1 | #!/usr/bin/env python |
---|
2 | """Auxiliary numerical tools |
---|
3 | |
---|
4 | """ |
---|
5 | |
---|
6 | |
---|
7 | #Establish which Numeric package to use |
---|
8 | #(this should move to somewhere central) |
---|
9 | try: |
---|
10 | from scipy import ArrayType, array, sum, innerproduct, ravel, sqrt, searchsorted, sort, concatenate, Float |
---|
11 | except: |
---|
12 | #print 'Could not find scipy - using Numeric' |
---|
13 | from Numeric import ArrayType, array, sum, innerproduct, ravel, sqrt, searchsorted, sort, concatenate, Float |
---|
14 | |
---|
15 | # Getting an infinite number to use when using Numeric |
---|
16 | INF = (array([1])/0.)[0] |
---|
17 | |
---|
18 | |
---|
19 | |
---|
20 | def angle(v1, v2=None): |
---|
21 | """Compute angle between 2D vectors v1 and v2. |
---|
22 | |
---|
23 | If v2 is not specified it will default |
---|
24 | to e1 (the unit vector in the x-direction) |
---|
25 | |
---|
26 | The angle is measured as a number in [0, 2pi] from v2 to v1. |
---|
27 | """ |
---|
28 | from math import acos, pi, sqrt |
---|
29 | |
---|
30 | # Prepare two Numeric vectors |
---|
31 | if v2 is None: |
---|
32 | v2 = [1.0, 0.0] # Unit vector along the x-axis |
---|
33 | |
---|
34 | v1 = ensure_numeric(v1, Float) |
---|
35 | v2 = ensure_numeric(v2, Float) |
---|
36 | |
---|
37 | # Normalise |
---|
38 | v1 = v1/sqrt(sum(v1**2)) |
---|
39 | v2 = v2/sqrt(sum(v2**2)) |
---|
40 | |
---|
41 | # Compute angle |
---|
42 | p = innerproduct(v1, v2) |
---|
43 | c = crossproduct_length(v1, v2) |
---|
44 | |
---|
45 | theta = acos(p) |
---|
46 | |
---|
47 | # Correct if v1 is in quadrant 3 or 4 with respect to v2 (as the x-axis) |
---|
48 | # If v2 was the unit vector [1,0] this would correspond to the test |
---|
49 | # if v1[1] < 0: theta = 2*pi-theta |
---|
50 | # In general we use the sign of the cross product length |
---|
51 | if c > 0: |
---|
52 | #Quadrant 3 or 4 |
---|
53 | theta = 2*pi-theta |
---|
54 | |
---|
55 | return theta |
---|
56 | |
---|
57 | |
---|
58 | def anglediff(v0, v1): |
---|
59 | """Compute difference between angle of vector v0 (x0, y0) and v1 (x1, y1). |
---|
60 | This is used for determining the ordering of vertices, |
---|
61 | e.g. for checking if they are counter clockwise. |
---|
62 | |
---|
63 | Always return a positive value |
---|
64 | """ |
---|
65 | |
---|
66 | from math import pi |
---|
67 | |
---|
68 | a0 = angle(v0) |
---|
69 | a1 = angle(v1) |
---|
70 | |
---|
71 | #Ensure that difference will be positive |
---|
72 | if a0 < a1: |
---|
73 | a0 += 2*pi |
---|
74 | |
---|
75 | return a0-a1 |
---|
76 | |
---|
77 | def normal_vector(v): |
---|
78 | """Normal vector to v |
---|
79 | """ |
---|
80 | |
---|
81 | return array([-v[1], v[0]], Float) |
---|
82 | |
---|
83 | |
---|
84 | def crossproduct_length(v1, v2): |
---|
85 | return v1[0]*v2[1]-v2[0]*v1[1] |
---|
86 | |
---|
87 | |
---|
88 | def mean(x): |
---|
89 | """Mean value of a vector |
---|
90 | """ |
---|
91 | return(float(sum(x))/len(x)) |
---|
92 | |
---|
93 | |
---|
94 | def cov(x, y=None): |
---|
95 | """Covariance of vectors x and y. |
---|
96 | |
---|
97 | If y is None: return cov(x, x) |
---|
98 | """ |
---|
99 | |
---|
100 | if y is None: |
---|
101 | y = x |
---|
102 | |
---|
103 | assert(len(x)==len(y)) |
---|
104 | N = len(x) |
---|
105 | |
---|
106 | cx = x - mean(x) |
---|
107 | cy = y - mean(y) |
---|
108 | |
---|
109 | p = innerproduct(cx,cy)/N |
---|
110 | return(p) |
---|
111 | |
---|
112 | |
---|
113 | def err(x, y=0, n=2, relative=True): |
---|
114 | """Relative error of ||x-y|| to ||y|| |
---|
115 | n = 2: Two norm |
---|
116 | n = None: Max norm |
---|
117 | |
---|
118 | If denominator evaluates to zero or if y is omitted, |
---|
119 | absolute error is returned |
---|
120 | """ |
---|
121 | |
---|
122 | x = ensure_numeric(x) |
---|
123 | if y: |
---|
124 | y = ensure_numeric(y) |
---|
125 | |
---|
126 | if n == 2: |
---|
127 | err = norm(x-y) |
---|
128 | if relative is True: |
---|
129 | try: |
---|
130 | err = err/norm(y) |
---|
131 | except: |
---|
132 | pass |
---|
133 | |
---|
134 | else: |
---|
135 | err = max(abs(x-y)) |
---|
136 | if relative is True: |
---|
137 | try: |
---|
138 | err = err/max(abs(y)) |
---|
139 | except: |
---|
140 | pass |
---|
141 | |
---|
142 | return err |
---|
143 | |
---|
144 | |
---|
145 | def norm(x): |
---|
146 | """2-norm of x |
---|
147 | """ |
---|
148 | |
---|
149 | y = ravel(x) |
---|
150 | p = sqrt(innerproduct(y,y)) |
---|
151 | return p |
---|
152 | |
---|
153 | |
---|
154 | def corr(x, y=None): |
---|
155 | """Correlation of x and y |
---|
156 | If y is None return autocorrelation of x |
---|
157 | """ |
---|
158 | |
---|
159 | from math import sqrt |
---|
160 | if y is None: |
---|
161 | y = x |
---|
162 | |
---|
163 | varx = cov(x) |
---|
164 | vary = cov(y) |
---|
165 | |
---|
166 | if varx == 0 or vary == 0: |
---|
167 | C = 0 |
---|
168 | else: |
---|
169 | C = cov(x,y)/sqrt(varx * vary) |
---|
170 | |
---|
171 | return(C) |
---|
172 | |
---|
173 | |
---|
174 | |
---|
175 | def ensure_numeric(A, typecode = None): |
---|
176 | """Ensure that sequence is a Numeric array. |
---|
177 | Inputs: |
---|
178 | A: Sequence. If A is already a Numeric array it will be returned |
---|
179 | unaltered |
---|
180 | If not, an attempt is made to convert it to a Numeric |
---|
181 | array |
---|
182 | typecode: Numeric type. If specified, use this in the conversion. |
---|
183 | If not, let Numeric decide |
---|
184 | |
---|
185 | This function is necessary as array(A) can cause memory overflow. |
---|
186 | """ |
---|
187 | |
---|
188 | if typecode is None: |
---|
189 | if type(A) == ArrayType: |
---|
190 | return A |
---|
191 | else: |
---|
192 | return array(A) |
---|
193 | else: |
---|
194 | if type(A) == ArrayType: |
---|
195 | if A.typecode == typecode: |
---|
196 | return array(A) #FIXME: Shouldn't this just return A? |
---|
197 | else: |
---|
198 | return A.astype(typecode) |
---|
199 | else: |
---|
200 | return array(A).astype(typecode) |
---|
201 | |
---|
202 | |
---|
203 | |
---|
204 | |
---|
205 | def histogram(a, bins): |
---|
206 | """Standard histogram straight from the Numeric manual |
---|
207 | """ |
---|
208 | |
---|
209 | n = searchsorted(sort(a), bins) |
---|
210 | n = concatenate( [n, [len(a)]] ) |
---|
211 | return n[1:]-n[:-1] |
---|
212 | |
---|
213 | |
---|
214 | |
---|
215 | #################################################################### |
---|
216 | #Python versions of function that are also implemented in numerical_tools_ext.c |
---|
217 | # |
---|
218 | |
---|
219 | def gradient_python(x0, y0, x1, y1, x2, y2, q0, q1, q2): |
---|
220 | """ |
---|
221 | """ |
---|
222 | |
---|
223 | det = (y2-y0)*(x1-x0) - (y1-y0)*(x2-x0) |
---|
224 | a = (y2-y0)*(q1-q0) - (y1-y0)*(q2-q0) |
---|
225 | a /= det |
---|
226 | |
---|
227 | b = (x1-x0)*(q2-q0) - (x2-x0)*(q1-q0) |
---|
228 | b /= det |
---|
229 | |
---|
230 | return a, b |
---|
231 | |
---|
232 | |
---|
233 | def gradient2_python(x0, y0, x1, y1, q0, q1): |
---|
234 | """Compute radient based on two points and enforce zero gradient |
---|
235 | in the direction orthogonal to (x1-x0), (y1-y0) |
---|
236 | """ |
---|
237 | |
---|
238 | #Old code |
---|
239 | #det = x0*y1 - x1*y0 |
---|
240 | #if det != 0.0: |
---|
241 | # a = (y1*q0 - y0*q1)/det |
---|
242 | # b = (x0*q1 - x1*q0)/det |
---|
243 | |
---|
244 | #Correct code (ON) |
---|
245 | det = (x1-x0)**2 + (y1-y0)**2 |
---|
246 | if det != 0.0: |
---|
247 | a = (x1-x0)*(q1-q0)/det |
---|
248 | b = (y1-y0)*(q1-q0)/det |
---|
249 | |
---|
250 | return a, b |
---|
251 | |
---|
252 | |
---|
253 | ############################################## |
---|
254 | #Initialise module |
---|
255 | |
---|
256 | from utilities import compile |
---|
257 | if compile.can_use_C_extension('util_ext.c'): |
---|
258 | from util_ext import gradient, gradient2 |
---|
259 | else: |
---|
260 | gradient = gradient_python |
---|
261 | gradient2 = gradient2_python |
---|
262 | |
---|
263 | |
---|
264 | if __name__ == "__main__": |
---|
265 | pass |
---|
266 | |
---|
267 | |
---|
268 | |
---|
269 | def angle_obsolete(v): |
---|
270 | """Compute angle between e1 (the unit vector in the x-direction) |
---|
271 | and the specified vector v. |
---|
272 | |
---|
273 | Return a number in [0, 2pi] |
---|
274 | """ |
---|
275 | from math import acos, pi, sqrt |
---|
276 | |
---|
277 | # Normalise v |
---|
278 | v = ensure_numeric(v, Float) |
---|
279 | v = v/sqrt(sum(v**2)) |
---|
280 | |
---|
281 | # Compute angle |
---|
282 | theta = acos(v[0]) |
---|
283 | |
---|
284 | if v[1] < 0: |
---|
285 | #Quadrant 3 or 4 |
---|
286 | theta = 2*pi-theta |
---|
287 | |
---|
288 | return theta |
---|
289 | |
---|