Greetings, I am trying to implement backward euler implicit method by newton-raphson iteration. The differential equation is for a simple planar pendulum. So the function for the pendulum is : (1) angularAcceleration (angle) = ( -gravity/length ) * sin(angle); and the update function for implicit integration is: (2) u_(t+1) = u_t + deltaTime * velocity_(t+1) velocity_(t+1) = velocity_t + deltaTime * angularAcceleration ( u_(t+1) ) My question is: given that newton's method is stated as x_(i+1) = x_i - f( x_i) / f ' (x_i) , what is the f in newton's method? Is it the residual of the pendulum function given by (1) or the residual of the update step given by (2)? Thanks in advance!