I am trying to solve a complex non-linear system. The problem is that there are many roots, and some of them are complex roots. From these roots I need to select only the one that the real part is between [0,1] and does not have complex part (ex: 0.23+0i). ex:
root1: 1.02 + 2i
root2: 0.23 + 1.23i
root3: 0.23 + 0i
...
Here is my system: tau1 and tau2 are the variables that I need to find. The equations are t1 and t2 that are dependent on tau1 and tau2
x0=0 # initial position
xf=30 # final position
x1= 10;
x2 = 20;
tf=20 # final time
tau_wp=[]
def f(tau_wp):
tau1, tau2 = tau_wp
a=(1-tau1)**5*(-10*tau2**3 +15*tau2**4 -6*tau2**5) + (1-tau1)**4*(20*tau2**3 -35*tau2**4 + 15*tau2**5) + (1-tau1)**3*(-10*tau2**3 +20*tau2**4 -10*tau2**5) + (tau2-tau1)**5
b=(1-tau2)**5*(-10*tau1**3 +15*tau1**4 -6*tau1**5) + (1-tau2)**4*(20*tau1**3 -35*tau1**4 + 15*tau1**5) + (1-tau2)**3*(-10*tau1**3 +20*tau1**4 -10*tau1**5)
den=a*b -36*tau2**5*(1-tau2)**5*tau1**5*(1-tau1)**5
p2=(-6*tau1**5*(1-tau1)**5*(xf-x0)*(10*tau2**3-15*tau2**4+6*tau2**5) \
-(xf-x0)*(10*tau1**3-15*tau1**4+6*tau1**5)*a \
+ (x1-x0)*a + (x2-x0)*(6*tau1**5*(1-tau1**5)))
p1=(a*b -36*tau2**5*(1-tau2)**5*tau1**5*(1-tau1)**5)*(-(xf-x0)*(10*tau1**3-15*tau1**4+6*tau1**5) +(x1-x0)) \
+ b*( (xf-x0)*(10*tau2**3 -15*tau2**4 +6*tau2**5)*(6*tau1**5*(1-tau1)**5) \
+ (xf-x0)*(10*tau1**3 -15*tau1**4 +6*tau1**5)*a \
- (x1-x0)*a - (x2-x0)*(6*tau1**5*(1-tau1)**5))
u0=(xf-x0)*(30*tau1**2-60*tau1**3 +30*tau1**4)+p1*tf**5/120*(60*tau1**9-270*tau1**8+480*tau1**7-420*tau1**6+180*tau1**5-30*tau1**4) \
+ p2*tf**5/120 * ((1-tau2)**5*(-30*tau1**2 +60*tau1**3 -30*tau1**4) + (1-tau2)**4*(60*tau1**2 - 140*tau1**3 +75*tau1**4) + \
(1-tau2)**3*(-30*tau1**2 +80*tau1**3 - 50*tau1**4))
u1=(xf-x0)*(30*tau2**2 - 60*tau2**3 + 30*tau2**4)+p1*tf**5/120*((1-tau1)**5*(-30*tau2**2 +60*tau1**3 -30*tau1**4) + \
(1-tau1)**4*(60*tau2**2 - 140*tau2**3 +75*tau2**4) + (1-tau1)**3*(-30*tau2**2 +80*tau2**3 - 50*tau2**4) \
+ 5*tau2**4 -20*tau2**3*tau1 +30*tau2**2*tau1**2 -20*tau2*tau1**3 +5*tau1**4) \
+ p2*tf**5/120*(60*tau2**9-270*tau2**8+480*tau2**7-420*tau2**6+180*tau2**5-30*tau2**4)
## system of nonlinear equations dependent on tau1 and tau2
t1=u0*p1 ### equation 1
t2=u1*p2 ### equation 2
return [t1,t2]
I tried to use fsolve, but with fsolve I couldn't get the complex part.
Is there any way to do this in python?
Thank you so much for your help!
It's a little confusing because you say that you "couldn't get the complex part" but in the question you say that you are looking for solutions where the imaginary part is 0 and the magnitude of the real part is between 0 and 1. If this is correct, then nsolve can solve this pair of equations if you give a good enough initial guess:
>>> from sympy import symbols
>>> v = symbols('tau1:3')
>>> nsolve(f(v), (tau1, tau2), (.5,.4))
Matrix([
[0.495387590772031],
[ 0.49736468918969]])
You can get a rough idea of where to look for roots by looking at the values of t1 and t2 for different values of tau1 and tau2. Since they should both be zero I look at the log of the sum of squares -- the smaller the better:
>>> Matrix(10,10,lambda i,j:
log(sqrt(sum([k.subs(tau1,i/10).subs(tau2,j/10)**2 for k in (t1,t2)]))).round())
Matrix([
[zoo, zoo, zoo, zoo, zoo, zoo, zoo, zoo, zoo, zoo],
[-17, -5, -6, -3, -2, -2, -3, -4, -5, -7],
[ -7, -2, -1, 0, 0, 0, 0, 0, -1, -2],
[ -1, 0, 3, 4, 3, 3, 4, 4, 3, 1],
[ 3, 4, 6, 7, 6, 4, 7, 7, 6, 4],
[ 5, 6, 8, 9, 9, 5, 9, 9, 8, 6],
[ 8, 8, 10, 11, 11, 6, 11, 11, 10, 8],
[ 9, 9, 11, 12, 12, 7, 12, 12, 11, 9],
[ 9, 10, 12, 13, 13, 7, 13, 13, 12, 10],
[ 8, 11, 13, 13, 13, 7, 13, 13, 12, 10]])
(The zoo values correspond to the trivial solution when tau1 is zero.