직접 해를 구하면,
다음은 RK4를 이용하여 구한 값과 실제 해에 대입한 값을 구하는 Python 코드이다. (h = 0.05, 0부터 20까지 계산)
import decimal
import math
def f(X, Y):
return decimal.Decimal((-X * Y) / (X * X - 9))
def Y(X):
return decimal.Decimal(4 / (math.sqrt(X * X - 9)))
h = decimal.Decimal('0.05')
x = []
y = []
x.append(decimal.Decimal('5.00'))
y.append(decimal.Decimal('1.00'))
for i in range(0, 21):
if i >= 1:
k1 = f(x[i - 1], y[i - 1])
k2 = f(x[i - 1] + h / 2, y[i - 1] + h * k1 / 2)
k3 = f(x[i - 1] + h / 2, y[i - 1] + h * k2 / 2)
k4 = f(x[i - 1] + h, y[i - 1] + h * k3)
x.append(x[i - 1] + h)
y.append(y[i - 1] + h * (k1 + 2 * k2 + 2 * k3 + k4) / 6)
print("x{} =".format(i), x[i], " RK4 y{} =".format(i), "%.10f" % y[i], end="")
print(" 참값 y{} =".format(i), "%.10f" % Y(x[i]), end="")
print(" 절대오차 {} =".format(i), "%.10f" % abs(y[i] - Y(x[i])), end="")
print(" 상대오차 {} =".format(i), "%.10f" % (abs(y[i] - Y(x[i])) * 100 / abs(Y(x[i]))), end="")
print("%")
x0 = 5.00 RK4 y0 = 1.0000000000 참값 y0 = 1.0000000000 절대오차 0 = 0.0000000000 상대오차 0 = 0.0000000000%
x1 = 5.05 RK4 y1 = 0.9846573354 참값 y1 = 0.9846573354 절대오차 1 = 0.0000000000 상대오차 1 = 0.0000000031%
x2 = 5.10 RK4 y2 = 0.9698572899 참값 y2 = 0.9698572899 절대오차 2 = 0.0000000001 상대오차 2 = 0.0000000059%
x3 = 5.15 RK4 y3 = 0.9555687905 참값 y3 = 0.9555687906 절대오차 3 = 0.0000000001 상대오차 3 = 0.0000000083%
x4 = 5.20 RK4 y4 = 0.9417632186 참값 y4 = 0.9417632187 절대오차 4 = 0.0000000001 상대오차 4 = 0.0000000104%
x5 = 5.25 RK4 y5 = 0.9284141650 참값 y5 = 0.9284141651 절대오차 5 = 0.0000000001 상대오차 5 = 0.0000000123%
x6 = 5.30 RK4 y6 = 0.9154972141 참값 y6 = 0.9154972142 절대오차 6 = 0.0000000001 상대오차 6 = 0.0000000140%
x7 = 5.35 RK4 y7 = 0.9029897537 참값 y7 = 0.9029897539 절대오차 7 = 0.0000000001 상대오차 7 = 0.0000000155%
x8 = 5.40 RK4 y8 = 0.8908708062 참값 y8 = 0.8908708064 절대오차 8 = 0.0000000002 상대오차 8 = 0.0000000169%
x9 = 5.45 RK4 y9 = 0.8791208790 참값 y9 = 0.8791208791 절대오차 9 = 0.0000000002 상대오차 9 = 0.0000000181%
x10 = 5.50 RK4 y10 = 0.8677218311 참값 y10 = 0.8677218313 절대오차 10 = 0.0000000002 상대오차 10 = 0.0000000192%
x11 = 5.55 RK4 y11 = 0.8566567548 참값 y11 = 0.8566567549 절대오차 11 = 0.0000000002 상대오차 11 = 0.0000000202%
x12 = 5.60 RK4 y12 = 0.8459098687 참값 y12 = 0.8459098689 절대오차 12 = 0.0000000002 상대오차 12 = 0.0000000211%
x13 = 5.65 RK4 y13 = 0.8354664231 참값 y13 = 0.8354664233 절대오차 13 = 0.0000000002 상대오차 13 = 0.0000000219%
x14 = 5.70 RK4 y14 = 0.8253126139 참값 y14 = 0.8253126141 절대오차 14 = 0.0000000002 상대오차 14 = 0.0000000226%
x15 = 5.75 RK4 y15 = 0.8154355061 참값 y15 = 0.8154355063 절대오차 15 = 0.0000000002 상대오차 15 = 0.0000000233%
x16 = 5.80 RK4 y16 = 0.8058229638 참값 y16 = 0.8058229640 절대오차 16 = 0.0000000002 상대오차 16 = 0.0000000239%
x17 = 5.85 RK4 y17 = 0.7964635879 참값 y17 = 0.7964635881 절대오차 17 = 0.0000000002 상대오차 17 = 0.0000000244%
x18 = 5.90 RK4 y18 = 0.7873466590 참값 y18 = 0.7873466592 절대오차 18 = 0.0000000002 상대오차 18 = 0.0000000249%
x19 = 5.95 RK4 y19 = 0.7784620858 참값 y19 = 0.7784620860 절대오차 19 = 0.0000000002 상대오차 19 = 0.0000000254%
x20 = 6.00 RK4 y20 = 0.7698003587 참값 y20 = 0.7698003589 절대오차 20 = 0.0000000002 상대오차 20 = 0.0000000258%
| x | y | 참값 | 절대오차 | 상대오차(%) |
|---|---|---|---|---|
| 5.00 | 1.0000000000 | 1.0000000000 | 0.0000000000 | 0.0000000000 |
| 5.05 | 0.9846573354 | 0.9846573354 | 0.0000000000 | 0.0000000031 |
| 5.10 | 0.9698572899 | 0.9698572899 | 0.0000000001 | 0.0000000059 |
| 5.15 | 0.9555687905 | 0.9555687906 | 0.0000000001 | 0.0000000083 |
| 5.20 | 0.9417632186 | 0.9417632187 | 0.0000000001 | 0.0000000104 |
| 5.25 | 0.9284141650 | 0.9284141651 | 0.0000000001 | 0.0000000123 |
| 5.30 | 0.9154972141 | 0.9154972142 | 0.0000000001 | 0.0000000140 |
| 5.35 | 0.9029897537 | 0.9029897539 | 0.0000000001 | 0.0000000155 |
| 5.40 | 0.8908708062 | 0.8908708064 | 0.0000000002 | 0.0000000169 |
| 5.45 | 0.8791208790 | 0.8791208791 | 0.0000000002 | 0.0000000181 |
| 5.50 | 0.8677218311 | 0.8677218313 | 0.0000000002 | 0.0000000192 |
| 5.55 | 0.8566567548 | 0.8566567549 | 0.0000000002 | 0.0000000202 |
| 5.60 | 0.8459098687 | 0.8459098689 | 0.0000000002 | 0.0000000211 |
| 5.65 | 0.8354664231 | 0.8354664233 | 0.0000000002 | 0.0000000219 |
| 5.70 | 0.8253126139 | 0.8253126141 | 0.0000000002 | 0.0000000226 |
| 5.75 | 0.8154355061 | 0.8154355063 | 0.0000000002 | 0.0000000233 |
| 5.80 | 0.8058229638 | 0.8058229640 | 0.0000000002 | 0.0000000239 |
| 5.85 | 0.7964635879 | 0.7964635881 | 0.0000000002 | 0.0000000244 |
| 5.90 | 0.7873466590 | 0.7873466592 | 0.0000000002 | 0.0000000249 |
| 5.95 | 0.7784620858 | 0.7784620860 | 0.0000000002 | 0.0000000254 |
| 6.00 | 0.7698003587 | 0.7698003589 | 0.0000000002 | 0.0000000258 |