hegvd (USM Version)¶

Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian positive-definite eigenproblem using a divide and conquer method. This routine belongs to the oneapi::mkl::lapack namespace.

Description ¶

The routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian positive-definite eigenproblem, of the form

A*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.

Here A and B are assumed to be Hermitian and B is also positive definite.

It uses a divide and conquer algorithm.

API ¶

Syntax¶

namespace oneapi::mkl::lapack {
  cl::sycl::event hegvd(cl::sycl::queue &queue,
  std::int64_t itype,
  mkl::job jobz,
  mkl::uplo uplo,
  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 = {})
}

hegvd (USM version) supports the following precision and devices.

T	Devices Supported
`std::complex<float>`	Host, CPU, GPU
`std::complex<double>`	Host, CPU, GPU

Input Parameters¶

queue

Device queue where calculations will be performed.

itype

Must be 1 or 2 or 3. Specifies the problem type to be solved:

if itype= 1, the problem type is A*x = lambda*B*x;

if itype= 2, the problem type is A*B*x = lambda*x;

if itype= 3, the problem type is B*A*x = lambda*x.

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.

uplo

Must be uplo::upper or uplo::lower.

If uplo = uplo::upper, a and b store the upper triangular part of A and B.

If uplo = uplo::lower, a and b stores the lower triangular part of A and B.

n

The order of the matrices A and B (0≤n).

a

Pointer to the array of size a(lda,*) containing the upper or lower triangle of the Hermitian matrix A, as specified by uplo.

The second dimension of a must be at least max(1, n).

lda

The leading dimension of a; at least max(1,n).

b

Pointer to the array of size b(ldb,*) containing the upper or lower triangle of the Hermitian matrix B, as specified by uplo.

The second dimension of b must be at least max(1, n).

ldb

The leading dimension of b; at least max(1,n).

scratchpad

Pointer to scratchpad memory to be used by the routine for storing intermediate results.

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 the hegvd_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 info = 0, a contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:

if itype= 1 or 2, Z^H*B*Z = I;

if itype= 3, Z^H*inv(B)*Z = I;

If jobz = job::novec, then on exit the upper triangle (if uplo = uplo::upper) or the lower triangle (if uplo = uplo::lower) of A, including the diagonal, is destroyed.

b

On exit, if info≤n, the part of b containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U^H*Uor B = L*L^H.

w

Pointer to the array of size at least n. If info = 0, contains the eigenvalues of the matrix A in ascending order. See also info.

Exceptions¶

Exception	Description
`mkl::lapack::exception`	This exception is thrown when problems occur during calculations. You can obtain the info code of the problem using the info() method of the exception object: If `info = -i`, the `i`-th parameter had an illegal value. For `info ≤ n`: If `info = i`, and `jobz = 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 `info = i`, and `jobz = job:vec`, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns `info/(n+1)` through `mod(info,n+1)`. For `info > n`: If `info = n + i`, for `1 ≤ i ≤ 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` is equal to the value passed as scratchpad size, and detail() returns non zero, then the passed scratchpad has an insufficient size, and the required size should not be less than the value returned by the detail() method of the exception object.

Exception

Description

mkl::lapack::exception

This exception is thrown when problems occur during calculations. You can obtain the info code of the problem using the info() method of the exception object:

If info = -i, the i-th parameter had an illegal value. For info ≤ n:

If info = i, and jobz = 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 info = i, and jobz = job:vec, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns info/(n+1) through mod(info,n+1). For info > n:

If info = n + i, for 1 ≤ i ≤ 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 is equal to the value passed as scratchpad size, and detail() returns non zero, then the passed scratchpad has an insufficient size, and the required size should not be less than the value returned by the detail() method of the exception object.

Return Values¶

Output event to wait on to ensure computation is complete.

hegvd hegvd_scratchpad_size

oneAPI Math Kernel Library (oneMKL) documentation