/
hab_split_find_roots.py
233 lines (199 loc) · 7.4 KB
/
hab_split_find_roots.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
# -*- coding: utf-8 -*-
#from sympy.mpmath import *
import sympy.mpmath as mp
#from numpy import *
from numpy import arange, array, vectorize, zeros, where, abs, concatenate, transpose
from pylab import plot, xlabel, ylabel, legend, scatter, show
from scipy.stats._support import unique
from scipy import optimize as op
from itertools import product as iproduct
import multiprocessing
# equation parameters
p = {
'a': 0.1,
'b': 0.,
't1': 2.,
't2': 2.,
'mJ': 0.1,
'mA': 0.001,
'DJ': 1.,
'DA': 1.,
'd': 100,
'L1': 1.
}
# grid parameters
scale = 0.01
minx = -10.
maxx = 10.
miny = -0.02
maxy = 10.
def F(x, p=p):
return (1 - p['b'] * mp.exp(-x * p['t2'])) * mp.exp(x * p['t1']) \
* (mp.cosh(mp.sqrt((p['mJ'] + x)/p['DJ']) * p['L1']) \
+ mp.sinh(mp.sqrt((p['mJ'] + x)/p['DJ']) * p['L1']) \
* mp.sqrt((p['mJ'] + x)/p['DJ']) / mp.sqrt((p['mA'] + x)/p['DA']) \
* mp.coth(mp.sqrt((p['mA'] + x)/p['DA']) * p['d'])) - p['a']
# identical to above, to be used in multprocessing. The function must be pickle-friendly.
def fmult(x, p=p):
return ( x[0], x[1], F(x[0] + 1j*x[1], p=p) )
def M(x, l, p=p):
return mp.exp(mp.sqrt((p['mJ'] + l)/p['DJ']) * x) + \
((1 - p['b'] * mp.exp(-x * p['t2'])) * mp.exp(x * p['t1']) - p['a'] * mp.exp(mp.sqrt((p['mJ'] + l)/p['DJ']) * p['L1'])) / \
((1 - p['b'] * mp.exp(-x * p['t2'])) * mp.exp(x * p['t1']) - p['a'] * mp.exp(- mp.sqrt((p['mJ'] + l)/p['DJ']) * p['L1'])) \
* mp.exp(- mp.sqrt((p['mJ'] + l)/p['DJ']) * x)
def N(x, l, p=p):
'''Incomplete!!'''
return 2. * mp.sinh(sqrt((p['mJ'] + l)/p['DJ']) * p['L1']) \
* (mp.exp(mp.sqrt((p['mA'] + l)/p['DA']) * x) + \
mp.exp(mp.sqrt((p['mA'] + l)/p['DA']) * (2*p['L2'] - x)))
def mp_real(x):
return vectorize(lambda y: y.real)(x)
def mp_imag(x):
return vectorize(lambda y: y.imag)(x)
def mp_to_float(x):
def get_real(y):
if type(y) == mpc:
if y.imag == mpf(0):
return y.real
else:
raise ValueError
else:
return mp.mpf(y)
return vectorize(get_real)(x)
def mp_solve(f, p, x0=0.01, limits=None, method='muller'):
if not limits:
return mp.findroot(lambda x: f(x, p=p), x0, solver=method)
else:
if method == 'muller':
method = 'anderson'
return mp.findroot(lambda x: f(x, p=p), limits, solver=method)
def solve(f, p, x0=0.01, limits=[1e-12, 5.], method='fsolve'):
if method == 'brentq':
return op.brentq(f, limits[0], limits[1], args=p)
elif method == 'fsolve':
return op.fsolve(f, x0=x0, args=p)
def make_grid(scale, minx, maxx, miny, maxy, mult=True):
gridx = arange(minx, maxx, scale)
gridy = arange(miny, maxy, scale)
if mult:
pool = multiprocessing.Pool(None)
tasks = list(iproduct(gridx, gridy))
results = []
r = pool.map_async(fmult, tasks, callback=results.append)
r.wait() # Wait on the results
rr = zeros((len(gridx), len(gridy)), dtype='object')
def insert(x):
rr[int((x[0] - minx)/scale)][int((x[1] - miny)/scale)] = x[2]
for x in results[0]:
insert(x)
return rr
else:
return array([ [ F(i + j*1j) for j in gridy ] for i in gridx ])
def roots_plot(r, part='real', scale=0.05, offset=10.):
if part == 'real':
p = mp_real(r) > zeros(r.shape)
elif part == 'imag':
p = mp_imag(r) > zeros(r.shape)
pT = where(p == True)
pF = where(p == False)
scatter(scale*pT[0] - 10, scale*pT[1] - offset, c='b')
scatter(scale*pF[0] - offset, scale*pF[1] - offset, c='r')
xlabel('Re')
ylabel('Im')
show()
def trace_roots(r, scale=0.05, offsetx=10., offsety=0):
rroots = []
iroots = []
for i in range(r.shape[0] - 1):
for j in range(r.shape[1] - 1):
if r[i][j].real * r[i+1][j].real < 0:
rroots.append([i, j])
rroots.append([i+1, j])
#rroots.append([i+0.5, j])
if r[i][j].real * r[i][j+1].real < 0:
rroots.append([i, j])
rroots.append([i, j+1])
#rroots.append([i, j+0.5])
if r[i][j].imag * r[i+1][j].imag < 0:
iroots.append([i, j])
iroots.append([i+1, j])
#iroots.append([i+0.5, j])
if r[i][j].imag * r[i][j+1].imag < 0:
iroots.append([i, j])
iroots.append([i, j+1])
#iroots.append([i, j+0.5])
rroots = array(rroots)
iroots = array(iroots)
return (intersect(rroots, iroots), rroots, iroots)
def get_roots(f, p, guesses, methods=['muller','secant']):
roots = []
for x in guesses:
for m in methods:
try:
r = mp_solve(f, p, x0=x, method=m)
except:
pass
else:
roots.append(r)
return mp_uniq(roots)
# definitely not working!
def intersect(a, b, axis=2):
from numpy import logical_and, logical_or
return unique(a[logical_or.reduce(logical_and.reduce(a == b[:,None], axis=axis))])
def mp_uniq(seq, tol=1e-16):
'''Uniqfies a list subject to a certain tolerance. Order preserving.
Adapted from http://www.peterbe.com/plog/uniqifiers-benchmark
'''
seq = array(seq)
noDupes = [seq[0]]
[ noDupes.append(i) for i in seq if min(abs(array(noDupes) - i)) > tol ]
return noDupes
def c_plot(x, *args, **kwargs):
as_ri = lambda y: [ y.real, y.imag if type(y) == mpc else 0 ]
z = array([ as_ri(y) for y in x ])
return plot(z[:,0], z[:,1], *args, **kwargs)
def get_root_seq(f, p, x0, params, methods=['secant']):
param = params[0]
ps = params[1]
roots = [x0]
for x in ps[1:]:
p[param] = x
roots.append(get_roots(f, p, [roots[-1]], methods)[0])
# reset params dict
p[param] = ps[0]
return [ ps, roots ]
def get_real_roots(f, p, limits=[-0., 20.], scale=1e-2, method='anderson'):
x = arange(limits[0], limits[1], scale)
v = mp_to_float(vectorize(f)(x))
r = []
for i in where(v[1:] * v[:-1] < 0)[0]:
r.append(mp_solve(f, p, limits=[x[i], x[i+1]], method=method))
return array(r)
def get_all_roots_seq(f, p, x0, params, limits=[0., 20.], scale=5*1e-3, methods=['secant']):
resultc = []
for x in x0:
resultc.append(get_root_seq(f, p, x, params, methods=['secant'])[1])
resultr = []
param = params[0]
ps = params[1]
for x in ps:
p[param] = x
resultr.append(get_real_roots(f, p, limits, scale))
# reset params dict
p[param] = ps[0]
return [ array([ps]), transpose(array(resultr)), array(resultc) ]
def zero_stable(f, p, extra_par={}, limits=[-0., 30.], res=1e-2):
old_p = p.copy()
p.update(extra_par)
x = arange(limits[0], limits[1], res)
v = vectorize(lambda y: f(y, p=p))(x)
p.update(old_p)
return v.max() * v.min() > 0
def find_bif(f=F, p=p, par='L1', plimits=[0.06, 0.07], res=1e-3, limits=[0., 1.]):
return op.bisect(lambda y: -0.5 + int(zero_stable(F, p, extra_par={par: y}, limits=limits, res=res)), plimits[0], plimits[1], xtol=res)
def go():
r = make_grid(scale, minx, maxx, miny, maxy)
tr = trace_roots(r)
guesses = tr[0][:,0]*scale + minx + 1j*tr[0][:,1]*scale + miny
roots = get_roots(F, p, guesses)
return (r, tr, guesses, roots)