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 25 of 58 Go Back Full Screen Close Quit Dacă în locul interpolării <strong>liniare</strong> se utilizează interpolarea pătratică se obt¸ine regula lui Simpson repetată . Varianta ei elementară, numită regula lui Simpson sau formula lui Simpson este xk+2 xk f(x)dx = h 3 (fk + 4fk+1 + fk+2) − 1 90 h5 f (4) (ξk), xk ≤ ξk ≤ xk+2, unde s-a presupus că f ∈ C 4 [a, b]. (22)
Introducere Derivare numerică Integrare numerică Cuadraturi adaptive Cuadraturi . . . Cuadraturi . . . Formule . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 25 of 58 Go Back Full Screen Close Quit Dacă în locul interpolării <strong>liniare</strong> se utilizează interpolarea pătratică se obt¸ine regula lui Simpson repetată . Varianta ei elementară, numită regula lui Simpson sau formula lui Simpson este xk+2 xk f(x)dx = h 3 (fk + 4fk+1 + fk+2) − 1 90 h5 f (4) (ξk), xk ≤ ξk ≤ xk+2, (22) unde s-a presupus că f ∈ C4 [a, b]. Să demonstrăm formula pentru restul formulei lui Simpson. Deoarece gradul de exactitate este 3, conform teoremei lui Peano avem unde K2(t) = 1 3! adică K2(t) = 1 6 (xk+1 − t) 4 4 (xk+2−t) 4 4 (xk+2−t) 4 4 R2(f) = xk+2 xk K2(t)f (4) (t) dt. − h (xk − t) 3 3 + + 4 (xk+1 − t) 3 + + (xk+2 − t) 3 + , − h 4 (xk+1 − t) 3 3 + (xk+2 − t) 3 , t ∈ [xk, xk+1] , − h 3 (xk+2 − t) 3 , t ∈ [xk+1, xk+2] . Se verifică ca pentru t ∈ [a, b], K2(t) ≤ 0 ¸si deci putem aplica corolarul la teorema lui Peano.
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Aproximarea functionalelor liniare

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