lanczos

lanczos(
    A,
    v0=None,
    deg=None,
    rtol=1e-08,
    orth=0,
    sparse_mat=False,
    return_basis=False,
    seed=None,
    dtype=None,
    **kwargs,
)

Lanczos method for symmetric tridiagonalization.

This function implements Paiges A27 variant (1) of the Lanczos method for tridiagonalizing linear operators, which builds a tridiagonal T from a symmetric A via an orthogonal change-of-basis Q: \begin{align*} K &= [v, Av, A^2 v, ..., A^{n-1}v] && \\ Q &= [q_1, q_2, ..., q_{n} ] \gets \mathrm{qr}(K) && \\ T &= Q^T A Q && \end{align*} Note that unlike other Lanczos implementations (e.g. SciPy’s eigsh), which includes e.g. restarting, & deflation steps, this method simply executes deg steps of the Lanczos method with orth vector reorthogonalizations (per step) and returns the entries of T.

This implementation supports varying degrees of re-orthogonalization. In particular, orth=0 corresponds to no re-orthogonalization, orth < deg corresponds to partial re-orthogonalization, and orth >= deg corresponds to full re-orthogonalization. The number of matvecs scales linearly with deg and the number of inner-products scales quadratically with orth.

Parameters

Name	Type	Description	Default
A	Union[np.ndarray, sparray, LinearOperator]	Symmetric operator to tridiagonalize.	required
v0	Optional[np.ndarray]	Initial vector to orthogonalize against.	None
deg	Optional[int]	Size of the Krylov subspace to expand.	None
rtol	float	Relative tolerance to consider the invariant subspace as converged.	1e-08
orth	int	Number of additional Lanczos vectors to orthogonalize against.	0
sparse_mat	bool	Whether to output the tridiagonal matrix as a sparse matrix.	False
return_basis	bool	If True, returns the Krylov basis vectors Q.	False
dtype	Optional[np.dtype]	The precision dtype to specialize the computation.	None

Returns

Name	Type	Description
	tuple	A tuple (a,b) parameterizing the diagonal and off-diagonal of the tridiagonal Jacobi matrix. If return_basis=True,
	tuple	the tuple (a,b), Q is returned, where Q represents an orthogonal basis for the degree-deg Krylov subspace.

See Also

• scipy.linalg.eigh_tridiagonal : Eigenvalue solver for real symmetric tridiagonal matrices.
• operator.matrix_function : Approximates the action of a matrix function via the Lanczos method.

References

1. Paige, Christopher C. “Computational variants of the Lanczos method for the eigenproblem.” IMA Journal of Applied Mathematics 10.3 (1972): 373-381.