LanczosEvolution

• full name: tenpy.linalg.krylov_based.LanczosEvolution

• parent module: tenpy.linalg.krylov_based

• type: class

Inheritance Diagram

Methods

LanczosEvolution.__init__(H, psi0, options)
LanczosEvolution.iadd_prefactor_other(w, ...)
LanczosEvolution.iscale_prefactor(w, scale)
LanczosEvolution.run(delta[, normalize])	Calculate expm(delta H).dot(psi0) using Lanczos.

class tenpy.linalg.krylov_based.LanczosEvolution(H, psi0, options)

Bases: LanczosGroundState

Calculate \(exp(delta H) |psi0>\) using Lanczos.

It turns out that the Lanczos algorithm is also good for calculating the matrix exponential applied to the starting vector. Instead of diagonalizing the tri-diagonal h and taking the ground state, we now calculate exp(delta h) e_0 in the Krylov ONB, where e_0 = (1, 0, 0, ...) corresponds to psi0 in the original basis.

Parameters:

• H – Hamiltonian, starting vector and parameters as defined in LanczosGroundState. The option :cfg:option`LanczosEvolution.P_tol` defines when convergence is reached, see _converged() for details.

• psi0 – Hamiltonian, starting vector and parameters as defined in LanczosGroundState. The option :cfg:option`LanczosEvolution.P_tol` defines when convergence is reached, see _converged() for details.

• options – Hamiltonian, starting vector and parameters as defined in LanczosGroundState. The option :cfg:option`LanczosEvolution.P_tol` defines when convergence is reached, see _converged() for details.

Options

config LanczosEvolution

option	summary
cutoff (from KrylovBased) in KrylovBased	Cutoff to abort if the norm of the new krylov vector is too small. [...]
E_shift (from KrylovBased) in KrylovBased	Shift the energy (=eigenvalues) by that amount *during* the Lanczos run by [...]
E_tol	Ignored.
min_gap	Ignored.
N_cache (from LanczosGroundState) in LanczosGroundState	The maximum number of `psi` to keep in memory during the first iteration. [...]
N_max (from KrylovBased) in KrylovBased	Maximum number of steps to perform.
N_min (from KrylovBased) in KrylovBased	Minimum number of steps to perform.
P_tol (from KrylovBased) in KrylovBased	Tolerance for the error estimate from the Ritz Residual, [...]
reortho (from KrylovBased) in KrylovBased	For poorly conditioned matrices, one can quickly loose orthogonality of the [...]

option E_tol

Ignored.

option min_gap

Ignored.

delta

Prefactor of H in the exponential.

Type:

float/complex

_result_norm

Norm of the resulting vector.

Type:

float

run(delta, normalize=None)

Calculate expm(delta H).dot(psi0) using Lanczos.

Parameters:

• delta (float/complex) – Time step by which we should evolve psi0: prefactor of H in the exponential. Note that the complex i is not included!

• normalize (bool) – Whether to normalize the resulting state. Defaults to np.real(delta) == 0.

Returns:

• psi_f (Array) – Best approximation for expm(delta H).dot(psi0). If Lanczos.E_shift is used, it’s an approximation for expm(delta (H + E_shift)).dot(psi).

• N (int) – Krylov space dimension used.