Numerical differentiation for high orders by an integration method

This paper mainly studies the numerical differentiation by integration method proposed first by Lanczos. New schemes of the Lanczos derivatives are put forward for reconstructing numerical derivatives for high orders from noise data. The convergence rate of these proposed methods is O ( δ 4 n + 4 ) as the noise level δ → 0 . Numerical examples show that the proposed methods are stable and efficient.