def in3(number):
t = sp.symbols('t')
if number == 1:
p0 = 1.5 # The interval, it's suppose that a < b
f = t**2 - 3 # The function
tol = 1e-5 # The max error acceptable
nmax = 10 # Max number iteractions
elif number == 2:
p0 = 3 # The interval, it's suppose that a < b
f = t**2 - 2 * t - 4 # The function
tol = 1e-5 # The max error acceptable
nmax = 10 # Max number iteractions
elif number == 3:
p0 = 1.5 # The interval, it's suppose that a < b
f = t**3 + 4 * t**2 - 10 # The function
tol = 1e-5 # The max error acceptable
nmax = 10 # Max number iteractions
f_ = sp.diff(f, t) # The function f' derivative of f
# The begin to start the calculations
feval = sp.lambdify(t, f, "numpy")
flatex = "$f(t) = " + aux.toLaTeX(f)
f = aux.Funcao(feval, flatex)
f_eval = sp.lambdify(t, f_, "numpy")
f_latex = "$f'(t) = " + aux.toLaTeX(f_)
f_ = aux.Funcao(f_eval, f_latex)
return p0, f, f_, tol, nmax
def in2(number):
f, t, x = master(number)
f_ = sp.diff(f, t)
feval = sp.lambdify(t, f, "numpy") # Transform the function to lambdify
flatex = "$f(t) = " + aux.toLaTeX(f)
f_eval = sp.lambdify(t, f_, "numpy") # Transform the function to lambdify
f_latex = "$f'(t) = " + aux.toLaTeX(f_)
f = aux.Funcao(feval, flatex)
f_ = aux.Funcao(f_eval, f_latex)
y = f.e(x)
y_ = f_.e(x)
return x, y, y_, f, f_
def in1(number):
f, t, x = master(number)
feval = sp.lambdify(t, f, "numpy") # Transform the function to lambdify
flatex = "$f(t) = " + aux.toLaTeX(f)
f = aux.Funcao(feval, flatex)
y = f.e(x)
return x, y, f # x e y são arrays, que indicam os pontos
def in10(number):
f, t, x = master(number)
f_ = sp.diff(f, t)
f_ = sp.lambdify(t, f_, "numpy")
FP = np.array((f_(x[0]), f_(x[-1])))
feval = sp.lambdify(t, f, "numpy") # Transform the function to lambdify
flatex = "$f(t) = " + aux.toLaTeX(f)
f = aux.Funcao(feval, flatex)
y = f.e(x)
return x, y, f, FP
def in2(number):
t = sp.symbols('t')
if 1 <= number <= 5:
p0 = 1.5 # The initial aproximation
f = t**3 + 4 * t**2 - 10 # The function that we want to calculate the roots
tol = 1e-5 # The max error acceptable
nmax = 10 # Max number iteractions
if number == 1:
g = t - t**3 - 4 * t**2 + 10 # It doesn't converg
elif number == 2:
g = sp.sqrt(4 * t - 10 / t) # It doesn't converg
elif number == 3:
g = sp.sqrt(
10 - t**3
) / 2 # The fixed point function, using f(x) = 0 we can get x = sqrt(10-x**3)/2
elif number == 4:
g = sp.sqrt(10 / (4 + t)) # It converges quite well
elif number == 5:
g = t - (t**3 + 4 * t**2 - 10) / (
3 * t**2 + 8 * t
) # It converges very well, we will see this funcion in the Newton's method
g_ = sp.diff(g, t) # The function g' derivative of g
feval = sp.lambdify(t, f, "numpy")
flatex = "$f(t) = " + aux.toLaTeX(f)
f = aux.Funcao(feval, flatex)
geval = sp.lambdify(t, g, "numpy")
glatex = "$g(t) = " + aux.toLaTeX(g)
g = aux.Funcao(geval, glatex)
g_eval = sp.lambdify(t, g_, "numpy")
g_latex = "$g'(t) = " + aux.toLaTeX(g_)
g_ = aux.Funcao(g_eval, g_latex)
return p0, f, g, g_, tol, nmax
def Lagrange(x, y):
# The variables
t = sp.symbols('t')
# The calculations
n = len(x)
L = []
for i in range(n):
Li = 1
for j in range(n):
if i != j:
Li *= (t-x[j])/(x[i]-x[j])
L.append(Li)
g = 0
for i in range(n):
g += L[i] * y[i]
g = sp.simplify(g)
L = aux.Funcao(sp.lambdify(t, g, "numpy"), '$L(t) = ' + aux.toLaTeX(g))
return L
def in1(number):
t = sp.symbols('t')
if number == 1:
a, b = 1, 2 # The interval, it's suppose that a < b
f = t**2 - 3 # The function
tol = 1e-5 # The max error acceptable
nmax = 10 # Max number iteractions
elif number == 2:
a, b = 2, 4 # The interval, it's suppose that a < b
f = t**2 - 2 * t - 4 # The function
tol = 1e-5 # The max error acceptable
nmax = 10 # Max number iteractions
elif number == 3:
a, b = 1, 2 # The interval, it's suppose that a < b
f = t**3 + 4 * t**2 - 10 # The function
tol = 1e-5 # The max error acceptable
nmax = 10 # Max number iteractions
# The begin to start the calculations
feval = sp.lambdify(t, f, "numpy")
flatex = "$f(t) = " + aux.toLaTeX(f)
f = aux.Funcao(feval, flatex)
return a, b, f, tol, nmax
def in1(number): f, t, a, b, n = master(number) feval = sp.lambdify(t, f, "numpy") flatex = "$f(t) = " + aux.toLaTeX(f) f = aux.Funcao(feval, flatex) return a, b, n, f