl::sycl::event syevd(cl::sycl::queue &queue, jobz jobz, oneapi::mkl::uplo upper_lower, std::int64_t n, T *a, std::int64_t lda, T *w, T *scratchpad, std::int64_t scratchpad_size, const cl::sycl::vector_class<cl::sycl::event> &events = {})
}

Input Parameters

queue

The queue where the routine should be executed.

jobz

Must be job::novec or job::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 or uplo::lower.

If upper_lower = job::upper, a stores the upper triangular part of \(A\).

If upper_lower = job::lower, a stores the lower triangular part of \(A\).

n

The order of the matrix \(A\) (\(0 \le n\)).

a

Pointer to array containing \(A\), size (lda,*). The second dimension of a must be at least \(\max(1, n)\).

lda

The leading dimension of a. Must be 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 syevd_scratchpad_size function.

events

List of events to wait for before starting computation. Defaults to empty list.

Output Parameters

a

If jobz = job::vec, then on exit this array is overwritten by the orthogonal matrix \(Z\) which contains the eigenvectors of \(A\).

w

Pointer to array of size at least \(n\). 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