# scipy.linalg.eigh_tridiagonal

scipy.linalg.eigh_tridiagonal(d, e, eigvals_only=False, select='a', select_range=None, check_finite=True, tol=0.0, lapack_driver='auto')

  Solve eigenvalue problem for a real symmetric tridiagonal matrix.

  Find eigenvalues w and optionally right eigenvectors v of `a`:

      a v[:,i] = w[i] v[:,i]
      v.H v    = identity

  For a real symmetric matrix `a` with diagonal elements d and off-diagonal elements e.

  Parameters:
    d ndarray, shape (ndim,)

      The diagonal elements of the array.

    e ndarray, shape (ndim-1,)

      The off-diagonal elements of the array.

    select{‘a’, ‘v’, ‘i’}, optional

      Which eigenvalues to calculate

      select | calculated
      --- | ---
      ‘a’ | All eigenvalues
      ‘v’ | Eigenvalues in the interval (min, max]
      ‘i’ | Eigenvalues with indices min <= i <= max

    select_range(min, max), optional

      Range of selected eigenvalues

    check_finite bool, optional

      Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

    tol float

      The absolute tolerance to which each eigenvalue is required (only used when ‘stebz’ is the lapack_driver). An eigenvalue (or cluster) is considered to have converged if it lies in an interval of this width. If <= 0. (default), the value `eps*|a|` is used where eps is the machine precision, and `|a|` is the 1-norm of the matrix `a`.

    lapack_driver str

      LAPACK function to use, can be ‘auto’, ‘stemr’, ‘stebz’, ‘sterf’, or ‘stev’. When ‘auto’ (default), it will use ‘stemr’ if `select='a'` and ‘stebz’ otherwise. When ‘stebz’ is used to find the eigenvalues and `eigvals_only=False`, then a second LAPACK call (to `?STEIN`) is used to find the corresponding eigenvectors. ‘sterf’ can only be used when `eigvals_only=True` and `select='a'`. ‘stev’ can only be used when `select='a'`.

  Returns:
    w(M,) ndarray

      The eigenvalues, in ascending order, each repeated according to its multiplicity.

    v(M, M) ndarray

      The normalized eigenvector corresponding to the eigenvalue `w[i]` is the column `v[:,i]`.

  Raises:
    LinAlgError

      If eigenvalue computation does not converge.

  See also

  `eigvalsh_tridiagonal`

    eigenvalues of symmetric/Hermitian tridiagonal matrices

  `eig`

    eigenvalues and right eigenvectors for non-symmetric arrays

  `eigh`

    eigenvalues and right eigenvectors for symmetric/Hermitian arrays

  `eig_banded`

    eigenvalues and right eigenvectors for symmetric/Hermitian band matrices

  Notes

  This function makes use of LAPACK `S/DSTEMR` routines.

  Examples

      >>> import numpy as np
      >>> from scipy.linalg import eigh_tridiagonal
      >>> d = 3*np.ones(4)
      >>> e = -1*np.ones(3)
      >>> w, v = eigh_tridiagonal(d, e)
      >>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1)
      >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
      True