  In the whole space $R^d$ ($d\ge 2$), we study homogenization of a
divergence-form matrix elliptic operator $L_\varepsilon$ of an arbitrary even
order larger than 2 with measurable $\varepsilon$-periodic coefficients, where
$\varepsilon$ is a small parameter. We constuct an approximation for the
resolvent of $L_\varepsilon$ with the remainder term of order $\varepsilon^2$
in the operator $L^2$-norm. We impose no regularity conditions on the operator
beyond ellipticity and boundedness of coefficients. We use two scale expansions
with correctors regularized by the Steklov smoothing.
