that have their roots in the right half plane. This is clear advantage over forward (standard) Euler and backward Euler. Forward Euler tends to have less damping than the real system when the negative real parts of the roots get near the imaginary axis and backward Euler may show the system be stable even when the roots are in the right half plane. Example The motion of a spring satisfying Hooke's law is given by The semi-implicit Euler for this equation is Substituting in the second equation with the expression given by the first equation, the iteration can be expressed in the following matrix form and since the determinant of the matrix is 1 the transformation is area-preserving. The iteration preserves the modified energy functional exactly, leading to stable periodic orbits (for sufficiently small step size) that deviate by from the exact orbits. The exact circular frequency increases in the numerical approximation by a factor of . References Numerical differential equations