Improvement of Gregory's formula using Particle Swarm Optimization
Consider the Gregory integration (G) formula
with end corrections where h Δ is the forward difference operator with step size h. In this study we prove that can be optimized by
minimizing some of the coefficient k a in the remainder term by
particle swarm optimization. Experimental tests prove that can be rendered a powerful formula for library use.
[1] G.C.ROTA, finite operator calculus, Academic press, Inc., 1975
[2] M.K.Belbahri, Generalized Gregory formula, Doctoral Thesis, Stevens Institute of Technology (1982).
[3] Eberhart, R.C. and Kennedy, J. (1995). A new optimizer using particles swarm theory', Sixth International Symposium on Micro Machine and Human Science, pp.39--43, Nagoya, Japan.
[4] K. E. Parsopoulos and M. N. Vrahatis, Modification of the Particle Swarm Optimizer for Locating all the Global Minima, V. Kurkova et
al., eds., Artificial Neural Networks and Genetic Algorithms, Springer,
New York, (2001), pp. 324-327.
[5] K. E. Parsopoulos et al., Stretching technique for obtaining global
minimizers through particle swarm optimization, Proc. of the PSO
Workshop, Indianapolis, USA, (2001b),pp. 22-29.
[6] P.C. Fourie, and Groenwold, A.A. Particle swarms in size and shape
optimization', Proceedings of the International Workshop on Multidisciplinary
Design Optimization, Pretoria, South Africa, August 7ÔÇö
10, (2000), pp.97--106.
[7] P.C. Fourie, and Groenwold, A.A. Particle swarms in topology
optimization', Extended Abstracts of the Fourth World Congress of
Structural and Multidisciplinary Optimization, Dalian, China, June 4--
8, (2001), pp.52, 53.
[8] R.C. Eberhart, et al. Computational Intelligence PC Tools, Academic
Press Professional, Boston. 1996.
[9] ] J. Kennedy, The behaviour of particles, Evol. Progr. VII (1998), 581-
587.
[10] J. Kennedy and R. C. Eberhart, Swarm Intelligence, Morgan
Kaufmann Publishers, San Francisco, 2001.
[11] Y. H. Shi and R. C. Eberhart, Fuzzy adaptive particle swarm
optimization, IEEE Int. Conf. on Evolutionary Computation, pp. 101-
106, (2001).
[12] Y. H. Shi and R. C. Eberhart, A modified particle swarm optimizer,
Proc. of the 1998 IEEE International Conference on Evolutionary
Computation, Anchorage, Alaska, May 4-9, 1998a.
[13] Y. H. Shi and R. C. Eberhart, Parameter selection in particle swarm
optimization, Evolutionary Programming VII, Lecture Notes in Computer Science, (1998b), pp. 591-600.
[14] M. Clerc, The swarm and the queen: towards a deterministic and
adaptive particle swarm optimization, Proceedings of the 1999 IEEE
Congress on Evolutionary Computation, Washington DC (1999),pp.1951--1957.
[15] R.C. Eberhart, and Shi, Y.. Parameter selection in particle swarm
optimization, in Porto, V.W., 1998
[16] T. I. Cristian, The particle swarm optimization algorithm: convergence
analysis and parameter selection, Information Processing Letters, Vol.
85, No. 6, (2003), pp.317-- 325.
[17] K. E. Parsopoulos et al., Objective function stretching to alleviate convergence to local minima, Nonlinear Analysis TMA 47 (2001a),
3419-3424.
[18] A. Zerarka and N. Khelil, A generalized integral quadratic method:
improvement of the solution for one dimensional Volterra integral equation using particle swarm optimization, Int. J. Simulation and
Process Modelling 2(1-2) (2006), 152-163.
[19] L. Djerou, M. Batouche, N. Khelil and A. Zerarka. Towards the Best
Points of Interpolation Using Particles Swarm Optimisation
Approach, IEEE Congress on Evolutionary Computation. Singapore September 25- 28, 2007
[1] G.C.ROTA, finite operator calculus, Academic press, Inc., 1975
[2] M.K.Belbahri, Generalized Gregory formula, Doctoral Thesis, Stevens Institute of Technology (1982).
[3] Eberhart, R.C. and Kennedy, J. (1995). A new optimizer using particles swarm theory', Sixth International Symposium on Micro Machine and Human Science, pp.39--43, Nagoya, Japan.
[4] K. E. Parsopoulos and M. N. Vrahatis, Modification of the Particle Swarm Optimizer for Locating all the Global Minima, V. Kurkova et
al., eds., Artificial Neural Networks and Genetic Algorithms, Springer,
New York, (2001), pp. 324-327.
[5] K. E. Parsopoulos et al., Stretching technique for obtaining global
minimizers through particle swarm optimization, Proc. of the PSO
Workshop, Indianapolis, USA, (2001b),pp. 22-29.
[6] P.C. Fourie, and Groenwold, A.A. Particle swarms in size and shape
optimization', Proceedings of the International Workshop on Multidisciplinary
Design Optimization, Pretoria, South Africa, August 7ÔÇö
10, (2000), pp.97--106.
[7] P.C. Fourie, and Groenwold, A.A. Particle swarms in topology
optimization', Extended Abstracts of the Fourth World Congress of
Structural and Multidisciplinary Optimization, Dalian, China, June 4--
8, (2001), pp.52, 53.
[8] R.C. Eberhart, et al. Computational Intelligence PC Tools, Academic
Press Professional, Boston. 1996.
[9] ] J. Kennedy, The behaviour of particles, Evol. Progr. VII (1998), 581-
587.
[10] J. Kennedy and R. C. Eberhart, Swarm Intelligence, Morgan
Kaufmann Publishers, San Francisco, 2001.
[11] Y. H. Shi and R. C. Eberhart, Fuzzy adaptive particle swarm
optimization, IEEE Int. Conf. on Evolutionary Computation, pp. 101-
106, (2001).
[12] Y. H. Shi and R. C. Eberhart, A modified particle swarm optimizer,
Proc. of the 1998 IEEE International Conference on Evolutionary
Computation, Anchorage, Alaska, May 4-9, 1998a.
[13] Y. H. Shi and R. C. Eberhart, Parameter selection in particle swarm
optimization, Evolutionary Programming VII, Lecture Notes in Computer Science, (1998b), pp. 591-600.
[14] M. Clerc, The swarm and the queen: towards a deterministic and
adaptive particle swarm optimization, Proceedings of the 1999 IEEE
Congress on Evolutionary Computation, Washington DC (1999),pp.1951--1957.
[15] R.C. Eberhart, and Shi, Y.. Parameter selection in particle swarm
optimization, in Porto, V.W., 1998
[16] T. I. Cristian, The particle swarm optimization algorithm: convergence
analysis and parameter selection, Information Processing Letters, Vol.
85, No. 6, (2003), pp.317-- 325.
[17] K. E. Parsopoulos et al., Objective function stretching to alleviate convergence to local minima, Nonlinear Analysis TMA 47 (2001a),
3419-3424.
[18] A. Zerarka and N. Khelil, A generalized integral quadratic method:
improvement of the solution for one dimensional Volterra integral equation using particle swarm optimization, Int. J. Simulation and
Process Modelling 2(1-2) (2006), 152-163.
[19] L. Djerou, M. Batouche, N. Khelil and A. Zerarka. Towards the Best
Points of Interpolation Using Particles Swarm Optimisation
Approach, IEEE Congress on Evolutionary Computation. Singapore September 25- 28, 2007
@article{"International Journal of Engineering, Mathematical and Physical Sciences:63261", author = "N. Khelil. L. Djerou and A. Zerarka and M. Batouche", title = "Improvement of Gregory's formula using Particle Swarm Optimization", abstract = "Consider the Gregory integration (G) formula
with end corrections where h Δ is the forward difference operator with step size h. In this study we prove that can be optimized by
minimizing some of the coefficient k a in the remainder term by
particle swarm optimization. Experimental tests prove that can be rendered a powerful formula for library use.", keywords = "Numerical integration, Gregory Formula, Particle Swarm optimization.", volume = "3", number = "10", pages = "867-3", }
