f(x) = P1(x) + (x-x0)(x-x1)/2! f''(ξ) |
P1(x) = f(x0)L1,0(x) + f(x1)L1,1(x) = f(x0)(x - x1)/(x0 - x1) + f(x1)(x - x0)/(x1 - x0) = f(x0)(x - x1)/(-h) + f(x1)(x - x0)/(h) = f(x0)(x - x1)/(-h) + f(x1)(x - x0)/(h) = f(x0)(x - x1)/(-h) + f(x0+h)(x - x0)/(h) |
f'(x) = ( f(x0 + h)- f(x0) )/h + [(x-x0)(x-x1)/2! f''(ξ)]' |
f'(x0) = ( f(x0 + h)- f(x0) )/h - f''(ξ)h/2 |
M = N(h) + K1 h2 |
M = N(h/2) + K1 h2/4 |
3M = 4N(h/2) - N(h) de donde tenemos un nuevo valor de M 4N(h/2) - N(h) M = ---------------- 3 |
4j-1Nj-1(h/2) - Nj-1(h) M = Nj = -------------------------- 4j-1 - 1 |
Intabf(x) = R1 = [ f(a) + f(b)] h1/2 - h1/12 h12 f''(ξ) |
Intabf(x) ~ R2 = [ f(a) + f(b) + 2f(a + h2) ] h2/2 = (1/2)[ R1 + h1 f(a + h1/2) ] |
Intabf(x) ~ Rk = (1/2)[ Rk-1 + hk-1 (f(a + hk-1/2) + f(a + hk-13/2) + + f(a + hk-13/2) + ... + f(a + hk-1(2(k-1)-1/2) ) ] |
4j-1Ri,j-1 - Ri-1,j-1 Ri,j = -------------------------- 4j-1 - 1 |
P0(x)= 1 P1(x)= x Pk(x)= Pk-1(x)(x - Bk) - Pk-2(x)Ck siendo Bk = (Intg x(Pk-1(x))2 dx) /(Int (Pk-1(x))2 dx) y Ck = (Int x Pk-1(x) Pk-2(x) dx) /(Int (Pk-2(x))2 dx) y las integrales se hacen entre [-1,1]. |
f(x) = Q(x)Pn(x) + R(x) |
Int f(x) dx = Int R(x) dx y expandiendo R en polinomios de Lagrange obtenemos Int f(x) dx = Sum ( R(xi) Int Li dx ) = Sum ( R(xi) ci) |
f(xi) = Q(xi)Pn(xi) + R(xi) = 0 + R(xi) = R(xi) |
Int f(x) dx = Sum ( f(xi) ci) |
t = 1/(b-a) (2x - a - b) por lo que Intab f(x) dx = Int-11 f([(b-a)t + b + a ]/2) (b-a)/2 dt |
Darío Mitnik