Eigenvalue Solver for Symmetric or Hermitian Tridiagonal Matrix#

HTEV (symmetric or Hermitian Tridiagonal EigenValues solver) computes the eigenvalues, and optionally the eigenvectors, of batched symmetric or Hermitian tridiagonal matrices using the QR algorithm.

\[A V = V \Lambda\]

where

  • \(A\) is an \(M \times M\) symmetric tridiagonal matrix,

  • \(\Lambda\) is an \(M \times M\) diagonal matrix containing the eigenvalues, and

  • \(V\) is an \(M \times M\) unitary matrix containing the eigenvectors.

cuSolverDx htev device functions are as follows (see Execution Methods):

// Compute eigenvalues only
__device__ void execute(data_type* d, data_type* e, status_type* info);
// Compute eigenvalues and the eigenvectors
__device__ void execute(data_type* d, data_type* e, data_type* V, status_type* info);
// Compute eigenvalues and the eigenvectors with runtime leading dimension of the matrix `V`
__device__ void execute(data_type* d, data_type* e, data_type* V, int ldv, status_type* info);

The symmetric tridiagonal matrix \(A\) is stored as two 1D arrays d and e, where d contains the diagonal elements, of size \(M\) per batch, and e contains the subdiagonal elements, of size \(M-1\) per batch. On exit, the function overwrites d with the eigenvalues in ascending order, and overwrites e with zeros.

If eigenvectors are requested, the full storage of matrix V, of size \(M \times M\), is required as input, with leading dimension \(\mathrm{ldv} \geq M\).

  • if job::all_vectors are selected for the Job operator, on exit the function fills V with the eigenvectors of the tridiagonal matrix.

  • if job::multiply_vectors are selected for the Job operator, on exit the function multiplies the input V by the computed eigenvectors of the tridiagonal matrix on the right, i.e., input_V * eigen_V, and overwrites V with the result.

The function returns status info for each batch, info = 0 if the function succeeds. If info = i > 0, then the algorithm failed to find all the eigenvalues in a total of \(30 M\) iterations, and i elements of e have not converged to zeros.

Note

cuSolverDx’s htev functions only support real data type of matrix A, its eigenvalues, and eigenvectors.

The function supports:

  1. if eigenvectors are requested, matrix V being column- or row-major layouts, see Arrangement operator, and

  2. job being either job::no_vectors, job::all_vectors or job::multiply_vectors, see Job operator. job::overwrite_vectors is not allowed for the htev function; doing that will result in a compile-time error.

Note

If eigenvectors are not requested, i.e., job::no_vectors is selected for the Job operator, then V is not used. As matrix A is implicitly defined with two vectors, d and e, the leading dimension, or arrangement of A or V, if defined, is ignored.