  We analyze in detail the error that arises from the linearization in
linearized augmented-plane-wave (LAPW) basis functions around predetermined
energies $E_l$ and show that it can lead to undesirable dependences of the
calculated results on method-inherent parameters such as energy parameters
$E_l$ and muffin-tin sphere radii. To overcome these dependences, we evaluate
approaches that eliminate the linearization error systematically by adding
local orbitals (LOs) to the basis set. We consider two kinds of LOs: (i)
constructed from solutions $u_l(r,E)$ to the scalar-relativistic approximation
of the radial Dirac equation with $E>E_l$ and (ii) constructed from second
energy derivatives $\partial^2 u_l(r,E) / \partial E^2$ at $E=E_l$. We find
that the latter eliminates the error most efficiently and yields the density
functional answer to many electronic and materials properties with very high
precision. Finally, we demonstrate that the so constructed LAPW+LO basis shows
a more favorable convergence behavior than the conventional LAPW basis due to a
better decoupling of muffin-tin and interstitial regions, similarly to the
related APW+lo approach, which requires an extra set of LOs to reach the same
total energy, though.
