def integral_4(x_min, x_max, n_particoes, equacao):
a = x_min
h = (x_max - x_min) / n_particoes
b = a + h
w1 = (18 + sqrt(30)) / 36
w2 = w1
w3 = (18 - sqrt(30)) / 36
w4 = w3
a1 = sqrt((3 - 2 * sqrt(6 / 5)) / 7)
a2 = -sqrt((3 - 2 * sqrt(6 / 5)) / 7)
a3 = sqrt((3 + 2 * sqrt(6 / 5)) / 7)
x4 = -sqrt((3 + 2 * sqrt(6 / 5)) / 7)
soma_part = 0
area = 0
for i in range(n_particoes):
area = (b - a) / 2 * (
w1 * calc(equacao, (b - a) / 2 * a1 +
(b + a) / 2) + w2 * calc(equacao, (b - a) / 2 * a2 +
(b + a) / 2) +
w3 * calc(equacao, (b - a) / 2 * a3 +
(b + a) / 2) + w4 * calc(equacao,
(b - a) / 2 * x4 + (b + a) / 2))
soma_part += area
a = b
b = b + h
return soma_part
def integral_3(x_min, x_max, n_particoes, equacao):
a = x_min
h = (x_max - x_min) / n_particoes
b = a + h
w1 = 5 / 9
w2 = 8 / 9
w3 = w1
a1 = -sqrt(3 / 5)
a2 = 0
a3 = sqrt(3 / 5)
soma_part = 0
area = 0
for i in range(n_particoes):
area = (b - a) / 2 * (w1 * calc(equacao,
(b - a) / 2 * a1 + (b + a) / 2) +
w2 * calc(equacao,
(b - a) / 2 * a2 + (b + a) / 2) +
w3 * calc(equacao,
(b - a) / 2 * a3 + (b + a) / 2))
soma_part += area
a = b
b = b + h
return soma_part
def integral_2(x_min, x_max, n_particoes, equacao):
a = x_min
h = (x_max - x_min) / n_particoes
b = a + h
soma_part = 0
area = 0
for i in range(n_particoes):
#area = (b-a)/2 * ( cos( (b-a)/2*-1/sqrt(3) + (b+a)/2 ) +
# cos( (b-a)/2*+1/sqrt(3) + (b+a)/2 ))
area = (b - a) / 2 * (calc(equacao,
(b - a) / 2 * -1 / sqrt(3) + (b + a) / 2) +
calc(equacao,
(b - a) / 2 * +1 / sqrt(3) + (b + a) / 2))
soma_part += area
a = b
b = b + h
return soma_part
def integral(x_inicial,
x_final,
equacao,
tolerancia=0.000001,
grau=1,
filosofia='fechada'):
''' Calcula a integral pelo metodo de Newton-Cotes com base em uma tolerancia '''
delta_x = x_final - x_inicial
I_anterior = 0
I = delta_x / 2 * (calc(equacao, x_inicial) + calc(equacao, x_final))
erro = abs((I - I_anterior) / I)
n = 1
contador = 1
while erro > tolerancia:
n = n * 2
contador += 1
I_anterior = I
I = integral_n(x_inicial, x_final, n, equacao, filosofia, grau)
erro = abs((I - I_anterior) / I)
return I, contador
def integral_n(a, b, n, equacao, filosofia, grau=1):
''' Calcula a integral pelo metodo de Newton-Cotes com base na quantidade de iteracoes n '''
I = 0
delta_x = (b - a) / n
if filosofia is 'fechada':
h = delta_x / grau
if filosofia is 'aberta':
h = delta_x / (grau + 2)
for k in range(n):
x_inicial = a + k * delta_x
x_final = x_inicial + delta_x
if filosofia is 'fechada' and grau == 1:
I += h/2 * \
( calc(equacao, x_inicial) \
+ calc(equacao, x_final))
if filosofia is 'fechada' and grau == 2:
I += h/3 * \
( calc(equacao, x_inicial) \
+ 4*calc(equacao, x_inicial + h) \
+ calc(equacao, x_final))
if filosofia is 'fechada' and grau == 3:
I += 3*h/8 * \
( calc(equacao, x_inicial) \
+ 3*calc(equacao, x_inicial + h) \
+ 3*calc(equacao, x_inicial + 2*h) \
+ calc(equacao, x_final))
if filosofia is 'fechada' and grau == 4:
I += 2*h/45 * \
( 7*calc(equacao, x_inicial) \
+ 32*calc(equacao, x_inicial + h) \
+ 12*calc(equacao, x_inicial + 2*h) \
+ 32*calc(equacao, x_inicial + 3*h) \
+ 7*calc(equacao, x_final))
if filosofia is 'aberta' and grau == 1:
I += delta_x/2 * \
( calc(equacao, x_inicial + h) \
+ calc(equacao, x_inicial + 2*h))
if filosofia is 'aberta' and grau == 2:
I += 4*h/3 * \
( 2*calc(equacao, x_inicial + h) \
- calc(equacao, x_inicial + 2*h) \
+ 2*calc(equacao, x_inicial + 3*h))
if filosofia is 'aberta' and grau == 3:
I += 5*h/24 * \
( 11*calc(equacao, x_inicial + h) \
+ calc(equacao, x_inicial + 2*h) \
+ calc(equacao, x_inicial + 3*h) \
+ 11*calc(equacao, x_inicial + 4*h))
if filosofia is 'aberta' and grau == 4:
I += 3*h/10 * \
( 11*calc(equacao, x_inicial + h) \
- 14*calc(equacao, x_inicial + 2*h) \
+ 26*calc(equacao, x_inicial + 3*h) \
- 14*calc(equacao, x_inicial + 4*h) \
+ 11*calc(equacao, x_inicial + 5*h))
return I