Improvement of Gregory's formula using Particle Swarm Optimization

Consider the Gregory integration (G) formula ( ) ( ) ( ) ( ) ( ) 0 0 0 0 ! k k n n k k k h h k k a f x dx f k f f n k − = ≥ ⎛ ⎞ = + Δ + Δ ⎜ ⎟ ⎝ ⎠ ∑ ∑ ∫ (1) with end corrections where h Δ is the forward difference operator with step size h. In this study we prove that (1) can be optimized by minimizing some of the coefficient k a in the remainder term by particle swarm optimization. Experimental tests prove that (1) can be rendered a powerful formula for library use. Keywords—Numerical integration, Gregory Formula, Particle Swarm optimization.