qit.utils.expv

qit.utils.expv(t, A, v, tol=1e-07, m=30, iteration='arnoldi')

Multiply a vector by an exponentiated matrix.

Approximates $exp(tA)v$ using a Krylov subspace technique. Efficient for large sparse matrices. The basis for the Krylov subspace is constructed using either Arnoldi or Lanczos iteration.

Parameters

• t (array[float]) – vector of nondecreasing time instances >= 0, len(t) == s

• A (array[complex]) – n*n matrix (usually sparse)

• v (array[complex]) – n-dimensional vector

• tol (float) – tolerance

• m (int) – Krylov subspace dimension, m <= n

• iteration (str) – iteration type, in (‘arnoldi’, ‘lanczos’). Lanczos is faster but requires a Hermitian A.

Returns

result vectors as a (s, n) array, total truncation error estimate, hump size

Return type

array[complex], float, float

The result is $W[i,:]\approx \exp (t[i]A)v$. The hump size is $\underset{s\in [0,t]}{max}\Vert \exp (sA)\Vert$

Uses the sparse algorithm from [28].