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')[source]#
- 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 - awith diagonal elements d and off-diagonal elements e.- Parameters:
- dndarray, shape (ndim,)
- The diagonal elements of the array. 
- endarray, shape (ndim-1,)
- The off-diagonal elements of the array. 
- eigvals_onlybool, 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_finitebool, 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. 
- tolfloat
- 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_driverstr
- 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=Trueand- 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/DSTEMRroutines.- 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