z1 = complex(0, 0)
z2 = complex(0, 0)
z3 = complex(0, 0)
z4 = complex(0, 0)
def complex_quadratic_formula(a, b, c):
import cmath
z1 = complex((-b + cmath.sqrt(b * b - 4 * a * c)) / (2 * a))
z2 = complex((-b - cmath.sqrt(b * b - 4 * a * c)) / (2 * a))
print("z1 = ", z1)
print("z2 = ", z2)
def real_quadratic_formula(a, b, c):
if b * b - 4 * a * c >= 0:
import math
z1 = float((-b + math.sqrt(b * b - 4 * a * c)) / (2 * a))
z2 = float((-b - math.sqrt(b * b - 4 * a * c)) / (2 * a))
else:
import cmath
z1 = complex((-b + cmath.sqrt(b * b - 4 * a * c)) / (2 * a))
z2 = complex((-b - cmath.sqrt(b * b - 4 * a * c)) / (2 * a))
print("z1 = ", z1)
print("z2 = ", z2)
def cubic_equation(A, B, C, D):
# https://en.wikipedia.org/wiki/Cubic_function
# http://www.wolframalpha.com/widgets/view.jsp?id=3f4366aeb9c157cf9a30c90693eafc55
import cmath
import math
A2 = float(A * A)
B2 = float(B * B)
B3 = float(B * B * B)
C2 = float(C * C)
C3 = float(C * C * C)
D2 = float(D * D)
Delta = float(18 * A * B * C * D - 4.0 * B3 * D
+ B2 * C2 - 4.0 * A * C3 - 27.0 * A2 * D2)
Delta0 = float(B2 - 3.0 * A * C)
Delta1 = float(2.0 * B3 - 9.0 * A * B * C + 27.0 * A2 * D)
r = cmath.sqrt(-27.0 * A2 * Delta)
c = complex(0.5 * (Delta1 + r)) ** (1.0 / 3.0)
d = complex(0.5 * (Delta1 - r)) ** (1.0 / 3.0)
if c.real == 0 and c.imag == 0:
c = d
i = complex(0.0, 1.0)
chi = complex(-0.5, 0) + i * 0.5 * math.sqrt(3)
import array
ar = array.array('f', [0,0,0])
ai = array.array('f', [0,0,0])
for k in range(0, 3, 1):
chik = chi ** k
z = -(B + chik * c + Delta0 / (chik * c)) / (3.0 * A)
ar[k] = z.real
ai[k] = z.imag;
z1 = ar[0] + ai[0] * complex(0, 1);
z2 = ar[1] + ai[1] * complex(0, 1);
z3 = ar[2] + ai[2] * complex(0, 1);
print("z1 = ", z1)
print("z2 = ", z2)
print("z3 = ", z3)
def quartic_equation(A, B, C, D, E):
# https://en.wikipedia.org/wiki/Quartic_function
# https://keisan.casio.com/exec/system/1181809416
import cmath
A2 = float(A * A)
A3 = float(A * A2)
A4 = float(A2 * A2)
B2 = float(B * B)
B3 = float(B * B2)
B4 = float(B2 * B2)
C2 = float(C * C)
C3 = float(C * C2)
C4 = float(C2 * C2)
D2 = float(D * D)
D3 = float(D * D2)
D4 = float(D2 * D2)
E2 = float(E * E)
E3 = float(E * E2)
E4 = float(E2 * E2)
Delta = float(256 * A3 * E3 - 192 * A2 * B * D * E2
- 128 * A2 * C2 * E2 + 144 * A2 * C * D2 * E
- 27 * A2 * D4 + 144 * A * B2 * C * E2
- 6 * A * B2 * D2 * E - 80 * A * B * C2 * D * E
+ 18 * A * B * C * D3 + 16 * A * C4 * E
- 4 * A * C3 * D2 - 27 * B4 * E2 + 18 * B3 * C * D * E
- 4 * B3 * D3 - 4 * B2 * C3 * E + B2 * C2 * D2)
Delta0 = float(C2 - 3 * B * D + 12 * A * E)
Delta1 = float(2 * C3 - 9 * B * C * D
+ 27 * B2 * E + 27 * A * D2 - 72 * A * C * E)
p = float((8 * A * C - 3 * B2) / (8 * A2))
q = float((B3 - 4 * A * B * C + 8 * A2 * D) / (8 * A3))
R = cmath.sqrt(Delta1 * Delta1 - 4 * Delta0 * Delta0 * Delta0)
P = cmath.sqrt(-27 * Delta)
Q = complex(0.5 * (Delta1 + P)) ** complex(1.0 / 3.0)
S = 0.5 * cmath.sqrt(((-2.0 * p) / 3.0)
+ (Q + Delta0 / Q) / (3 * A))
a = complex(-B / (4 * A), 0)
s = 0.5 * cmath.sqrt(-4 * S * S - 2 * p + q / S)
z1 = a - S + s
z2 = a - S - s
s = 0.5 * cmath.sqrt(-4 * S * S - 2 * p - q / S)
z3 = a + S + s
z4 = a + S - s
print("z1 = ", z1)
print("z2 = ", z2)
print("z3 = ", z3)
print("z4 = ", z4)
option = int(1)
print("Menu")
print("1 Solve a linear equation")
print("2 Solve a quadratic eqution")
print("3 Solve a cubic equation")
print("4 Solve a quartic equation")
print("5 Quit")
option = int(input("Option 1 to 5: "))
while option != 5:
if option >= 1:
a = float(input("a = "))
b = float(input("b = "))
if option >= 2:
c = float(input("c = "))
if option >= 3:
d = float(input("d = "))
if option == 4:
e = float(input("e = "))
if option == 1:
z1 = -b / a
print("z1 = ", z1)
elif option == 2:
real_quadratic_formula(a, b, c)
elif option == 3:
cubic_equation(a, b, c, d)
elif option == 4:
quartic_equation(a, b, c, d, e)
option = int(input("Option 1 to 5: "))
Numerical Explorations 