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scipy.linalg.

# eig_banded

scipy.linalg.eig_banded(a_band, lower=False, eigvals_only=False, overwrite_a_band=False, select='a', select_range=None, max_ev=0, check_finite=True)

  Solve real symmetric or complex Hermitian band matrix eigenvalue problem.

  Find eigenvalues w and optionally right eigenvectors v of a:

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

  The matrix a is stored in a_band either in lower diagonal or upper diagonal ordered form:

  > a_band[u + i - j, j] == a[i,j] (if upper form; i <= j) a_band[ i - j, j] == a[i,j] (if lower form; i >= j)

  where u is the number of bands above the diagonal.

  Example of a_band (shape of a is (6,6), u=2):

      upper form:
      *   *   a02 a13 a24 a35
      *   a01 a12 a23 a34 a45
      a00 a11 a22 a33 a44 a55

      lower form:
      a00 a11 a22 a33 a44 a55
      a10 a21 a32 a43 a54 *
      a20 a31 a42 a53 *   *

  Cells marked with * are not used.

  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.

  Parameters:
    a_band(u+1, M) array_like

      The bands of the M by M matrix a.

    lower bool, optional

      Is the matrix in the lower form. (Default is upper form)

    eigvals_only bool, optional

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

    overwrite_a_band bool, optional

      Discard data in a_band (may enhance performance)

    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

    max_ev int, optional

      For select==’v’, maximum number of eigenvalues expected. For other values of select, has no meaning.

      In doubt, leave this parameter untouched.

    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.

  Returns:
    w(M,) ndarray

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

    v(M, M) float or complex 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

  `eigvals_banded`

    eigenvalues for symmetric/Hermitian band matrices

  `eig`

    eigenvalues and right eigenvectors of general arrays.

  `eigh`

    eigenvalues and right eigenvectors for symmetric/Hermitian arrays

  `eigh_tridiagonal`

    eigenvalues and right eigenvectors for symmetric/Hermitian tridiagonal matrices

  Examples

      >>> import numpy as np
      >>> from scipy.linalg import eig_banded
      >>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]])
      >>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]])
      >>> w, v = eig_banded(Ab, lower=True)
      >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
      True
      >>> w = eig_banded(Ab, lower=True, eigvals_only=True)
      >>> w
      array([-4.26200532, -2.22987175,  3.95222349, 12.53965359])

  Request only the eigenvalues between `[-3, 4]`

      >>> w, v = eig_banded(Ab, lower=True, select='v', select_range=[-3, 4])
      >>> w
      array([-2.22987175,  3.95222349])