Aproximarea functionalelor liniare

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$P 0.1 Stabilişti natura şsi suma seriilor: a) Σ 1 n\ ; b) Σ arctan 1 n# + n ...$

Introducere Derivare numerică Integrare numerică Cuadraturi adaptive Cuadraturi . . . Cuadraturi . . . Formule . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 45 of 58 Go Back Full Screen Close Quit Formulele de tip interpolator sunt chiar formulele obt¸inute prin interpolare, adică pentru care n wkf(tk) = k=1 sau echivalent unde b wk = a b a Ln−1(f; t1, . . . , tn, t)w(t)dt (34) ℓk(t)w(t)dt, k = 1, 2, . . . , n, (35) ℓk(t) = n l=1 l=k t − tl tk − tl (36) sunt polinoamele fundamentale Lagrange asociate nodurilor t1, t2, . . . , tn. Faptul că (32) are gradul de exactitate d = n − 1 este evident, deoarece pentru orice f ∈ Pn−1 Ln−1(f; ·) ≡ f(·) în (34). Reciproc, dacă (32) are gradul de exactitate d = n − 1, atunci luând f(t) = ℓr(t) în (33) ne dă b a adică (35). ℓr(t)w(t)dt = n wkℓr(tk) = wr, r = 1, 2, . . . , n, k=1
Introducere Derivare numerică Integrare numerică Cuadraturi adaptive Cuadraturi . . . Cuadraturi . . . Formule . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 45 of 58 Go Back Full Screen Close Quit Formulele de tip interpolator sunt chiar formulele obt¸inute prin interpolare, adică pentru care n wkf(tk) = k=1 sau echivalent unde b wk = a b a Ln−1(f; t1, . . . , tn, t)w(t)dt (34) ℓk(t)w(t)dt, k = 1, 2, . . . , n, (35) ℓk(t) = n l=1 l=k t − tl tk − tl (36) sunt polinoamele fundamentale Lagrange asociate nodurilor t1, t2, . . . , tn. Faptul că (32) are gradul de exactitate d = n − 1 este evident, deoarece pentru orice f ∈ Pn−1 Ln−1(f; ·) ≡ f(·) în (34). Reciproc, dacă (32) are gradul de exactitate d = n − 1, atunci luând f(t) = ℓr(t) în (33) ne dă b a adică (35). ℓr(t)w(t)dt = n wkℓr(tk) = wr, r = 1, 2, . . . , n, k=1
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Aproximarea functionalelor liniare

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