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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.

  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.

    eigvals_only bool, optional

      Compute only the eigenvalues and no eigenvectors. (Default: calculate also eigenvectors)

    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’, ‘stev’, or ‘stevd’. When ‘auto’ (default), it will use ‘stevd’ 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]`. Only returned if `eigvals_only=False`.

  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