[en] The exact radial solution inside the muffin-tin sphere is replaced by its Taylor expansion with respect to the energy, truncated after the quadratic term. Making use of it the energy independent augmented plane waves are formed which lead to the secular equations linear in energy. The method resembles the currently used linearized APW method but yields higher accuracy. The analysis of solution inside one muffin-tin sphere shows that the eigenvalue error is proportional to (E-E₀)⁶ as compared with (E-E₀)⁴ for LAPW. The error of eigenfunctions is (E-E₀)³ ((E-E₀)² for LAPW). These conclusions are confirmed by direct numerical calculation of band structure of Cu and Al. (author)