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scipy.linalg.

# eigvalsh_tridiagonal

scipy.linalg.eigvalsh_tridiagonal(d, e, 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 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.

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details. Note that calls with zero-size batches are unsupported and will raise a `ValueError`.

  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 `lapack_driver='stebz'`). 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’, ‘stev’, or ‘stevd’. When ‘auto’ (default), it will use ‘stevd’ if `select='a'` and ‘stebz’ otherwise. ‘sterf’ and ‘stev’ can only be used when `select='a'`.

  Returns:
    w(M,) ndarray

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

  Raises:
    LinAlgError

      If eigenvalue computation does not converge.

  See also

  `eigh_tridiagonal`

    eigenvalues and right eigenvectors for symmetric/Hermitian tridiagonal matrices

  Examples

      >>> import numpy as np
      >>> from scipy.linalg import eigvalsh_tridiagonal, eigvalsh
      >>> d = 3*np.ones(4)
      >>> e = -1*np.ones(3)
      >>> w = eigvalsh_tridiagonal(d, e)
      >>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1)
      >>> w2 = eigvalsh(A)  # Verify with other eigenvalue routines
      >>> np.allclose(w - w2, np.zeros(4))
      True