quspin.tools.lanczos.lanczos_full

quspin.tools.lanczos.lanczos_full(A, v0, m, full_ortho=False, out=None, eps=None)[source]

Creates Lanczos basis; diagonalizes Krylov subspace in Lanczos basis.

Given a hermitian matrix A of size \(n\times n\) and an integer m, the Lanczos algorithm computes

  • an \(n\times m\) matrix \(Q\), and

  • a real symmetric tridiagonal matrix \(T=Q^\dagger A Q\) of size \(m\times m\). The matrix \(T\) can be represented via its eigendecomposition (E,V): \(T=V\mathrm{diag}(E)V^T\).

This function computes the triple \((E,V,Q^T)\).

NOTE: This function returns \(Q^T;\,Q^T\) is (in general) different from \(Q^\dagger\).

Parameters:
ALinearOperator, hamiltonian, numpy.ndarray, or object with a ‘dot’ method and a ‘dtype’ method.

Python object representing a linear map to compute the Lanczos approximation to the largest eigenvalues/vectors of. Must contain a dot-product method, used as A.dot(v) and a dtype method, used as A.dtype, e.g. hamiltonian, quantum_operator, quantum_LinearOperator, sparse or dense matrix.

v0array_like, (n,)

initial vector to start the Lanczos algorithm from.

mint

Number of Lanczos vectors (size of the Krylov subspace)

full_orthobool, optional

perform a QR decomposition on Q_T generated from the standard lanczos iteration to remove any loss of orthogonality due to numerical precision.

outnumpy.ndarray, optional

Array to store the Lanczos vectors in (e.g. Q). in memory efficient way.

epsfloat, optional

Used to cutoff lanczos iteration when off diagonal matrix elements of T drops below this value.

Returns:
tuple(E,V,Q_T)

  • E : (m,) numpy.ndarray: eigenvalues of Krylov subspace tridiagonal matrix \(T\).

  • V : (m,m) numpy.ndarray: eigenvectors of Krylov subspace tridiagonal matrix \(T\).

  • Q_T : (m,n) numpy.ndarray: matrix containing the m Lanczos vectors. This is \(Q^T\) (not \(Q^\dagger\))!

Notes

  • performs classical lanczos algorithm for hermitian matrices and cannot handle degeneracies when calculating eigenvalues.

  • the function allows for full orthogonalization, see full_ortho. The resulting \(T\) will not neccesarily be tridiagonal.

  • V is always real-valued, since \(T\) is real and symmetric.

  • A must have a ‘dot’ method to perform calculation,

  • The ‘out’ argument to pass back the results of the matrix-vector product will be used if the ‘dot’ function supports this argument.

Examples

>>> E, V, Q_T = lanczos_full(H,v0,20)