heevd#

Computes all eigenvalues and, optionally, all eigenvectors of a complex Hermitian matrix using divide and conquer algorithm.

Description

heevd supports the following precisions.

T

std::complex<float>

std::complex<double>

T
`std::complex<float>`
`std::complex<double>`

The routine computes all the eigenvalues, and optionally all the eigenvectors, of a complex Hermitian matrix \(A\). In other words, it can compute the spectral factorization of \(A\) as: \(A = Z\Lambda Z^H\).

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

\(Az_i = \lambda_i z_i\) 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.

heevd (Buffer Version)#

Syntax

namespace oneapi::mkl::lapack {
  void heevd(sycl::queue &queue, oneapi::mkl::job jobz, oneapi::mkl::uplo upper_lower, std::int64_t n, butter<T,1> &a, std::int64_t lda, sycl::buffer<realT,1> &w, 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 heevd_scratchpad_size function.

Output Parameters

a: If jobz = job::vec, then on exit this buffer is overwritten by the unitary 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.

heevd (USM Version)#