def func(t, args):
# args=[f,y0,x,h,arg,i]
cop = [_ for _ in args[1]]
i = args[5]
cop[i] = t
c = call_func(args[0], args[4], cop, args[2] + args[3])
if type(c) != tuple and type(c) != list: c = [c]
return -t + args[1][i] + args[3] * c[i]
def rk2(f, y0, x0, xstop, N, args=None):
h = (xstop - x0) / N
result = [y0]
c = [x0]
iter = 0
while iter < N:
iter += 1
d = call_func(f, args, result[-1], x0)
k1_by_2 = h * d / 2
yn = result[-1] + k1_by_2
d = call_func(f, args, yn, x0 + h / 2)
k2 = h * d
yn = result[-1] + k2
result.append(yn)
x0 += h
c.append(x0)
return c, result
def rk2(f, y0, x0, xstop, N=1000, args=None):
h = (xstop - x0) / N
if type(y0) != list and type(y0) != tuple: y0 = [y0]
result = [y0]
c = [x0]
while abs(x0 - xstop) >= abs(h):
d = call_func(f, args, result[-1], x0)
if type(d) != list and type(d) != tuple: d = [d]
k1_by_2 = mul(h / 2, d)
yn = add(result[-1], k1_by_2)
d = call_func(f, args, yn, x0 + h / 2)
if type(d) != list: d = [d]
k2 = mul(h, d)
yn = add(result[-1], k2)
result.append(yn)
x0 += h
c.append(x0)
return result, c
def predictor_corrector(f, y0, x0, xstop, N, args=None):
h = (xstop - x0) / N
result = [y0]
c = [x0]
iter = 0
while iter < N:
iter += 1
d = call_func(f, args, result[-1], x0)
k1 = h * d
yp = result[-1] + k1 #predicted
d = call_func(f, args, yp, x0 + h)
k2_p = h * d #predicted k2
k = (k2_p + k1) / 2 # h(f(y0,x0) +f(y1_p,x0+h)
yc = result[-1] + k #corrected
result.append(yc)
x0 += h
c.append(x0)
return c, result
def predictor_corrector(f, y0, x0, xstop, N=1000, args=None):
h = (xstop - x0) / N
if type(y0) != list: y0 = [y0]
result = [y0]
c = [x0]
while abs(x0 - xstop) >= abs(h):
d = call_func(f, args, result[-1], x0)
if type(d) != list and type(d) != tuple: d = [d]
k1 = mul(h, d)
yp = add(result[-1], k1) #predicted
d = call_func(f, args, yp, x0 + h)
if type(d) != list: d = [d]
k2_p = mul(h, d)
k_into_2 = add(k2_p, k1) # h(f(y0,x0) +f(y1_p,x0+h)
k = mul(0.5, k_into_2)
yc = add(result[-1], k) #corrected
result.append(yc)
x0 += h
c.append(x0)
return result, c
def rk4(f, y0, x0, xstop, N, args=None):
h = (xstop - x0) / N
if type(y0) != list and type(y0) != tuple: y0 = [y0]
prev_result = y0
y = [[prev_result[i]] for i in range(len(y0))]
c = [x0]
iter = 0
while iter < N:
iter += 1
#finding k1/2 and k1/6
d = call_func(f, args, prev_result, x0)
if type(d) != list and type(d) != tuple: d = [d]
k1_by_2 = mul(h / 2, d)
k1_by_6 = mul(h / 6, d)
#finding k2/2 and k2/3
yn = add(prev_result, k1_by_2)
d = call_func(f, args, yn, x0 + h / 2)
if type(d) != list: d = [d]
k2_by_2 = mul(h / 2, d)
k2_by_3 = mul(h / 3, d)
#finding k3
yn = add(prev_result, k2_by_2)
d = call_func(f, args, yn, x0 + h / 2)
if type(d) != list: d = [d]
k3 = mul(h, d)
k3_by_3 = mul(h / 3, d)
#finding k4
yn = add(prev_result, k3)
d = call_func(f, args, yn, x0 + h)
if type(d) != list: d = [d]
k4_by_6 = mul(h / 6, d)
#finding the correct yn
yn = add(prev_result, k1_by_6, k2_by_3, k3_by_3, k4_by_6)
prev_result = yn
for i in range(len(y0)):
y[i].append(prev_result[i])
x0 += h
c.append(x0)
return c, y
def euler_forward(f, y0, x0, xstop, N=1000, args=None):
h = (xstop - x0) / N
if type(y0) != list: y0 = [y0]
result = [y0]
c = [x0]
while abs(x0 - xstop) >= abs(h):
d = call_func(f, args, result[-1], x0)
if type(d) != list: d = [d]
k1 = mul(h, d)
yn = add(result[-1], k1)
result.append(yn)
x0 += h
c.append(x0)
return result, c
def euler_forward(f, y0, x0, xstop, N, args=None):
h = (xstop - x0) / N
result = [y0]
c = [x0]
iter = 0
while iter < N:
iter += 1
d = call_func(f, args, result[-1], x0)
k1 = h * d
yn = result[-1] + k1
result.append(yn)
x0 += h
c.append(x0)
return c, result
def rk4(f, y0, x0, xstop, N=1000, args=None):
h = (xstop - x0) / N
if type(y0) != list and type(y0) != tuple: y0 = [y0]
result = [y0]
c = [x0]
while abs(x0 - xstop) >= abs(h):
#finding k1 and k1/2
d = call_func(f, args, result[-1], x0)
if type(d) != list and type(d) != tuple: d = [d]
k1 = mul(h, d)
k1_by_2 = mul(h / 2, d)
k1_by_6 = mul(h / 6, d)
#finding k2 and k2/2
yn = add(result[-1], k1_by_2)
d = call_func(f, args, yn, x0 + h / 2)
if type(d) != list: d = [d]
k2 = mul(h, d)
k2_by_2 = mul(h / 2, d)
k2_by_3 = mul(h / 3, d)
#finding k3
yn = add(result[-1], k2_by_2)
d = call_func(f, args, yn, x0 + h / 2)
if type(d) != list: d = [d]
k3 = mul(h, d)
k3_by_3 = mul(h / 3, d)
#finding k4
yn = add(result[-1], k3)
d = call_func(f, args, yn, x0 + h)
if type(d) != list: d = [d]
k4 = mul(h, d)
k4_by_6 = mul(h / 6, d)
#finding the correct yn
yn = add(result[-1], k1_by_6, k2_by_3, k3_by_3, k4_by_6)
result.append(yn)
x0 += h
c.append(x0)
return result, c
def euler_backward(f, y0, x0, xstop, N, args=None):
h = (xstop - x0) / N
result = [y0]
xpar = [x0]
iter = 0
while iter < N:
iter += 1
list_of_args = [f, result[-1], x0, h, args]
sol = newton_raphson(func, result[-1], 0.0001, list_of_args)
d = call_func(f, args, sol, x0 + h)
d1 = h * d
d2 = d1 + result[-1]
result.append(d2)
x0 += h
xpar.append(x0)
return xpar, result
def euler_backward(f, y0, x0, xstop, N=1000, args=None):
h = (xstop - x0) / N
if type(y0) != tuple and type(y0) != list: y0 = [y0]
result = [y0]
xpar = [x0]
while (abs(xstop - x0) >= abs(h)):
list_of_args = [f, result[-1], x0, h, args, 0]
newtonn = []
for i in range(len(y0)):
list_of_args[5] = i
sol = newton_raphson(func, result[-1][i], 0.0001, list_of_args)
newtonn.append(sol)
#result.append(result[-1]+h*call_func(f,args,newton,x0+h))
d = call_func(f, args, newtonn, x0 + h)
if type(d) != tuple and type(d) != list: d = [d]
d1 = mul(h, d)
d2 = add(d1, result[-1])
result.append(d2)
x0 += h
xpar.append(x0)
return result, xpar
def func(t, args):
# args=[f,y0,x,h,arg]
c = call_func(args[0], args[4], t, args[2] + args[3])
return -t + args[1] + args[3] * c