/
hermite.py
1853 lines (1503 loc) · 56.5 KB
/
hermite.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
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
"""
Objects for dealing with Hermite series.
This module provides a number of objects (mostly functions) useful for
dealing with Hermite series, including a `Hermite` class that
encapsulates the usual arithmetic operations. (General information
on how this module represents and works with such polynomials is in the
docstring for its "parent" sub-package, `numpy.polynomial`).
Constants
---------
- `hermdomain` -- Hermite series default domain, [-1,1].
- `hermzero` -- Hermite series that evaluates identically to 0.
- `hermone` -- Hermite series that evaluates identically to 1.
- `hermx` -- Hermite series for the identity map, ``f(x) = x``.
Arithmetic
----------
- `hermmulx` -- multiply a Hermite series in ``P_i(x)`` by ``x``.
- `hermadd` -- add two Hermite series.
- `hermsub` -- subtract one Hermite series from another.
- `hermmul` -- multiply two Hermite series.
- `hermdiv` -- divide one Hermite series by another.
- `hermval` -- evaluate a Hermite series at given points.
- `hermval2d` -- evaluate a 2D Hermite series at given points.
- `hermval3d` -- evaluate a 3D Hermite series at given points.
- `hermgrid2d` -- evaluate a 2D Hermite series on a Cartesian product.
- `hermgrid3d` -- evaluate a 3D Hermite series on a Cartesian product.
Calculus
--------
- `hermder` -- differentiate a Hermite series.
- `hermint` -- integrate a Hermite series.
Misc Functions
--------------
- `hermfromroots` -- create a Hermite series with specified roots.
- `hermroots` -- find the roots of a Hermite series.
- `hermvander` -- Vandermonde-like matrix for Hermite polynomials.
- `hermvander2d` -- Vandermonde-like matrix for 2D power series.
- `hermvander3d` -- Vandermonde-like matrix for 3D power series.
- `hermgauss` -- Gauss-Hermite quadrature, points and weights.
- `hermweight` -- Hermite weight function.
- `hermcompanion` -- symmetrized companion matrix in Hermite form.
- `hermfit` -- least-squares fit returning a Hermite series.
- `hermtrim` -- trim leading coefficients from a Hermite series.
- `hermline` -- Hermite series of given straight line.
- `herm2poly` -- convert a Hermite series to a polynomial.
- `poly2herm` -- convert a polynomial to a Hermite series.
Classes
-------
- `Hermite` -- A Hermite series class.
See also
--------
`numpy.polynomial`
"""
from __future__ import division, absolute_import, print_function
import warnings
import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
__all__ = [
'hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline', 'hermadd',
'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow', 'hermval',
'hermder', 'hermint', 'herm2poly', 'poly2herm', 'hermfromroots',
'hermvander', 'hermfit', 'hermtrim', 'hermroots', 'Hermite',
'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', 'hermvander2d',
'hermvander3d', 'hermcompanion', 'hermgauss', 'hermweight']
hermtrim = pu.trimcoef
def poly2herm(pol):
"""
poly2herm(pol)
Convert a polynomial to a Hermite series.
Convert an array representing the coefficients of a polynomial (relative
to the "standard" basis) ordered from lowest degree to highest, to an
array of the coefficients of the equivalent Hermite series, ordered
from lowest to highest degree.
Parameters
----------
pol : array_like
1-D array containing the polynomial coefficients
Returns
-------
c : ndarray
1-D array containing the coefficients of the equivalent Hermite
series.
See Also
--------
herm2poly
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy.polynomial.hermite import poly2herm
>>> poly2herm(np.arange(4))
array([ 1. , 2.75 , 0.5 , 0.375])
"""
[pol] = pu.as_series([pol])
deg = len(pol) - 1
res = 0
for i in range(deg, -1, -1):
res = hermadd(hermmulx(res), pol[i])
return res
def herm2poly(c):
"""
Convert a Hermite series to a polynomial.
Convert an array representing the coefficients of a Hermite series,
ordered from lowest degree to highest, to an array of the coefficients
of the equivalent polynomial (relative to the "standard" basis) ordered
from lowest to highest degree.
Parameters
----------
c : array_like
1-D array containing the Hermite series coefficients, ordered
from lowest order term to highest.
Returns
-------
pol : ndarray
1-D array containing the coefficients of the equivalent polynomial
(relative to the "standard" basis) ordered from lowest order term
to highest.
See Also
--------
poly2herm
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy.polynomial.hermite import herm2poly
>>> herm2poly([ 1. , 2.75 , 0.5 , 0.375])
array([ 0., 1., 2., 3.])
"""
from .polynomial import polyadd, polysub, polymulx
[c] = pu.as_series([c])
n = len(c)
if n == 1:
return c
if n == 2:
c[1] *= 2
return c
else:
c0 = c[-2]
c1 = c[-1]
# i is the current degree of c1
for i in range(n - 1, 1, -1):
tmp = c0
c0 = polysub(c[i - 2], c1*(2*(i - 1)))
c1 = polyadd(tmp, polymulx(c1)*2)
return polyadd(c0, polymulx(c1)*2)
#
# These are constant arrays are of integer type so as to be compatible
# with the widest range of other types, such as Decimal.
#
# Hermite
hermdomain = np.array([-1, 1])
# Hermite coefficients representing zero.
hermzero = np.array([0])
# Hermite coefficients representing one.
hermone = np.array([1])
# Hermite coefficients representing the identity x.
hermx = np.array([0, 1/2])
def hermline(off, scl):
"""
Hermite series whose graph is a straight line.
Parameters
----------
off, scl : scalars
The specified line is given by ``off + scl*x``.
Returns
-------
y : ndarray
This module's representation of the Hermite series for
``off + scl*x``.
See Also
--------
polyline, chebline
Examples
--------
>>> from numpy.polynomial.hermite import hermline, hermval
>>> hermval(0,hermline(3, 2))
3.0
>>> hermval(1,hermline(3, 2))
5.0
"""
if scl != 0:
return np.array([off, scl/2])
else:
return np.array([off])
def hermfromroots(roots):
"""
Generate a Hermite series with given roots.
The function returns the coefficients of the polynomial
.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
in Hermite form, where the `r_n` are the roots specified in `roots`.
If a zero has multiplicity n, then it must appear in `roots` n times.
For instance, if 2 is a root of multiplicity three and 3 is a root of
multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
roots can appear in any order.
If the returned coefficients are `c`, then
.. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x)
The coefficient of the last term is not generally 1 for monic
polynomials in Hermite form.
Parameters
----------
roots : array_like
Sequence containing the roots.
Returns
-------
out : ndarray
1-D array of coefficients. If all roots are real then `out` is a
real array, if some of the roots are complex, then `out` is complex
even if all the coefficients in the result are real (see Examples
below).
See Also
--------
polyfromroots, legfromroots, lagfromroots, chebfromroots,
hermefromroots.
Examples
--------
>>> from numpy.polynomial.hermite import hermfromroots, hermval
>>> coef = hermfromroots((-1, 0, 1))
>>> hermval((-1, 0, 1), coef)
array([ 0., 0., 0.])
>>> coef = hermfromroots((-1j, 1j))
>>> hermval((-1j, 1j), coef)
array([ 0.+0.j, 0.+0.j])
"""
if len(roots) == 0:
return np.ones(1)
else:
[roots] = pu.as_series([roots], trim=False)
roots.sort()
p = [hermline(-r, 1) for r in roots]
n = len(p)
while n > 1:
m, r = divmod(n, 2)
tmp = [hermmul(p[i], p[i+m]) for i in range(m)]
if r:
tmp[0] = hermmul(tmp[0], p[-1])
p = tmp
n = m
return p[0]
def hermadd(c1, c2):
"""
Add one Hermite series to another.
Returns the sum of two Hermite series `c1` + `c2`. The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Hermite series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the Hermite series of their sum.
See Also
--------
hermsub, hermmul, hermdiv, hermpow
Notes
-----
Unlike multiplication, division, etc., the sum of two Hermite series
is a Hermite series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."
Examples
--------
>>> from numpy.polynomial.hermite import hermadd
>>> hermadd([1, 2, 3], [1, 2, 3, 4])
array([ 2., 4., 6., 4.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c1[:c2.size] += c2
ret = c1
else:
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def hermsub(c1, c2):
"""
Subtract one Hermite series from another.
Returns the difference of two Hermite series `c1` - `c2`. The
sequences of coefficients are from lowest order term to highest, i.e.,
[1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Hermite series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Hermite series coefficients representing their difference.
See Also
--------
hermadd, hermmul, hermdiv, hermpow
Notes
-----
Unlike multiplication, division, etc., the difference of two Hermite
series is a Hermite series (without having to "reproject" the result
onto the basis set) so subtraction, just like that of "standard"
polynomials, is simply "component-wise."
Examples
--------
>>> from numpy.polynomial.hermite import hermsub
>>> hermsub([1, 2, 3, 4], [1, 2, 3])
array([ 0., 0., 0., 4.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def hermmulx(c):
"""Multiply a Hermite series by x.
Multiply the Hermite series `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
Notes
-----
The multiplication uses the recursion relationship for Hermite
polynomials in the form
.. math::
xP_i(x) = (P_{i + 1}(x)/2 + i*P_{i - 1}(x))
Examples
--------
>>> from numpy.polynomial.hermite import hermmulx
>>> hermmulx([1, 2, 3])
array([ 2. , 6.5, 1. , 1.5])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
# The zero series needs special treatment
if len(c) == 1 and c[0] == 0:
return c
prd = np.empty(len(c) + 1, dtype=c.dtype)
prd[0] = c[0]*0
prd[1] = c[0]/2
for i in range(1, len(c)):
prd[i + 1] = c[i]/2
prd[i - 1] += c[i]*i
return prd
def hermmul(c1, c2):
"""
Multiply one Hermite series by another.
Returns the product of two Hermite series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Hermite series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Hermite series coefficients representing their product.
See Also
--------
hermadd, hermsub, hermdiv, hermpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Hermite polynomial basis set. Thus, to express
the product as a Hermite series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial.hermite import hermmul
>>> hermmul([1, 2, 3], [0, 1, 2])
array([ 52., 29., 52., 7., 6.])
"""
# s1, s2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c = c2
xs = c1
else:
c = c1
xs = c2
if len(c) == 1:
c0 = c[0]*xs
c1 = 0
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = hermsub(c[-i]*xs, c1*(2*(nd - 1)))
c1 = hermadd(tmp, hermmulx(c1)*2)
return hermadd(c0, hermmulx(c1)*2)
def hermdiv(c1, c2):
"""
Divide one Hermite series by another.
Returns the quotient-with-remainder of two Hermite series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Hermite series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarrays
Of Hermite series coefficients representing the quotient and
remainder.
See Also
--------
hermadd, hermsub, hermmul, hermpow
Notes
-----
In general, the (polynomial) division of one Hermite series by another
results in quotient and remainder terms that are not in the Hermite
polynomial basis set. Thus, to express these results as a Hermite
series, it is necessary to "reproject" the results onto the Hermite
basis set, which may produce "unintuitive" (but correct) results; see
Examples section below.
Examples
--------
>>> from numpy.polynomial.hermite import hermdiv
>>> hermdiv([ 52., 29., 52., 7., 6.], [0, 1, 2])
(array([ 1., 2., 3.]), array([ 0.]))
>>> hermdiv([ 54., 31., 52., 7., 6.], [0, 1, 2])
(array([ 1., 2., 3.]), array([ 2., 2.]))
>>> hermdiv([ 53., 30., 52., 7., 6.], [0, 1, 2])
(array([ 1., 2., 3.]), array([ 1., 1.]))
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0:
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
if lc1 < lc2:
return c1[:1]*0, c1
elif lc2 == 1:
return c1/c2[-1], c1[:1]*0
else:
quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
rem = c1
for i in range(lc1 - lc2, - 1, -1):
p = hermmul([0]*i + [1], c2)
q = rem[-1]/p[-1]
rem = rem[:-1] - q*p[:-1]
quo[i] = q
return quo, pu.trimseq(rem)
def hermpow(c, pow, maxpower=16):
"""Raise a Hermite series to a power.
Returns the Hermite series `c` raised to the power `pow`. The
argument `c` is a sequence of coefficients ordered from low to high.
i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.``
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to
high.
pow : integer
Power to which the series will be raised
maxpower : integer, optional
Maximum power allowed. This is mainly to limit growth of the series
to unmanageable size. Default is 16
Returns
-------
coef : ndarray
Hermite series of power.
See Also
--------
hermadd, hermsub, hermmul, hermdiv
Examples
--------
>>> from numpy.polynomial.hermite import hermpow
>>> hermpow([1, 2, 3], 2)
array([ 81., 52., 82., 12., 9.])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
power = int(pow)
if power != pow or power < 0:
raise ValueError("Power must be a non-negative integer.")
elif maxpower is not None and power > maxpower:
raise ValueError("Power is too large")
elif power == 0:
return np.array([1], dtype=c.dtype)
elif power == 1:
return c
else:
# This can be made more efficient by using powers of two
# in the usual way.
prd = c
for i in range(2, power + 1):
prd = hermmul(prd, c)
return prd
def hermder(c, m=1, scl=1, axis=0):
"""
Differentiate a Hermite series.
Returns the Hermite series coefficients `c` differentiated `m` times
along `axis`. At each iteration the result is multiplied by `scl` (the
scaling factor is for use in a linear change of variable). The argument
`c` is an array of coefficients from low to high degree along each
axis, e.g., [1,2,3] represents the series ``1*H_0 + 2*H_1 + 3*H_2``
while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) +
2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is
``y``.
Parameters
----------
c : array_like
Array of Hermite series coefficients. If `c` is multidimensional the
different axis correspond to different variables with the degree in
each axis given by the corresponding index.
m : int, optional
Number of derivatives taken, must be non-negative. (Default: 1)
scl : scalar, optional
Each differentiation is multiplied by `scl`. The end result is
multiplication by ``scl**m``. This is for use in a linear change of
variable. (Default: 1)
axis : int, optional
Axis over which the derivative is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
der : ndarray
Hermite series of the derivative.
See Also
--------
hermint
Notes
-----
In general, the result of differentiating a Hermite series does not
resemble the same operation on a power series. Thus the result of this
function may be "unintuitive," albeit correct; see Examples section
below.
Examples
--------
>>> from numpy.polynomial.hermite import hermder
>>> hermder([ 1. , 0.5, 0.5, 0.5])
array([ 1., 2., 3.])
>>> hermder([-0.5, 1./2., 1./8., 1./12., 1./16.], m=2)
array([ 1., 2., 3.])
"""
c = np.array(c, ndmin=1, copy=1)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
cnt, iaxis = [int(t) for t in [m, axis]]
if cnt != m:
raise ValueError("The order of derivation must be integer")
if cnt < 0:
raise ValueError("The order of derivation must be non-negative")
if iaxis != axis:
raise ValueError("The axis must be integer")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
n = len(c)
if cnt >= n:
c = c[:1]*0
else:
for i in range(cnt):
n = n - 1
c *= scl
der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
for j in range(n, 0, -1):
der[j - 1] = (2*j)*c[j]
c = der
c = np.moveaxis(c, 0, iaxis)
return c
def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
"""
Integrate a Hermite series.
Returns the Hermite series coefficients `c` integrated `m` times from
`lbnd` along `axis`. At each iteration the resulting series is
**multiplied** by `scl` and an integration constant, `k`, is added.
The scaling factor is for use in a linear change of variable. ("Buyer
beware": note that, depending on what one is doing, one may want `scl`
to be the reciprocal of what one might expect; for more information,
see the Notes section below.) The argument `c` is an array of
coefficients from low to high degree along each axis, e.g., [1,2,3]
represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]]
represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) +
2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
Parameters
----------
c : array_like
Array of Hermite series coefficients. If c is multidimensional the
different axis correspond to different variables with the degree in
each axis given by the corresponding index.
m : int, optional
Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
Integration constant(s). The value of the first integral at
``lbnd`` is the first value in the list, the value of the second
integral at ``lbnd`` is the second value, etc. If ``k == []`` (the
default), all constants are set to zero. If ``m == 1``, a single
scalar can be given instead of a list.
lbnd : scalar, optional
The lower bound of the integral. (Default: 0)
scl : scalar, optional
Following each integration the result is *multiplied* by `scl`
before the integration constant is added. (Default: 1)
axis : int, optional
Axis over which the integral is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
S : ndarray
Hermite series coefficients of the integral.
Raises
------
ValueError
If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
``np.ndim(scl) != 0``.
See Also
--------
hermder
Notes
-----
Note that the result of each integration is *multiplied* by `scl`.
Why is this important to note? Say one is making a linear change of
variable :math:`u = ax + b` in an integral relative to `x`. Then
:math:`dx = du/a`, so one will need to set `scl` equal to
:math:`1/a` - perhaps not what one would have first thought.
Also note that, in general, the result of integrating a C-series needs
to be "reprojected" onto the C-series basis set. Thus, typically,
the result of this function is "unintuitive," albeit correct; see
Examples section below.
Examples
--------
>>> from numpy.polynomial.hermite import hermint
>>> hermint([1,2,3]) # integrate once, value 0 at 0.
array([ 1. , 0.5, 0.5, 0.5])
>>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0
array([-0.5 , 0.5 , 0.125 , 0.08333333, 0.0625 ])
>>> hermint([1,2,3], k=1) # integrate once, value 1 at 0.
array([ 2. , 0.5, 0.5, 0.5])
>>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1
array([-2. , 0.5, 0.5, 0.5])
>>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1)
array([ 1.66666667, -0.5 , 0.125 , 0.08333333, 0.0625 ])
"""
c = np.array(c, ndmin=1, copy=1)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if not np.iterable(k):
k = [k]
cnt, iaxis = [int(t) for t in [m, axis]]
if cnt != m:
raise ValueError("The order of integration must be integer")
if cnt < 0:
raise ValueError("The order of integration must be non-negative")
if len(k) > cnt:
raise ValueError("Too many integration constants")
if np.ndim(lbnd) != 0:
raise ValueError("lbnd must be a scalar.")
if np.ndim(scl) != 0:
raise ValueError("scl must be a scalar.")
if iaxis != axis:
raise ValueError("The axis must be integer")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
k = list(k) + [0]*(cnt - len(k))
for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
c[0] += k[i]
else:
tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
tmp[0] = c[0]*0
tmp[1] = c[0]/2
for j in range(1, n):
tmp[j + 1] = c[j]/(2*(j + 1))
tmp[0] += k[i] - hermval(lbnd, tmp)
c = tmp
c = np.moveaxis(c, 0, iaxis)
return c
def hermval(x, c, tensor=True):
"""
Evaluate an Hermite series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
with themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
hermval2d, hermgrid2d, hermval3d, hermgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.hermite import hermval
>>> coef = [1,2,3]
>>> hermval(1, coef)
11.0
>>> hermval([[1,2],[3,4]], coef)
array([[ 11., 51.],
[ 115., 203.]])
"""
c = np.array(c, ndmin=1, copy=0)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
x2 = x*2
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - c1*(2*(nd - 1))
c1 = tmp + c1*x2
return c0 + c1*x2
def hermval2d(x, y, c):
"""
Evaluate a 2-D Hermite series at points (x, y).
This function returns the values:
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * H_i(x) * H_j(y)
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars and they
must have the same shape after conversion. In either case, either `x`
and `y` or their elements must support multiplication and addition both
with themselves and with the elements of `c`.
If `c` is a 1-D array a one is implicitly appended to its shape to make
it 2-D. The shape of the result will be c.shape[2:] + x.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points `(x, y)`,
where `x` and `y` must have the same shape. If `x` or `y` is a list
or tuple, it is first converted to an ndarray, otherwise it is left
unchanged and if it isn't an ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term
of multi-degree i,j is contained in ``c[i,j]``. If `c` has
dimension greater than two the remaining indices enumerate multiple
sets of coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points formed with
pairs of corresponding values from `x` and `y`.
See Also
--------
hermval, hermgrid2d, hermval3d, hermgrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
try:
x, y = np.array((x, y), copy=0)
except Exception:
raise ValueError('x, y are incompatible')
c = hermval(x, c)
c = hermval(y, c, tensor=False)
return c