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PROJET ANALYSE NUMÉRIQUE         MATLAB Comparaison expérimentale de la convergence des             formules de quadrature
INTRODUCTION•                   Polynômes de Lagrange               Transformée de Fourrier discrète
PLAN DE PRÉSENTATIONI.    Présentation et implémentation des formules de quadrature      A.     Newton-Cotes      B.     G...
A.NEWTON-COTES•
A. NEWTON-COTES•                      Les poids sont calculables à priori
A. NEWTON-COTES    Symétrie  Vectorisation
B.GAUSS-LEGENDRE•          Polynômes de Legendre                                   Trop                                  c...
•    B.GAUSS-LEGENDRE
B.GAUSS-LEGENDRE  Vectorisation
C.CLENSHAW-CURTIS•                                                  Difficile                    Transformée de Fourrier d...
C.CLENSHAW-CURTIS  Vectorisation
COMPARAISON DES CRITERES THEORIQUES   NC            GL           CC n/n+1         2n+1            n
PLAN DE PRÉSENTATIONI.    Présentation et implémentation des formules de quadrature      A.     Newton-Cotes      B.     G...
A.RÉPARTITION DES NOEUDS
A.RÉPARTITION DES NOEUDS
B.NEWTON-COTES -CONVERGENCE ENFONCTION DEL’INTÉGRAND F        Continuité ?
B.NEWTON-COTES: CONVERGENCE EN FONCTION DE L’INTEGRAND F
B.NEWTON-COTES -CONVERGENCE ENFONCTION DEL’INTÉGRAND F        Continuité ?
B.NEWTON-COTES: CONVERGENCE EN FONCTION DE L’INTEGRAND F•   Newton-Cotes ne convergent pas en général pour tout intégrand ...
C.GAUSS-LEGENDREVS CLENSHAW-CURTIS
C.GAUSS-LEGENDRE    VS CLENSHAW-CURTISFacteur relatif de 2
C.GAUSS-LEGENDREVS CLENSHAW-CURTIS
C.GAUSS-LEGENDRE           VS CLENSHAW-CURTIS Gauss-           Clenshaw-Legendre            Curtis
C. GAUSS-LEGENDRE VS CLENSHAW-CURTISCAS DE LA FONCTION 7
C.GAUSS-LEGENDRE           VS CLENSHAW-CURTIS Gauss-           Clenshaw-Legendre            Curtis
C.GAUSS-LEGENDREVS CLENSHAW-CURTIS
D.PREMIERE CONCLUSION                                               f non           f analytique                      anal...
D.ERREUR              D’INTERPOLATION,                  POLYNOME DE                     MEILLEUR                APPROXIMAT...
PLAN DE PRÉSENTATIONI.    Présentation et implémentation des formules de quadrature      A.     Newton-Cotes      B.     G...
CONCLUSION                                               f non           f analytique                      analytique  GL ...
QUESTIONS ?I.    Présentation et implémentation des formules de quadrature      A.     Newton-Cotes      B.     Gauss-Lege...
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Experimental comparison of quadrature formulas convergence

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Numerical Analysis : Experimental comparison of quadrature formulas convergence (Matlab)

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Experimental comparison of quadrature formulas convergence

  1. 1. PROJET ANALYSE NUMÉRIQUE MATLAB Comparaison expérimentale de la convergence des formules de quadrature
  2. 2. INTRODUCTION• Polynômes de Lagrange Transformée de Fourrier discrète
  3. 3. PLAN DE PRÉSENTATIONI. Présentation et implémentation des formules de quadrature A. Newton-Cotes B. Gauss-Legendre C. Clenshaw-CurtisII. Etude numérique de convergence A. Répartition des noeuds B. Newton-Cotes : convergence en fonction de l’intégrand C. Gauss-Legendre vs Clenshaw-Curtis : complexité, précision et vitesse de convergence D. Une première conclusion E. Comparaison avec l’erreur d’interpolation du polynôme de meilleur approximation
  4. 4. A.NEWTON-COTES•
  5. 5. A. NEWTON-COTES• Les poids sont calculables à priori
  6. 6. A. NEWTON-COTES Symétrie Vectorisation
  7. 7. B.GAUSS-LEGENDRE• Polynômes de Legendre Trop coûteux !!!
  8. 8. • B.GAUSS-LEGENDRE
  9. 9. B.GAUSS-LEGENDRE Vectorisation
  10. 10. C.CLENSHAW-CURTIS• Difficile Transformée de Fourrier discrète
  11. 11. C.CLENSHAW-CURTIS Vectorisation
  12. 12. COMPARAISON DES CRITERES THEORIQUES NC GL CC n/n+1 2n+1 n
  13. 13. PLAN DE PRÉSENTATIONI. Présentation et implémentation des formules de quadrature A. Newton-Cotes B. Gauss-Legendre C. Clenshaw-CurtisII. Etude numérique de convergence A. Répartition des noeuds B. Newton-Cotes : convergence en fonction de l’intégrand C. Gauss-Legendre vs Clenshaw-Curtis : complexité, précision et vitesse de convergence D. Une première conclusion E. Comparaison avec l’erreur d’interpolation du polynôme de meilleur approximation
  14. 14. A.RÉPARTITION DES NOEUDS
  15. 15. A.RÉPARTITION DES NOEUDS
  16. 16. B.NEWTON-COTES -CONVERGENCE ENFONCTION DEL’INTÉGRAND F Continuité ?
  17. 17. B.NEWTON-COTES: CONVERGENCE EN FONCTION DE L’INTEGRAND F
  18. 18. B.NEWTON-COTES -CONVERGENCE ENFONCTION DEL’INTÉGRAND F Continuité ?
  19. 19. B.NEWTON-COTES: CONVERGENCE EN FONCTION DE L’INTEGRAND F• Newton-Cotes ne convergent pas en général pour tout intégrand f continu• Newton-Cotes converge f analytique dans un voisinnage de l’intervalle de l’intégration assez grand. Pas d’erreurs d’arrondi Formules composites, Gauss-Legendre,• Pour n grand Clenshaw Curtis…
  20. 20. C.GAUSS-LEGENDREVS CLENSHAW-CURTIS
  21. 21. C.GAUSS-LEGENDRE VS CLENSHAW-CURTISFacteur relatif de 2
  22. 22. C.GAUSS-LEGENDREVS CLENSHAW-CURTIS
  23. 23. C.GAUSS-LEGENDRE VS CLENSHAW-CURTIS Gauss- Clenshaw-Legendre Curtis
  24. 24. C. GAUSS-LEGENDRE VS CLENSHAW-CURTISCAS DE LA FONCTION 7
  25. 25. C.GAUSS-LEGENDRE VS CLENSHAW-CURTIS Gauss- Clenshaw-Legendre Curtis
  26. 26. C.GAUSS-LEGENDREVS CLENSHAW-CURTIS
  27. 27. D.PREMIERE CONCLUSION f non f analytique analytique GL : vitesse CC : vitesse Vitesse de de de convergence convergence convergence Précision GL : CC : précision précision
  28. 28. D.ERREUR D’INTERPOLATION, POLYNOME DE MEILLEUR APPROXIMATION Gauss- Clenshaw -Legendre Clenshaw- Curtis Curtis
  29. 29. PLAN DE PRÉSENTATIONI. Présentation et implémentation des formules de quadrature A. Newton-Cotes B. Gauss-Legendre C. Clenshaw-CurtisII. Etude numérique de convergence A. Répartition des noeuds B. Newton-Cotes : convergence en fonction de l’intégrand C. Gauss-Legendre vs Clenshaw-Curtis : complexité, précision et vitesse de convergence D. Une première conclusion E. Comparaison avec l’erreur d’interpolation du polynôme de meilleur approximation
  30. 30. CONCLUSION f non f analytique analytique GL : vitesse CC : vitesse Vitesse de de de convergence convergence convergence Précision GL : CC : précision précision
  31. 31. QUESTIONS ?I. Présentation et implémentation des formules de quadrature A. Newton-Cotes B. Gauss-Legendre C. Clenshaw-CurtisII. Etude numérique de convergence A. Répartition des noeuds B. Newton-Cotes : convergence en fonction de l’intégrand C. Gauss-Legendre vs Clenshaw-Curtis : complexité, précision et vitesse de convergence D. Une première conclusion E. Comparaison avec l’erreur d’interpolation du polynôme de meilleur approximation

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