[SciPy-user] odeint and complex coupled differential equations

[Nils Wagner]

Mark Everitt wrote:
> Hi everyone,
>
> I'm having a problem with odeint and complex numbers. I have a  
> coupled differential equation:
>
> def hyper10101(c,t,d,g):
> 	dc = array(zeros(11,Complex32))
> 	dc[0]   = 1j*d[2]*c[0]  + 2j*d[4]*c[0]  - 1j*g[4]*c[1]*sqrt(2)
> 	dc[1]   = 1j*d[2]*c[1]  + 1j*d[4]*c[1]  - 1j*g[4]*c[0]*sqrt(2) - 1j*g 
> [5]*c[2]*exp(1j*(d[5]-d[0])*t)
> 	dc[2]   = 1j*d[0]*c[2]  + 1j*d[2]*c[2]  + 1j*d[4]*c[2] - 1j*g[5]*c[1] 
> *exp(1j*(d[0]-d[5])*t) - 1j*g[0]*c[3]
> 	dc[3]   = 1j*d[2]*c[3]  + 1j*d[4]*c[3]  - 1j*g[0]*c[2] - 1j*g[1]*c[4]
> 	dc[4]   = 1j*d[1]*c[4]  + 1j*d[2]*c[4]  + 1j*d[4]*c[4] - 1j*g[1]*c 
> [3] - 1j*g[2]*c[5]
> 	dc[5]   = 1j*d[1]*c[5]  + 1j*d[4]*c[5]  - 1j*g[2]*c[4] - 1j*g[3]*c[6];
> 	dc[6]   = 1j*d[1]*c[6]  + 1j*d[3]*c[6]  + 1j*d[4]*c[6] - 1j*g[3]*c 
> [5] - 1j*g[4]*c[7] #- 1j*g[6]*c[9]
> 	dc[7]   = 1j*d[1]*c[7]  + 1j*d[3]*c[7]  - 1j*g[4]*c[6] - 1j*g[5]*c[8] 
> *exp(1j*(d[5]-d[0])*t)
> 	dc[8]   = 1j*d[0]*c[8]  + 1j*d[1]*c[8]  + 1j*d[3]*c[8] - 1j*g[5]*c[7] 
> *exp(1j*(d[0]-d[5])*t) - 1j*g[0]*c[9]
> 	dc[9]   = 1j*d[1]*c[9]  + 1j*d[3]*c[9]  - 1j*g[0]*c[8] - 1j*g[1]*c 
> [10]*sqrt(2)
> 	dc[10] = 2j*d[1]*c[10] + 1j*d[3]*c[10] - 1j*g[1]*c[9]*sqrt(2)
> 	return dc
>
> d = array([d1,d2,d3,d4,d5,d6])
> g = array([gab,gbc,gcd,gde,gef,gfa])
>
> And the elements of these are floats. I call odeint like this:
>
> Y = odeint(hyper10101,c10101,tt,args=(d,g))
>
> This returns the error:
>
> TypeError: array cannot be safely cast to required type
> odepack.error: Result from function call is not a proper array of  
> floats.
>
> Are complex numbers broken for this, or am I just missing something?
>
> Mark
>
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>   
AFAIK. you can't use odeint with complex inputs, but you can double the
size of your ODE

\dot{z} = f(z,t)

\dot{x}+i \dot{y} = \Re{f} + i \Im{f}

where i = \sqrt{-1}

\dot{x} = \Re{f(z,t)}
\dot{y} = \Im{f(z,t)}

The initial conditions are

x_0 = \Re{z_0}
y_0=\Im{z_0}

Nils