def _rk4(red, x0, x1, y0, a):
"""
Runge-Kutta 4th order numerical method.
"""
A = sympify(x0)._to_mpmath(mp.prec)
B = sympify(x1)._to_mpmath(mp.prec)
y_0 = [sympify(i)._to_mpmath(mp.prec) for i in y0]
h = B - A
f_0_n = 0
f_1_n = 0
f_2_n = 0
f_3_n = 0
f_0 = y_0[1:]
for i in range(a):
f_0_n += sympify(DMFsubs(red[i], A, mpm=True))._to_mpmath(mp.prec) * y_0[i]
f_0.append(f_0_n)
f_1 = [y_0[i] + f_0[i] * h / 2 for i in range(1, a)]
for i in range(a):
f_1_n += sympify(DMFsubs(red[i], A + h / 2, mpm=True))._to_mpmath(mp.prec) * (
y_0[i] + f_0[i] * h / 2
)
f_1.append(f_1_n)
f_2 = [y_0[i] + f_1[i] * h / 2 for i in range(1, a)]
for i in range(a):
f_2_n += sympify(DMFsubs(red[i], A + h / 2, mpm=True))._to_mpmath(mp.prec) * (
y_0[i] + f_1[i] * h / 2
)
f_2.append(f_2_n)
f_3 = [y_0[i] + f_2[i] * h for i in range(1, a)]
for i in range(a):
f_3_n += sympify(DMFsubs(red[i], A + h, mpm=True))._to_mpmath(mp.prec) * (
y_0[i] + f_2[i] * h
)
f_3.append(f_3_n)
sol = []
for i in range(a):
sol.append(y_0[i] + h * (f_0[i] + 2 * f_1[i] + 2 * f_2[i] + f_3[i]) / 6)
return sol
def _rk4(red, x0, x1, y0, a):
A = sympify(x0)._to_mpmath(mp.prec)
B = sympify(x1)._to_mpmath(mp.prec)
y_0 = [sympify(i)._to_mpmath(mp.prec) for i in y0]
h = B - A
f_0_n = 0
f_1_n = 0
f_2_n = 0
f_3_n = 0
f_0 = y_0[1:]
for i in range(a):
f_0_n += sympify(DMFsubs(red[i], A))._to_mpmath(mp.prec) * y_0[i]
f_0.append(f_0_n)
f_1 = [y_0[i] + f_0[i] * h / 2 for i in range(1, a)]
for i in range(a):
f_1_n += sympify(DMFsubs(red[i], A + h / 2))._to_mpmath(
mp.prec) * (y_0[i] + f_0[i] * h / 2)
f_1.append(f_1_n)
f_2 = [y_0[i] + f_1[i] * h / 2 for i in range(1, a)]
for i in range(a):
f_2_n += sympify(DMFsubs(red[i], A + h / 2))._to_mpmath(
mp.prec) * (y_0[i] + f_1[i] * h / 2)
f_2.append(f_2_n)
f_3 = [y_0[i] + f_2[i] * h for i in range(1, a)]
for i in range(a):
f_3_n += sympify(DMFsubs(red[i], A + h))._to_mpmath(
mp.prec) * (y_0[i] + f_2[i] * h)
f_3.append(f_3_n)
sol = []
for i in range(a):
sol.append(y_0[i] + h *
(f_0[i] + 2 * f_1[i] + 2 * f_2[i] + f_3[i]) / 6)
return sol
def _euler(red, x0, x1, y0, a):
A = sympify(x0)._to_mpmath(mp.prec)
B = sympify(x1)._to_mpmath(mp.prec)
y_0 = [sympify(i)._to_mpmath(mp.prec) for i in y0]
h = B - A
f_0 = y_0[1:]
f_0_n = 0
for i in range(a):
f_0_n += sympify(DMFsubs(red[i], A))._to_mpmath(mp.prec) * y_0[i]
f_0.append(f_0_n)
sol = []
for i in range(a):
sol.append(y_0[i] + h * f_0[i])
return sol
def _euler(red, x0, x1, y0, a):
"""
Euler's method for numerical integration.
From x0 to x1 with initial values given at x0 as vector y0.
"""
A = sympify(x0)._to_mpmath(mp.prec)
B = sympify(x1)._to_mpmath(mp.prec)
y_0 = [sympify(i)._to_mpmath(mp.prec) for i in y0]
h = B - A
f_0 = y_0[1:]
f_0_n = 0
for i in range(a):
f_0_n += sympify(DMFsubs(red[i], A))._to_mpmath(mp.prec) * y_0[i]
f_0.append(f_0_n)
sol = []
for i in range(a):
sol.append(y_0[i] + h * f_0[i])
return sol