codekingpro/portable-devtools
114k
1"""Tests for hermite_e module.
2
3"""
4from functools import reduce
5
6import numpy as np
7import numpy.polynomial.hermite_e as herme
8from numpy.polynomial.polynomial import polyval
9from numpy.testing import assert_, assert_almost_equal, assert_equal, assert_raises
10
11He0 = np.array([1])
12He1 = np.array([0, 1])
13He2 = np.array([-1, 0, 1])
14He3 = np.array([0, -3, 0, 1])
15He4 = np.array([3, 0, -6, 0, 1])
16He5 = np.array([0, 15, 0, -10, 0, 1])
17He6 = np.array([-15, 0, 45, 0, -15, 0, 1])
18He7 = np.array([0, -105, 0, 105, 0, -21, 0, 1])
19He8 = np.array([105, 0, -420, 0, 210, 0, -28, 0, 1])
20He9 = np.array([0, 945, 0, -1260, 0, 378, 0, -36, 0, 1])
21
22Helist = [He0, He1, He2, He3, He4, He5, He6, He7, He8, He9]
23
24
25def trim(x):
26 return herme.hermetrim(x, tol=1e-6)
27
28
29class TestConstants:
30
31 def test_hermedomain(self):
32 assert_equal(herme.hermedomain, [-1, 1])
33
34 def test_hermezero(self):
35 assert_equal(herme.hermezero, [0])
36
37 def test_hermeone(self):
38 assert_equal(herme.hermeone, [1])
39
40 def test_hermex(self):
41 assert_equal(herme.hermex, [0, 1])
42
43
44class TestArithmetic:
45 x = np.linspace(-3, 3, 100)
46
47 def test_hermeadd(self):
48 for i in range(5):
49 for j in range(5):
50 msg = f"At i={i}, j={j}"
51 tgt = np.zeros(max(i, j) + 1)
52 tgt[i] += 1
53 tgt[j] += 1
54 res = herme.hermeadd([0] * i + [1], [0] * j + [1])
55 assert_equal(trim(res), trim(tgt), err_msg=msg)
56
57 def test_hermesub(self):
58 for i in range(5):
59 for j in range(5):
60 msg = f"At i={i}, j={j}"
61 tgt = np.zeros(max(i, j) + 1)
62 tgt[i] += 1
63 tgt[j] -= 1
64 res = herme.hermesub([0] * i + [1], [0] * j + [1])
65 assert_equal(trim(res), trim(tgt), err_msg=msg)
66
67 def test_hermemulx(self):
68 assert_equal(herme.hermemulx([0]), [0])
69 assert_equal(herme.hermemulx([1]), [0, 1])
70 for i in range(1, 5):
71 ser = [0] * i + [1]
72 tgt = [0] * (i - 1) + [i, 0, 1]
73 assert_equal(herme.hermemulx(ser), tgt)
74
75 def test_hermemul(self):
76 # check values of result
77 for i in range(5):
78 pol1 = [0] * i + [1]
79 val1 = herme.hermeval(self.x, pol1)
80 for j in range(5):
81 msg = f"At i={i}, j={j}"
82 pol2 = [0] * j + [1]
83 val2 = herme.hermeval(self.x, pol2)
84 pol3 = herme.hermemul(pol1, pol2)
85 val3 = herme.hermeval(self.x, pol3)
86 assert_(len(pol3) == i + j + 1, msg)
87 assert_almost_equal(val3, val1 * val2, err_msg=msg)
88
89 def test_hermediv(self):
90 for i in range(5):
91 for j in range(5):
92 msg = f"At i={i}, j={j}"
93 ci = [0] * i + [1]
94 cj = [0] * j + [1]
95 tgt = herme.hermeadd(ci, cj)
96 quo, rem = herme.hermediv(tgt, ci)
97 res = herme.hermeadd(herme.hermemul(quo, ci), rem)
98 assert_equal(trim(res), trim(tgt), err_msg=msg)
99
100 def test_hermepow(self):
101 for i in range(5):
102 for j in range(5):
103 msg = f"At i={i}, j={j}"
104 c = np.arange(i + 1)
105 tgt = reduce(herme.hermemul, [c] * j, np.array([1]))
106 res = herme.hermepow(c, j)
107 assert_equal(trim(res), trim(tgt), err_msg=msg)
108
109
110class TestEvaluation:
111 # coefficients of 1 + 2*x + 3*x**2
112 c1d = np.array([4., 2., 3.])
113 c2d = np.einsum('i,j->ij', c1d, c1d)
114 c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d)
115
116 # some random values in [-1, 1)
117 x = np.random.random((3, 5)) * 2 - 1
118 y = polyval(x, [1., 2., 3.])
119
120 def test_hermeval(self):
121 # check empty input
122 assert_equal(herme.hermeval([], [1]).size, 0)
123
124 # check normal input)
125 x = np.linspace(-1, 1)
126 y = [polyval(x, c) for c in Helist]
127 for i in range(10):
128 msg = f"At i={i}"
129 tgt = y[i]
130 res = herme.hermeval(x, [0] * i + [1])
131 assert_almost_equal(res, tgt, err_msg=msg)
132
133 # check that shape is preserved
134 for i in range(3):
135 dims = [2] * i
136 x = np.zeros(dims)
137 assert_equal(herme.hermeval(x, [1]).shape, dims)
138 assert_equal(herme.hermeval(x, [1, 0]).shape, dims)
139 assert_equal(herme.hermeval(x, [1, 0, 0]).shape, dims)
140
141 def test_hermeval2d(self):
142 x1, x2, x3 = self.x
143 y1, y2, y3 = self.y
144
145 # test exceptions
146 assert_raises(ValueError, herme.hermeval2d, x1, x2[:2], self.c2d)
147
148 # test values
149 tgt = y1 * y2
150 res = herme.hermeval2d(x1, x2, self.c2d)
151 assert_almost_equal(res, tgt)
152
153 # test shape
154 z = np.ones((2, 3))
155 res = herme.hermeval2d(z, z, self.c2d)
156 assert_(res.shape == (2, 3))
157
158 def test_hermeval3d(self):
159 x1, x2, x3 = self.x
160 y1, y2, y3 = self.y
161
162 # test exceptions
163 assert_raises(ValueError, herme.hermeval3d, x1, x2, x3[:2], self.c3d)
164
165 # test values
166 tgt = y1 * y2 * y3
167 res = herme.hermeval3d(x1, x2, x3, self.c3d)
168 assert_almost_equal(res, tgt)
169
170 # test shape
171 z = np.ones((2, 3))
172 res = herme.hermeval3d(z, z, z, self.c3d)
173 assert_(res.shape == (2, 3))
174
175 def test_hermegrid2d(self):
176 x1, x2, x3 = self.x
177 y1, y2, y3 = self.y
178
179 # test values
180 tgt = np.einsum('i,j->ij', y1, y2)
181 res = herme.hermegrid2d(x1, x2, self.c2d)
182 assert_almost_equal(res, tgt)
183
184 # test shape
185 z = np.ones((2, 3))
186 res = herme.hermegrid2d(z, z, self.c2d)
187 assert_(res.shape == (2, 3) * 2)
188
189 def test_hermegrid3d(self):
190 x1, x2, x3 = self.x
191 y1, y2, y3 = self.y
192
193 # test values
194 tgt = np.einsum('i,j,k->ijk', y1, y2, y3)
195 res = herme.hermegrid3d(x1, x2, x3, self.c3d)
196 assert_almost_equal(res, tgt)
197
198 # test shape
199 z = np.ones((2, 3))
200 res = herme.hermegrid3d(z, z, z, self.c3d)
201 assert_(res.shape == (2, 3) * 3)
202
203
204class TestIntegral:
205
206 def test_hermeint(self):
207 # check exceptions
208 assert_raises(TypeError, herme.hermeint, [0], .5)
209 assert_raises(ValueError, herme.hermeint, [0], -1)
210 assert_raises(ValueError, herme.hermeint, [0], 1, [0, 0])
211 assert_raises(ValueError, herme.hermeint, [0], lbnd=[0])
212 assert_raises(ValueError, herme.hermeint, [0], scl=[0])
213 assert_raises(TypeError, herme.hermeint, [0], axis=.5)
214
215 # test integration of zero polynomial
216 for i in range(2, 5):
217 k = [0] * (i - 2) + [1]
218 res = herme.hermeint([0], m=i, k=k)
219 assert_almost_equal(res, [0, 1])
220
221 # check single integration with integration constant
222 for i in range(5):
223 scl = i + 1
224 pol = [0] * i + [1]
225 tgt = [i] + [0] * i + [1 / scl]
226 hermepol = herme.poly2herme(pol)
227 hermeint = herme.hermeint(hermepol, m=1, k=[i])
228 res = herme.herme2poly(hermeint)
229 assert_almost_equal(trim(res), trim(tgt))
230
231 # check single integration with integration constant and lbnd
232 for i in range(5):
233 scl = i + 1
234 pol = [0] * i + [1]
235 hermepol = herme.poly2herme(pol)
236 hermeint = herme.hermeint(hermepol, m=1, k=[i], lbnd=-1)
237 assert_almost_equal(herme.hermeval(-1, hermeint), i)
238
239 # check single integration with integration constant and scaling
240 for i in range(5):
241 scl = i + 1
242 pol = [0] * i + [1]
243 tgt = [i] + [0] * i + [2 / scl]
244 hermepol = herme.poly2herme(pol)
245 hermeint = herme.hermeint(hermepol, m=1, k=[i], scl=2)
246 res = herme.herme2poly(hermeint)
247 assert_almost_equal(trim(res), trim(tgt))
248
249 # check multiple integrations with default k
250 for i in range(5):
251 for j in range(2, 5):
252 pol = [0] * i + [1]
253 tgt = pol[:]
254 for k in range(j):
255 tgt = herme.hermeint(tgt, m=1)
256 res = herme.hermeint(pol, m=j)
257 assert_almost_equal(trim(res), trim(tgt))
258
259 # check multiple integrations with defined k
260 for i in range(5):
261 for j in range(2, 5):
262 pol = [0] * i + [1]
263 tgt = pol[:]
264 for k in range(j):
265 tgt = herme.hermeint(tgt, m=1, k=[k])
266 res = herme.hermeint(pol, m=j, k=list(range(j)))
267 assert_almost_equal(trim(res), trim(tgt))
268
269 # check multiple integrations with lbnd
270 for i in range(5):
271 for j in range(2, 5):
272 pol = [0] * i + [1]
273 tgt = pol[:]
274 for k in range(j):
275 tgt = herme.hermeint(tgt, m=1, k=[k], lbnd=-1)
276 res = herme.hermeint(pol, m=j, k=list(range(j)), lbnd=-1)
277 assert_almost_equal(trim(res), trim(tgt))
278
279 # check multiple integrations with scaling
280 for i in range(5):
281 for j in range(2, 5):
282 pol = [0] * i + [1]
283 tgt = pol[:]
284 for k in range(j):
285 tgt = herme.hermeint(tgt, m=1, k=[k], scl=2)
286 res = herme.hermeint(pol, m=j, k=list(range(j)), scl=2)
287 assert_almost_equal(trim(res), trim(tgt))
288
289 def test_hermeint_axis(self):
290 # check that axis keyword works
291 c2d = np.random.random((3, 4))
292
293 tgt = np.vstack([herme.hermeint(c) for c in c2d.T]).T
294 res = herme.hermeint(c2d, axis=0)
295 assert_almost_equal(res, tgt)
296
297 tgt = np.vstack([herme.hermeint(c) for c in c2d])
298 res = herme.hermeint(c2d, axis=1)
299 assert_almost_equal(res, tgt)
300
301 tgt = np.vstack([herme.hermeint(c, k=3) for c in c2d])
302 res = herme.hermeint(c2d, k=3, axis=1)
303 assert_almost_equal(res, tgt)
304
305
306class TestDerivative:
307
308 def test_hermeder(self):
309 # check exceptions
310 assert_raises(TypeError, herme.hermeder, [0], .5)
311 assert_raises(ValueError, herme.hermeder, [0], -1)
312
313 # check that zeroth derivative does nothing
314 for i in range(5):
315 tgt = [0] * i + [1]
316 res = herme.hermeder(tgt, m=0)
317 assert_equal(trim(res), trim(tgt))
318
319 # check that derivation is the inverse of integration
320 for i in range(5):
321 for j in range(2, 5):
322 tgt = [0] * i + [1]
323 res = herme.hermeder(herme.hermeint(tgt, m=j), m=j)
324 assert_almost_equal(trim(res), trim(tgt))
325
326 # check derivation with scaling
327 for i in range(5):
328 for j in range(2, 5):
329 tgt = [0] * i + [1]
330 res = herme.hermeder(
331 herme.hermeint(tgt, m=j, scl=2), m=j, scl=.5)
332 assert_almost_equal(trim(res), trim(tgt))
333
334 def test_hermeder_axis(self):
335 # check that axis keyword works
336 c2d = np.random.random((3, 4))
337
338 tgt = np.vstack([herme.hermeder(c) for c in c2d.T]).T
339 res = herme.hermeder(c2d, axis=0)
340 assert_almost_equal(res, tgt)
341
342 tgt = np.vstack([herme.hermeder(c) for c in c2d])
343 res = herme.hermeder(c2d, axis=1)
344 assert_almost_equal(res, tgt)
345
346
347class TestVander:
348 # some random values in [-1, 1)
349 x = np.random.random((3, 5)) * 2 - 1
350
351 def test_hermevander(self):
352 # check for 1d x
353 x = np.arange(3)
354 v = herme.hermevander(x, 3)
355 assert_(v.shape == (3, 4))
356 for i in range(4):
357 coef = [0] * i + [1]
358 assert_almost_equal(v[..., i], herme.hermeval(x, coef))
359
360 # check for 2d x
361 x = np.array([[1, 2], [3, 4], [5, 6]])
362 v = herme.hermevander(x, 3)
363 assert_(v.shape == (3, 2, 4))
364 for i in range(4):
365 coef = [0] * i + [1]
366 assert_almost_equal(v[..., i], herme.hermeval(x, coef))
367
368 def test_hermevander2d(self):
369 # also tests hermeval2d for non-square coefficient array
370 x1, x2, x3 = self.x
371 c = np.random.random((2, 3))
372 van = herme.hermevander2d(x1, x2, [1, 2])
373 tgt = herme.hermeval2d(x1, x2, c)
374 res = np.dot(van, c.flat)
375 assert_almost_equal(res, tgt)
376
377 # check shape
378 van = herme.hermevander2d([x1], [x2], [1, 2])
379 assert_(van.shape == (1, 5, 6))
380
381 def test_hermevander3d(self):
382 # also tests hermeval3d for non-square coefficient array
383 x1, x2, x3 = self.x
384 c = np.random.random((2, 3, 4))
385 van = herme.hermevander3d(x1, x2, x3, [1, 2, 3])
386 tgt = herme.hermeval3d(x1, x2, x3, c)
387 res = np.dot(van, c.flat)
388 assert_almost_equal(res, tgt)
389
390 # check shape
391 van = herme.hermevander3d([x1], [x2], [x3], [1, 2, 3])
392 assert_(van.shape == (1, 5, 24))
393
394
395class TestFitting:
396
397 def test_hermefit(self):
398 def f(x):
399 return x * (x - 1) * (x - 2)
400
401 def f2(x):
402 return x**4 + x**2 + 1
403
404 # Test exceptions
405 assert_raises(ValueError, herme.hermefit, [1], [1], -1)
406 assert_raises(TypeError, herme.hermefit, [[1]], [1], 0)
407 assert_raises(TypeError, herme.hermefit, [], [1], 0)
408 assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0)
409 assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0)
410 assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0)
411 assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]])
412 assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1, 1])
413 assert_raises(ValueError, herme.hermefit, [1], [1], [-1,])
414 assert_raises(ValueError, herme.hermefit, [1], [1], [2, -1, 6])
415 assert_raises(TypeError, herme.hermefit, [1], [1], [])
416
417 # Test fit
418 x = np.linspace(0, 2)
419 y = f(x)
420 #
421 coef3 = herme.hermefit(x, y, 3)
422 assert_equal(len(coef3), 4)
423 assert_almost_equal(herme.hermeval(x, coef3), y)
424 coef3 = herme.hermefit(x, y, [0, 1, 2, 3])
425 assert_equal(len(coef3), 4)
426 assert_almost_equal(herme.hermeval(x, coef3), y)
427 #
428 coef4 = herme.hermefit(x, y, 4)
429 assert_equal(len(coef4), 5)
430 assert_almost_equal(herme.hermeval(x, coef4), y)
431 coef4 = herme.hermefit(x, y, [0, 1, 2, 3, 4])
432 assert_equal(len(coef4), 5)
433 assert_almost_equal(herme.hermeval(x, coef4), y)
434 # check things still work if deg is not in strict increasing
435 coef4 = herme.hermefit(x, y, [2, 3, 4, 1, 0])
436 assert_equal(len(coef4), 5)
437 assert_almost_equal(herme.hermeval(x, coef4), y)
438 #
439 coef2d = herme.hermefit(x, np.array([y, y]).T, 3)
440 assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
441 coef2d = herme.hermefit(x, np.array([y, y]).T, [0, 1, 2, 3])
442 assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
443 # test weighting
444 w = np.zeros_like(x)
445 yw = y.copy()
446 w[1::2] = 1
447 y[0::2] = 0
448 wcoef3 = herme.hermefit(x, yw, 3, w=w)
449 assert_almost_equal(wcoef3, coef3)
450 wcoef3 = herme.hermefit(x, yw, [0, 1, 2, 3], w=w)
451 assert_almost_equal(wcoef3, coef3)
452 #
453 wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, 3, w=w)
454 assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
455 wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w)
456 assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
457 # test scaling with complex values x points whose square
458 # is zero when summed.
459 x = [1, 1j, -1, -1j]
460 assert_almost_equal(herme.hermefit(x, x, 1), [0, 1])
461 assert_almost_equal(herme.hermefit(x, x, [0, 1]), [0, 1])
462 # test fitting only even Legendre polynomials
463 x = np.linspace(-1, 1)
464 y = f2(x)
465 coef1 = herme.hermefit(x, y, 4)
466 assert_almost_equal(herme.hermeval(x, coef1), y)
467 coef2 = herme.hermefit(x, y, [0, 2, 4])
468 assert_almost_equal(herme.hermeval(x, coef2), y)
469 assert_almost_equal(coef1, coef2)
470
471
472class TestCompanion:
473
474 def test_raises(self):
475 assert_raises(ValueError, herme.hermecompanion, [])
476 assert_raises(ValueError, herme.hermecompanion, [1])
477
478 def test_dimensions(self):
479 for i in range(1, 5):
480 coef = [0] * i + [1]
481 assert_(herme.hermecompanion(coef).shape == (i, i))
482
483 def test_linear_root(self):
484 assert_(herme.hermecompanion([1, 2])[0, 0] == -.5)
485
486
487class TestGauss:
488
489 def test_100(self):
490 x, w = herme.hermegauss(100)
491
492 # test orthogonality. Note that the results need to be normalized,
493 # otherwise the huge values that can arise from fast growing
494 # functions like Laguerre can be very confusing.
495 v = herme.hermevander(x, 99)
496 vv = np.dot(v.T * w, v)
497 vd = 1 / np.sqrt(vv.diagonal())
498 vv = vd[:, None] * vv * vd
499 assert_almost_equal(vv, np.eye(100))
500
501 # check that the integral of 1 is correct
502 tgt = np.sqrt(2 * np.pi)
503 assert_almost_equal(w.sum(), tgt)
504
505
506class TestMisc:
507
508 def test_hermefromroots(self):
509 res = herme.hermefromroots([])
510 assert_almost_equal(trim(res), [1])
511 for i in range(1, 5):
512 roots = np.cos(np.linspace(-np.pi, 0, 2 * i + 1)[1::2])
513 pol = herme.hermefromroots(roots)
514 res = herme.hermeval(roots, pol)
515 tgt = 0
516 assert_(len(pol) == i + 1)
517 assert_almost_equal(herme.herme2poly(pol)[-1], 1)
518 assert_almost_equal(res, tgt)
519
520 def test_hermeroots(self):
521 assert_almost_equal(herme.hermeroots([1]), [])
522 assert_almost_equal(herme.hermeroots([1, 1]), [-1])
523 for i in range(2, 5):
524 tgt = np.linspace(-1, 1, i)
525 res = herme.hermeroots(herme.hermefromroots(tgt))
526 assert_almost_equal(trim(res), trim(tgt))
527
528 def test_hermetrim(self):
529 coef = [2, -1, 1, 0]
530
531 # Test exceptions
532 assert_raises(ValueError, herme.hermetrim, coef, -1)
533
534 # Test results
535 assert_equal(herme.hermetrim(coef), coef[:-1])
536 assert_equal(herme.hermetrim(coef, 1), coef[:-3])
537 assert_equal(herme.hermetrim(coef, 2), [0])
538
539 def test_hermeline(self):
540 assert_equal(herme.hermeline(3, 4), [3, 4])
541
542 def test_herme2poly(self):
543 for i in range(10):
544 assert_almost_equal(herme.herme2poly([0] * i + [1]), Helist[i])
545
546 def test_poly2herme(self):
547 for i in range(10):
548 assert_almost_equal(herme.poly2herme(Helist[i]), [0] * i + [1])
549
550 def test_weight(self):
551 x = np.linspace(-5, 5, 11)
552 tgt = np.exp(-.5 * x**2)
553 res = herme.hermeweight(x)
554 assert_almost_equal(res, tgt)
555