  We study greedy approximation in uniformly smooth Banach spaces. The Weak
Chebyshev Greedy Algorithm (WCGA) is defined for any Banach space $X$ and a
dictionary $\mathcal{D}$, and provides nonlinear $n$-term approximation with
respect to $\mathcal{D}$. In this paper we study the Approximate Weak Chebyshev
Greedy Algorithm (AWCGA) -- a modification of the WCGA that was introduces by
V.N. Temlyakov. In the AWCGA we are allowed to calculate $n$-term approximation
with a perturbation in computing the norming functional and a relative error in
calculating the approximant. Such permission is natural for the numerical
applications and simplifies realization of the algorithm. We obtain conditions
that are necessary and sufficient for the convergence of the AWCGA for any
element of $X$. In particular, we show that if perturbations and errors are
from $\ell_1$ space then the conditions for the convergence of the AWCGA are
the same as for the WCGA. For specifically chosen perturbations and errors we
estimate the rate of convergence for any element $f$ from the closure of the
convex hull of $\mathcal{D}$ and demonstrate that in special cases the AWCGA
performs as well as the WCGA.
