def woodbury_chol(self):
        """
        return $L_{W}$ where L is the lower triangular Cholesky decomposition of the Woodbury matrix
        $$
        L_{W}L_{W}^{\top} = W^{-1}
        W^{-1} := \texttt{Woodbury inv}
        $$
        """
        if self._woodbury_chol is None:
            # compute woodbury chol from
            if self._woodbury_inv is not None:
                winv = np.atleast_3d(self._woodbury_inv)
                self._woodbury_chol = np.zeros(winv.shape)
                for p in range(winv.shape[-1]):
                    self._woodbury_chol[:, :, p] = pdinv(winv[:, :, p])[2]
                    # Li = jitchol(self._woodbury_inv)
                    # self._woodbury_chol, _ = dtrtri(Li)
                    # W, _, _, _, = pdinv(self._woodbury_inv)
                    # symmetrify(W)
                    # self._woodbury_chol = jitchol(W)
            # try computing woodbury chol from cov
            elif self._covariance is not None:
                raise NotImplementedError("TODO: check code here")
                B = self._K - self._covariance
                tmp, _ = dpotrs(self.K_chol, B)
                self._woodbury_inv, _ = dpotrs(self.K_chol, tmp.T)
                _, _, self._woodbury_chol, _ = pdinv(self._woodbury_inv)
            else:
                raise ValueError("insufficient information to compute posterior")
        return self._woodbury_chol