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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