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'''
Owner: AerospaceResearch.net
About: This module hosts the functions used for finding roots of a cubic equation
Note: Please try to maintain proper documentation
Logic Description:
'''
import math
import copy
from visma.io.checks import getVariables
from visma.io.parser import tokensToString
from visma.functions.structure import Expression
from visma.functions.constant import Constant, Zero
from visma.functions.variable import Variable
from visma.functions.operator import Binary, Plus, Minus, Sqrt
from visma.simplify.simplify import simplifyEquation, moveRTokensToLTokens
from visma.config.values import ROUNDOFF
def getRootsCubic(coeffs):
""" Applies an implementation of Cardano's Method (https://en.wikipedia.org/wiki/Cubic_function) on the coefficients
of the cubic equation roots
Arguments:
coeffs {list} -- list of coefficients of the equation
Returns:
roots {list} -- list of roots of cubic equation
(each element of roots {list} is a list of two elements, where 1st one denotes real part & second part shows imaginary part)
animation {list} -- list of equation solving process
comments {list} -- list of comments in equation solving process
"""
from visma.solvers.polynomial.roots import cubeRoot
roots = []
animations = []
comments = []
a = coeffs[3]
b = coeffs[2]
c = coeffs[1]
d = coeffs[0]
f = ((3*c/a) - (b**2/a**2))/3
g = ((2*(b**3)/(a**3)) - (9*b*c/(a**2)) + (27*d/a))/27
h = ((g**2)/4) + ((f**3)/27)
animations += [[]]
comments += [['Value of determinants [f, g, h] are ' + str(f) + ', ' + str(g) + ', ' + str(h)]]
if h <= 0:
if h == 0 and g == 0 and f == 0:
# All three (real) roots exist and are equal
animations += [[]]
comments += [['Hence, three equal real roots exist.']]
res = cubeRoot(d/a)
valueX1 = [-res, 0]
valueX2 = [-res, 0]
valueX3 = [-res, 0]
roots.append(valueX1)
else:
# All three (real) roots exist
animations += [[]]
comments += [['Hence, three equal non-equal real roots exist.']]
i = (((g**2)/4) - h) ** (1./2.)
j = cubeRoot(i)
k = math.acos(-g/(2*i))
L = j * (-1)
M = math.cos(k/3)
N = math.sqrt(3) * (math.sin(k/3))
P = -(b/(3*a))
valueX1 = [2*j*(math.cos(k/3)) - (b/(3*a)), 0]
valueX2 = [L*(M + N) + P, 0]
valueX3 = [L*(M - N) + P, 0]
roots.extend([valueX1, valueX2, valueX3])
else:
# Only one (real) root exists
animations += [[]]
comments += [['Hence, one real root exists']]
R = -(g/2) + h ** (1./2.)
S = cubeRoot(R)
T = -(g/2) - (h ** (1./2.))
U = cubeRoot(T)
valueX1 = [(S + U) - (b/(3*a)), 0]
valueRealX2 = -(S + U)/2 - (b/(3*a))
valueImagX2 = (S - U)*(3 ** (1./2.))/2
valueX2 = [valueRealX2, valueImagX2]
valueRealX3 = -(S + U)/2 - (b/(3*a))
valueImagX3 = -(S - U)*(3 ** (1./2.))/2
valueX3 = [valueRealX3, valueImagX3]
roots.extend([valueX1, valueX2, valueX3])
return roots, animations, comments
def cubicRoots(lTokens, rTokens):
'''Used to get roots of a cubic equation
This functions also translates roots {list} into final result of solution
Argument:
lTokens {list} -- list of LHS tokens
rTokens {list} -- list of RHS tokens
Returns:
lTokens {list} -- list of LHS tokens
rTokens {list} -- list of RHS tokens
{empty list}
token_string {string} -- final result stored in a string
animation {list} -- list of equation solving process
comments {list} -- list of comments in equation solving process
'''
from visma.solvers.polynomial.roots import getCoefficients
animations = []
comments = []
lTokens, rTokens, _, token_string, animNew1, commentNew1 = simplifyEquation(lTokens, rTokens)
animations.extend(animNew1)
comments.extend(commentNew1)
if len(rTokens) > 0:
lTokens, rTokens = moveRTokensToLTokens(lTokens, rTokens)
coeffs = getCoefficients(lTokens, rTokens, 3)
var = getVariables(lTokens)
roots, animNew2, commentNew2 = getRootsCubic(coeffs)
animations.extend(animNew2)
comments.extend(commentNew2)
tokens1 = []
expression1 = Expression(coefficient=1, power=3)
variable = Variable(1, var[0], 1)
tokens1.append(variable)
if roots[0][1] == 0:
binary = Binary()
if roots[0][0] < 0:
roots[0][0] *= -1
binary.value = '+'
else:
binary.value = '-'
tokens1.append(binary)
constant = Constant(round(roots[0][0], ROUNDOFF), 1)
tokens1.append(constant)
expression1.tokens = tokens1
lTokens = [expression1, Binary('*')]
if len(roots) > 1:
expression1.power = 1
for _, root in enumerate(roots[1:]):
tokens2 = []
expression2 = Expression(coefficient=1, power=1)
variable = Variable(1, var[0], 1)
tokens2.append(variable)
binary = Binary()
if root[1] == 0:
if root[0] < 0:
root[0] *= -1
binary.value = '+'
else:
binary.value = '-'
tokens2.append(binary)
constant = Constant(round(root[0], ROUNDOFF), 1)
tokens2.append(constant)
else:
binary.value = '-'
tokens2.append(binary)
expressionResult = Expression(coefficient=1, power=1)
tokensResult = []
real = Constant(round(root[0], ROUNDOFF), 1)
tokensResult.append(real)
imaginary = Constant(round(root[1], ROUNDOFF), 1)
if imaginary.value < 0:
tokensResult.append(Minus())
imaginary.value = abs(imaginary.value)
tokensResult.append(imaginary)
else:
tokensResult.extend([Plus(), imaginary])
sqrt = Sqrt(Constant(2, 1), Constant(-1, 1))
tokensResult.append(Binary('*'))
tokensResult.append(sqrt)
expressionResult.tokens = tokensResult
tokens2.append(expressionResult)
expression2.tokens = tokens2
lTokens.extend([expression2, Binary('*')])
lTokens.pop()
rTokens = [Zero()]
tokenToStringBuilder = copy.deepcopy(lTokens)
tokLen = len(lTokens)
equalTo = Binary()
equalTo.scope = [tokLen]
equalTo.value = '='
tokenToStringBuilder.append(equalTo)
tokenToStringBuilder.extend(rTokens)
token_string = tokensToString(tokenToStringBuilder)
animations.append(copy.deepcopy(tokenToStringBuilder))
comments.append([])
return lTokens, rTokens, [], token_string, animations, comments
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