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.- 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:
- 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’, ‘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=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