Download e-book for iPad: Introduction à l'analyse numérique by Jacques Rappaz, Marco Picasso

By Jacques Rappaz, Marco Picasso

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F (x0 + 2h) = f (x0 ) + f (x0 ) ✰✺❞❆✹ ✖✙✢❊✳✵✩❘✩✥✚✜❂✜✩●✤❅✹❨✤❅✚✸✤❊✹✛✰✣✮❃✦✙✔✣✽ ✮✜✩ ✙✴ ◗ ✶✜⑥❬✶❍✽✸✮✙✰✣✚✜✩❝✰✲❂✸✤❊✽ ★✪✮✙✰✣✮✿✩ ✵ η1 ∈ ]x0 , x0 + 2h[ η2 ∈ ]x0 − 2h, x0 [ η3 ∈ ]x0 , x0 + h[ ★✯✤ η4 ∈ ]x0 − h, x0 [ ◗ δh4 f (x0 ) = f (4) (x0 )h4 ✷✰✲✹❨✤ ✄ + h0 > 0 ♠❴✰✣✚✜✢❝✤❊✰✣✚✙✤ 4 64 (6) f (η1 ) + f (6) (η2 ) − f (6) (η3 ) + f (6) (η4 ) 6! 6! ✚✙✮✐✮✙✰✲✱❥❂✙✢❅★▼✔▲✢❅❂✙✹❨✤❅✢❅✔✣✹❁✢❅★❢★❍✤✫✩✥✰✲✹❨✤ C= h ≤ h0 h6 . 17 max f (6) (x) . 90 x∈[x0 −2h0 ,x0 +2h0 ] ✽✙✮✜✰✣✚✜✩❘✔ ❖ ✰✣✮✜✩❝✦✸✰✲✮✜✾ δh4 f (x0 ) − f (4) (x0 ) ≤ Ch2 .

90 x∈[x0 −2h0 ,x0 +2h0 ] ✽✙✮✜✰✣✚✜✩❘✔ ❖ ✰✣✮✜✩❝✦✸✰✲✮✜✾ δh4 f (x0 ) − f (4) (x0 ) ≤ Ch2 . h4 ✸✘✻✦❢ ✔♥✰✲✂❡✮✙❬❭✜✯✰✣✮✙✏❍✮✜✭✪❦▲✾❍★✣★✫✤❊✽✿❦▲✹✛✰✣✩❊✜ ✰✣✮❃✹❁✿❊✤ ✦✸❲ ✭❍✿ ❧✿✮✙✄✸✹❁✰✣★❣✹❁✤ ✖✜✔✣★❍✢ ✤ ✯ ✦✸✰✣✮✜✚✙✮✙✮✙✭✪★✈✩✪◗ ❬❭✰✲✄✷✮✜✰✲✹❨✾s✤ ✤❅✹❁✰✲✮✠✤❅✢❊✰✲✹✛✩♥❬❭✰✲✹✛✩✗✽ ✾✯✰✣✮✼✤❊✹✛✮☎✄✙✱✴★✯✮✼✤q✦✸★❍✭✯✤❢✢❅✹ ✩❊❖ ✰✣✔✣✹❁❂✙✤ ✘❁★ f : R → R x0 ∈ R h>0 ⑥✲◗ ❆✭✯✢❅✹❁❧✜★✯✢❃✺✼✚✙★ ★❍✤ g(x) = f (x0 ) + x1 = x0 + h x2 = x0 + 2h ∆2 f (x0 ) ∆h f (x0 ) (x − x0 ) + h 2 (x − x0 )(x − x1 ). h 2h ❊✤ ★✪✖❛✘✛✩❆✰✣✚✙✺✲✢✚✜★ g(xj ) = f (xj ) ξ0 ∈ [x0 , x1 ] ξ1 ∈ [x1 , x2 ] j = 0, 1, 2 f (ξ0 ) = g (ξ0 ) , g ★❍✤✐★✯✮ ✦✸✭✪✦✸✚✜✹❁✢❅★⑩✺✼✚❋❏ ✹✛✘❢★✯■✸✹✛✩✥✤❊★ f (ξ1 ) = g (ξ1 ).

H2 h3 h1 f (x0 + h) = f (x0 ) + f (x0 ) + f (x0 ) + f (x0 ) 1! 2! 3! h5 h6 h4 (5) (6) (4) + f (x0 ) + f (x0 ) + f (η3 ) , 4! 5! 6! h1 h2 h3 f (x0 − h) = f (x0 ) − f (x0 ) + f (x0 ) − f (x0 ) 1! 2! 3! 4 5 h h h6 + f (4) (x0 ) − f (5) (x0 ) + f (6) (η4 ) , 4! 5! 6! f (x0 + 2h) = f (x0 ) + f (x0 ) ✰✺❞❆✹ ✖✙✢❊✳✵✩❘✩✥✚✜❂✜✩●✤❅✹❨✤❅✚✸✤❊✹✛✰✣✮❃✦✙✔✣✽ ✮✜✩ ✙✴ ◗ ✶✜⑥❬✶❍✽✸✮✙✰✣✚✜✩❝✰✲❂✸✤❊✽ ★✪✮✙✰✣✮✿✩ ✵ η1 ∈ ]x0 , x0 + 2h[ η2 ∈ ]x0 − 2h, x0 [ η3 ∈ ]x0 , x0 + h[ ★✯✤ η4 ∈ ]x0 − h, x0 [ ◗ δh4 f (x0 ) = f (4) (x0 )h4 ✷✰✲✹❨✤ ✄ + h0 > 0 ♠❴✰✣✚✜✢❝✤❊✰✣✚✙✤ 4 64 (6) f (η1 ) + f (6) (η2 ) − f (6) (η3 ) + f (6) (η4 ) 6!

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