with differential global optimization one of the roots is found:
In [10]: differential_evolution(f,((-2,2), (-2,2), (-2,2), (-2,2), (-2,2),
(-2,2
....: ), (-2,2), (-2,2), (-2,2), (-2,2), (-2,2), (-2,2), (-2,2), (-2,2),
(-2,
....: 2), (-2,2),(-2,2), (-2,2),(-2,2), (-2,2), (-2,2)))
Out[10]:
fun: 1.8444985066496224e-09
jac: array([ 8.23037829e-06, -3.25546909e-05, -1.20137119e-05,
-3.40010940e-06, -3.18740293e-05, -3.19651128e-05,
1.18561859e-05, -5.92874802e-05, -4.87322296e-05,
1.17806110e-06, 2.53454759e-05, 6.31250916e-06,
-1.75807827e-06, 1.92369061e-06, 6.78304593e-05,
1.76279553e-05, 1.43592260e-05, 4.05940515e-05,
-4.35746205e-05, -2.54559468e-05, 3.37541106e-05])
message: 'Maximum number of iterations has been exceeded.'
nfev: 315667
nit: 1000
success: False
x: array([ 0.62224365, 1.46881062, -1.47343122, -0.85611247,
0.09195341,
0.423449 , -0.20369088, -0.76820833, -0.41410515, 0.66288976,
-1.01724249, -1.97129718, 0.63188554, 0.69403993, 0.91437698,
-0.2956973 , -0.60242029, 0.74482681, 0.45377653, 0.19310465,
0.98123914])
In [11]: res=_
In [12]: res.x
Out[12]:
array([ 0.62224365, 1.46881062, -1.47343122, -0.85611247, 0.09195341,
0.423449 , -0.20369088, -0.76820833, -0.41410515, 0.66288976,
-1.01724249, -1.97129718, 0.63188554, 0.69403993, 0.91437698,
-0.2956973 , -0.60242029, 0.74482681, 0.45377653, 0.19310465,
0.98123914])
In [13]: equations(res.x)
Out[13]:
(-2.8165037928157277e-06,
-9.5617945984893815e-06,
-1.4916310316442916e-05,
-4.1531045657239307e-06,
-5.3874567714773391e-06,
-1.5715540758742819e-05,
2.8150894887946087e-06,
7.9860279372789833e-06,
7.2868002708448287e-06,
9.2345080527356238e-06,
-2.9513712621720423e-05,
-1.2176049990486604e-05)
On Friday, April 15, 2016 at 12:32:35 AM UTC-5, Denis Akhiyarov wrote:
>
> something like this:
>
> ~ $ ptipython
> Python 3.5.1 |Anaconda 4.0.0 (64-bit)| (default, Dec 7 2015, 11:16:01)
> Type "copyright", "credits" or "license" for more information.
>
>
> IPython 4.1.2 -- An enhanced Interactive Python.
> ? -> Introduction and overview of IPython's features.
> %quickref -> Quick reference.
> help -> Python's own help system.
> object? -> Details about 'object', use 'object??' for extra details.
>
>
> In [1]: def equations(p):
> ...: (a, b, c, d, e, f,
> ...: g, h, i, j, k, l,
> ...: m, n, o, p, q, r, s, t, u)=p
> ...: return (1 - a*j*p + d*l*q + g*n*r,
> ...: 0 - b*j*p + e*l*q + h*n*r,
> ...: 0 - c*j*p + f*l*q + i*n*r,
> ...: 0 - a*k*p + d*m*q + g*o*r,
> ...: 1 - b*k*p + e*m*q + h*o*r,
> ...: 0 - c*k*p + f*m*q + i*o*r,
> ...: 0 - a*j*s + d*l*t + g*n*u,
> ...: 1 - b*j*s + e*l*t + h*n*u,
> ...: 0 - c*j*s + f*l*t + i*n*u,
> ...: 0 - a*k*s + d*m*t + g*o*u,
> ...: 0 - b*k*s + e*m*t + h*o*u,
> ...: 1 - c*k*s + f*m*t + i*o*u)
>
>
> In [2]: from scipy.optimize import fmin
>
>
> In [3]: import numpy as np
>
>
> In [4]: def f(p):return np.abs(np.sum(np.array(equations(p))**2)-0)
>
>
> In [5]: fmin(f,(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
> , 1))
> Warning: Maximum number of function evaluations has been exceeded.
> Out[5]:
> array([-0.89996132, 3.24921453, 2.05127943, -2.03527783, 1.69544993,
> 1.40438226, 1.92348097, 1.06625688, 0.24227183, 0.08901109,
> 0.09085688, 2.91819752, 0.6803978 , 4.09234832, 1.90642669,
> 2.95941855, 0.17154083, -0.02346012, 1.44161738, -0.02367669,
> -0.03451558])
>
>
> In [6]: from scipy.optimize import fmin_l_bfgs_b
>
>
> In [7]: fmin_l_bfgs_b(f,(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
> , 1, 1, 1, 1))
> ---------------------------------------------------------------------------
> IndexError Traceback (most recent call last
> )
> <ipython-input-7-b040792986e3> in <module>()
> ----> 1 fmin_l_bfgs_b(f,(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
> , 1, 1, 1, 1))
>
>
> /home/dta/anaconda3/lib/python3.5/site-packages/scipy/optimize/lbfgsb.py
> in fmin_l_bfgs_b(func, x0, fprime, args, approx_grad, bounds, m, factr,
> pgtol, epsilon, iprint, maxfun, maxiter, disp, callback, maxls)
> 191
> 192 res = _minimize_lbfgsb(fun, x0, args=args, jac=jac, bounds=
> bounds,
> --> 193 **opts)
> 194 d = {'grad': res['jac'],
> 195 'task': res['message'],
>
>
> /home/dta/anaconda3/lib/python3.5/site-packages/scipy/optimize/lbfgsb.py
> in _minimize_lbfgsb(fun, x0, args, jac, bounds, disp, maxcor, ftol, gtol,
> eps, maxfun, maxiter, iprint, callback, maxls, **unknown_options)
> 328 # minimization routine wants f and g at the
> current x
> 329 # Overwrite f and g:
> --> 330 f, g = func_and_grad(x)
> 331 elif task_str.startswith(b'NEW_X'):
> 332 # new iteration
>
>
> /home/dta/anaconda3/lib/python3.5/site-packages/scipy/optimize/lbfgsb.py
> in func_and_grad(x)
> 276 else:
> 277 def func_and_grad(x):
> --> 278 f = fun(x, *args)
> 279 g = jac(x, *args)
> 280 return f, g
>
>
> /home/dta/anaconda3/lib/python3.5/site-packages/scipy/optimize/optimize.py
> in function_wrapper(*wrapper_args)
> 287 def function_wrapper(*wrapper_args):
> 288 ncalls[0] += 1
> --> 289 return function(*(wrapper_args + args))
> 290
> 291 return ncalls, function_wrapper
>
>
> /home/dta/anaconda3/lib/python3.5/site-packages/scipy/optimize/optimize.py
> in __call__(self, x, *args)
> 62 self.x = numpy.asarray(x).copy()
> 63 fg = self.fun(x, *args)
> ---> 64 self.jac = fg[1]
> 65 return fg[0]
> 66
>
>
> IndexError: invalid index to scalar variable.
>
>
> In [8]: from scipy.optimize import fmin_bfgs
>
>
> In [9]: fmin_bfgs(f,(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
> 1, 1, 1))
> Optimization terminated successfully.
> Current function value: 2.500000
> Iterations: 12
> Function evaluations: 483
> Gradient evaluations: 21
> Out[9]:
> array([ 0.99750085, 1.13125434, 0.99750097, 0.84397729, 0.75589231,
> 0.84398644, 0.8439863 , 0.75589223, 0.84397754, 1.07736067,
> 1.07736081, 0.73335887, 0.73335281, 0.73335274, 0.73335887,
> 1.07736068, 0.73335893, 0.73335289, 1.07736063, 0.73335281,
> 0.73335887])
>
>
>
> On Wednesday, April 13, 2016 at 11:26:48 PM UTC-5, Fmwyso null wrote:
>>
>> Sorry for the vagueness of "solution set", I should have specified that
>> all I require is a unique numerical solution.
>>
>> If optimization can provide this optimal solution (with minimized norm2),
>> that would be great.
>>
>> Otherwise, if Sympy could provide a set of symbolic solutions, I could
>> construct my own minimized norm2 solution.
>>
>> On Wednesday, April 13, 2016 at 8:31:12 PM UTC-7, Denis Akhiyarov wrote:
>>>
>>> Are you trying to find set of symbolic solutions by elimination or a
>>> unique numerical solution by optimization, since it is indeterminate?
>>>
>>>
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