  Scattering lengths for two pseudoscalar meson systems, $\pi\pi(I=2)$,
$KK(I=1)$ and $\pi K(I=3/2,\ 1/2)$, are calculated from lattice QCD by using
the finite size formula. We perform the calculation with $N_f=2+1$ gauge
configurations generated on $32^3 \times 64$ lattice using the Iwasaki gauge
action and non-perturbatively ${\cal O}(a)$-improved Wilson action at $a^{-1} =
2.19$ GeV. The quark masses correspond to $m_\pi = 0.17 - 0.71$ GeV. For $\pi
K(I=1/2)$ system, we use the variational method with the two operators,
$\bar{s}u$ and $\pi K$, to separate the contamination from the higher states.
In order to obtain the scattering length at the physical quark mass, we fit our
results at the several quark masses with the formula of the ${\cal O}(p^4)$
chiral perturbation theory (ChPT) and that including the effects of the
discretization error from the Wilson fermion, Wilson chiral perturbation theory
(WChPT). We found that the mass dependence of our results near $m_\pi=0.17$ GeV
are described well by WChPT but not by ChPT. The scattering lengths at the
physical point are given as $a_0^{(2)} m_\pi =-0.04243(22)(43)$, $a_0^{(1)} m_K
=-0.312(17)(31)$, $a_0^{(3/2)}\mu_{\pi K}=-0.0477(27)(20)$ and
$a_0^{(1/2)}\mu_{\pi K}=0.150(16)(37)$. Possible systematic errors are also
discussed.
