Culegere de probleme de Analiz˘a numeric˘a

140 Aproximarea funct¸ionalelor liniare
b
= −
a
= −f(m+2) (α)
(m+2)!
φm+1(x) f(m+2) (ξx)
dx =
(m+2)!
b
Integrând din nou prin părt¸i se obt¸ine
b
a
a
φm+1(x)dx = −
Luândx = x0 +sh s¸i utilizând lema 2
φm+1(x)dx a < α < b
b
Rm(f) = f(m+2) (ξ)
(m+2)! hm+3
a
xϕn+1(x)dx > 0
m
0
sψm+1(s)ds < 0
Deoarece f (m+2) (ξ) = 0 când f ∈ Pm+1 ⇒ r = m+1 pentrumpar.
Cazul m impar
Rm(f) =
b−h
a
ϕm+1(x)[x0,...,xm,x;f]dx+
b
+ ϕm+1(x)[x0,...,xm,x;f]dx
b−h
ϕm+1(x) = ϕm(x)(x−xm)
Deci b−h
ϕm+1(x)[x0,...,xm,x;f]dx =
=
b−h
a
a
dφm
dx ([x0,...,xm−1,x;f]−[x0,...,xm;f])dx
m impar ⇒ φm(b−h) = 0. Integrând prin părt¸i se obt¸ine
b−h
a
= − f(m+1) (ξ ′ )
(m+1)!
Aplicăm Teorema 1 de medie
− f(m+1) (ξ ′′ )
(m+1)!
φm+1(x)[x0,...,xm,x;f]dx =
b−h
φm(x)dx = Kf
a
(m+1) (ξ ′ )
a < ξ ′ < b−h
b
ϕm+1(x)dx = Lf
b−h
(m+1) (ξ ′′ )