File size: 44,376 Bytes
7885a28 |
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 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 |
# mypy: disable-error-code="attr-defined"
import os
import pytest
import math
import numpy as np
from numpy.testing import assert_allclose
from scipy.conftest import array_api_compatible
import scipy._lib._elementwise_iterative_method as eim
from scipy._lib._array_api_no_0d import xp_assert_close, xp_assert_equal
from scipy._lib._array_api import array_namespace, xp_size, xp_ravel, xp_copy, is_numpy
from scipy import special, stats
from scipy.integrate import quad_vec, nsum, tanhsinh as _tanhsinh
from scipy.integrate._tanhsinh import _pair_cache
from scipy.stats._discrete_distns import _gen_harmonic_gt1
def norm_pdf(x, xp=None):
xp = array_namespace(x) if xp is None else xp
return 1/(2*xp.pi)**0.5 * xp.exp(-x**2/2)
def norm_logpdf(x, xp=None):
xp = array_namespace(x) if xp is None else xp
return -0.5*math.log(2*xp.pi) - x**2/2
def _vectorize(xp):
# xp-compatible version of np.vectorize
# assumes arguments are all arrays of the same shape
def decorator(f):
def wrapped(*arg_arrays):
shape = arg_arrays[0].shape
arg_arrays = [xp_ravel(arg_array) for arg_array in arg_arrays]
res = []
for i in range(math.prod(shape)):
arg_scalars = [arg_array[i] for arg_array in arg_arrays]
res.append(f(*arg_scalars))
return res
return wrapped
return decorator
@array_api_compatible
@pytest.mark.usefixtures("skip_xp_backends")
@pytest.mark.skip_xp_backends(
'array_api_strict', reason='Currently uses fancy indexing assignment.'
)
@pytest.mark.skip_xp_backends(
'jax.numpy', reason='JAX arrays do not support item assignment.'
)
class TestTanhSinh:
# Test problems from [1] Section 6
def f1(self, t):
return t * np.log(1 + t)
f1.ref = 0.25
f1.b = 1
def f2(self, t):
return t ** 2 * np.arctan(t)
f2.ref = (np.pi - 2 + 2 * np.log(2)) / 12
f2.b = 1
def f3(self, t):
return np.exp(t) * np.cos(t)
f3.ref = (np.exp(np.pi / 2) - 1) / 2
f3.b = np.pi / 2
def f4(self, t):
a = np.sqrt(2 + t ** 2)
return np.arctan(a) / ((1 + t ** 2) * a)
f4.ref = 5 * np.pi ** 2 / 96
f4.b = 1
def f5(self, t):
return np.sqrt(t) * np.log(t)
f5.ref = -4 / 9
f5.b = 1
def f6(self, t):
return np.sqrt(1 - t ** 2)
f6.ref = np.pi / 4
f6.b = 1
def f7(self, t):
return np.sqrt(t) / np.sqrt(1 - t ** 2)
f7.ref = 2 * np.sqrt(np.pi) * special.gamma(3 / 4) / special.gamma(1 / 4)
f7.b = 1
def f8(self, t):
return np.log(t) ** 2
f8.ref = 2
f8.b = 1
def f9(self, t):
return np.log(np.cos(t))
f9.ref = -np.pi * np.log(2) / 2
f9.b = np.pi / 2
def f10(self, t):
return np.sqrt(np.tan(t))
f10.ref = np.pi * np.sqrt(2) / 2
f10.b = np.pi / 2
def f11(self, t):
return 1 / (1 + t ** 2)
f11.ref = np.pi / 2
f11.b = np.inf
def f12(self, t):
return np.exp(-t) / np.sqrt(t)
f12.ref = np.sqrt(np.pi)
f12.b = np.inf
def f13(self, t):
return np.exp(-t ** 2 / 2)
f13.ref = np.sqrt(np.pi / 2)
f13.b = np.inf
def f14(self, t):
return np.exp(-t) * np.cos(t)
f14.ref = 0.5
f14.b = np.inf
def f15(self, t):
return np.sin(t) / t
f15.ref = np.pi / 2
f15.b = np.inf
def error(self, res, ref, log=False, xp=None):
xp = array_namespace(res, ref) if xp is None else xp
err = abs(res - ref)
if not log:
return err
with np.errstate(divide='ignore'):
return xp.log10(err)
def test_input_validation(self, xp):
f = self.f1
zero = xp.asarray(0)
f_b = xp.asarray(f.b)
message = '`f` must be callable.'
with pytest.raises(ValueError, match=message):
_tanhsinh(42, zero, f_b)
message = '...must be True or False.'
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, log=2)
message = '...must be real numbers.'
with pytest.raises(ValueError, match=message):
_tanhsinh(f, xp.asarray(1+1j), f_b)
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, atol='ekki')
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, rtol=pytest)
message = '...must be non-negative and finite.'
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, rtol=-1)
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, atol=xp.inf)
message = '...may not be positive infinity.'
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, rtol=xp.inf, log=True)
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, atol=xp.inf, log=True)
message = '...must be integers.'
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, maxlevel=object())
# with pytest.raises(ValueError, match=message): # unused for now
# _tanhsinh(f, zero, f_b, maxfun=1+1j)
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, minlevel="migratory coconut")
message = '...must be non-negative.'
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, maxlevel=-1)
# with pytest.raises(ValueError, match=message): # unused for now
# _tanhsinh(f, zero, f_b, maxfun=-1)
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, minlevel=-1)
message = '...must be True or False.'
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, preserve_shape=2)
message = '...must be callable.'
with pytest.raises(ValueError, match=message):
_tanhsinh(f, zero, f_b, callback='elderberry')
@pytest.mark.parametrize("limits, ref", [
[(0, math.inf), 0.5], # b infinite
[(-math.inf, 0), 0.5], # a infinite
[(-math.inf, math.inf), 1.], # a and b infinite
[(math.inf, -math.inf), -1.], # flipped limits
[(1, -1), stats.norm.cdf(-1.) - stats.norm.cdf(1.)], # flipped limits
])
def test_integral_transforms(self, limits, ref, xp):
# Check that the integral transforms are behaving for both normal and
# log integration
limits = [xp.asarray(limit) for limit in limits]
dtype = xp.asarray(float(limits[0])).dtype
ref = xp.asarray(ref, dtype=dtype)
res = _tanhsinh(norm_pdf, *limits)
xp_assert_close(res.integral, ref)
logres = _tanhsinh(norm_logpdf, *limits, log=True)
xp_assert_close(xp.exp(logres.integral), ref, check_dtype=False)
# Transformation should not make the result complex unnecessarily
xp_test = array_namespace(*limits) # we need xp.isdtype
assert (xp_test.isdtype(logres.integral.dtype, "real floating") if ref > 0
else xp_test.isdtype(logres.integral.dtype, "complex floating"))
xp_assert_close(xp.exp(logres.error), res.error, atol=1e-16, check_dtype=False)
# 15 skipped intentionally; it's very difficult numerically
@pytest.mark.skip_xp_backends(np_only=True,
reason='Cumbersome to convert everything.')
@pytest.mark.parametrize('f_number', range(1, 15))
def test_basic(self, f_number, xp):
f = getattr(self, f"f{f_number}")
rtol = 2e-8
res = _tanhsinh(f, 0, f.b, rtol=rtol)
assert_allclose(res.integral, f.ref, rtol=rtol)
if f_number not in {14}: # mildly underestimates error here
true_error = abs(self.error(res.integral, f.ref)/res.integral)
assert true_error < res.error
if f_number in {7, 10, 12}: # succeeds, but doesn't know it
return
assert res.success
assert res.status == 0
@pytest.mark.skip_xp_backends(np_only=True,
reason="Distributions aren't xp-compatible.")
@pytest.mark.parametrize('ref', (0.5, [0.4, 0.6]))
@pytest.mark.parametrize('case', stats._distr_params.distcont)
def test_accuracy(self, ref, case, xp):
distname, params = case
if distname in {'dgamma', 'dweibull', 'laplace', 'kstwo'}:
# should split up interval at first-derivative discontinuity
pytest.skip('tanh-sinh is not great for non-smooth integrands')
if (distname in {'studentized_range', 'levy_stable'}
and not int(os.getenv('SCIPY_XSLOW', 0))):
pytest.skip('This case passes, but it is too slow.')
dist = getattr(stats, distname)(*params)
x = dist.interval(ref)
res = _tanhsinh(dist.pdf, *x)
assert_allclose(res.integral, ref)
@pytest.mark.parametrize('shape', [tuple(), (12,), (3, 4), (3, 2, 2)])
def test_vectorization(self, shape, xp):
# Test for correct functionality, output shapes, and dtypes for various
# input shapes.
rng = np.random.default_rng(82456839535679456794)
a = xp.asarray(rng.random(shape))
b = xp.asarray(rng.random(shape))
p = xp.asarray(rng.random(shape))
n = math.prod(shape)
def f(x, p):
f.ncall += 1
f.feval += 1 if (xp_size(x) == n or x.ndim <= 1) else x.shape[-1]
return x**p
f.ncall = 0
f.feval = 0
@_vectorize(xp)
def _tanhsinh_single(a, b, p):
return _tanhsinh(lambda x: x**p, a, b)
res = _tanhsinh(f, a, b, args=(p,))
refs = _tanhsinh_single(a, b, p)
xp_test = array_namespace(a) # need xp.stack, isdtype
attrs = ['integral', 'error', 'success', 'status', 'nfev', 'maxlevel']
for attr in attrs:
ref_attr = xp_test.stack([getattr(ref, attr) for ref in refs])
res_attr = xp_ravel(getattr(res, attr))
xp_assert_close(res_attr, ref_attr, rtol=1e-15)
assert getattr(res, attr).shape == shape
assert xp_test.isdtype(res.success.dtype, 'bool')
assert xp_test.isdtype(res.status.dtype, 'integral')
assert xp_test.isdtype(res.nfev.dtype, 'integral')
assert xp_test.isdtype(res.maxlevel.dtype, 'integral')
assert xp.max(res.nfev) == f.feval
# maxlevel = 2 -> 3 function calls (2 initialization, 1 work)
assert xp.max(res.maxlevel) >= 2
assert xp.max(res.maxlevel) == f.ncall
def test_flags(self, xp):
# Test cases that should produce different status flags; show that all
# can be produced simultaneously.
def f(xs, js):
f.nit += 1
funcs = [lambda x: xp.exp(-x**2), # converges
lambda x: xp.exp(x), # reaches maxiter due to order=2
lambda x: xp.full_like(x, xp.nan)] # stops due to NaN
res = []
for i in range(xp_size(js)):
x = xs[i, ...]
j = int(xp_ravel(js)[i])
res.append(funcs[j](x))
return xp.stack(res)
f.nit = 0
args = (xp.arange(3, dtype=xp.int64),)
a = xp.asarray([xp.inf]*3)
b = xp.asarray([-xp.inf] * 3)
res = _tanhsinh(f, a, b, maxlevel=5, args=args)
ref_flags = xp.asarray([0, -2, -3], dtype=xp.int32)
xp_assert_equal(res.status, ref_flags)
def test_flags_preserve_shape(self, xp):
# Same test as above but using `preserve_shape` option to simplify.
def f(x):
res = [xp.exp(-x[0]**2), # converges
xp.exp(x[1]), # reaches maxiter due to order=2
xp.full_like(x[2], xp.nan)] # stops due to NaN
return xp.stack(res)
a = xp.asarray([xp.inf] * 3)
b = xp.asarray([-xp.inf] * 3)
res = _tanhsinh(f, a, b, maxlevel=5, preserve_shape=True)
ref_flags = xp.asarray([0, -2, -3], dtype=xp.int32)
xp_assert_equal(res.status, ref_flags)
def test_preserve_shape(self, xp):
# Test `preserve_shape` option
def f(x, xp):
return xp.stack([xp.stack([x, xp.sin(10 * x)]),
xp.stack([xp.cos(30 * x), x * xp.sin(100 * x)])])
ref = quad_vec(lambda x: f(x, np), 0, 1)
res = _tanhsinh(lambda x: f(x, xp), xp.asarray(0), xp.asarray(1),
preserve_shape=True)
dtype = xp.asarray(0.).dtype
xp_assert_close(res.integral, xp.asarray(ref[0], dtype=dtype))
def test_convergence(self, xp):
# demonstrate that number of accurate digits doubles each iteration
dtype = xp.float64 # this only works with good precision
def f(t):
return t * xp.log(1 + t)
ref = xp.asarray(0.25, dtype=dtype)
a, b = xp.asarray(0., dtype=dtype), xp.asarray(1., dtype=dtype)
last_logerr = 0
for i in range(4):
res = _tanhsinh(f, a, b, minlevel=0, maxlevel=i)
logerr = self.error(res.integral, ref, log=True, xp=xp)
assert (logerr < last_logerr * 2 or logerr < -15.5)
last_logerr = logerr
def test_options_and_result_attributes(self, xp):
# demonstrate that options are behaving as advertised and status
# messages are as intended
xp_test = array_namespace(xp.asarray(1.)) # need xp.atan
def f(x):
f.calls += 1
f.feval += xp_size(xp.asarray(x))
return x**2 * xp_test.atan(x)
f.ref = xp.asarray((math.pi - 2 + 2 * math.log(2)) / 12, dtype=xp.float64)
default_rtol = 1e-12
default_atol = f.ref * default_rtol # effective default absolute tol
# Keep things simpler by leaving tolerances fixed rather than
# having to make them dtype-dependent
a = xp.asarray(0., dtype=xp.float64)
b = xp.asarray(1., dtype=xp.float64)
# Test default options
f.feval, f.calls = 0, 0
ref = _tanhsinh(f, a, b)
assert self.error(ref.integral, f.ref) < ref.error < default_atol
assert ref.nfev == f.feval
ref.calls = f.calls # reference number of function calls
assert ref.success
assert ref.status == 0
# Test `maxlevel` equal to required max level
# We should get all the same results
f.feval, f.calls = 0, 0
maxlevel = int(ref.maxlevel)
res = _tanhsinh(f, a, b, maxlevel=maxlevel)
res.calls = f.calls
assert res == ref
# Now reduce the maximum level. We won't meet tolerances.
f.feval, f.calls = 0, 0
maxlevel -= 1
assert maxlevel >= 2 # can't compare errors otherwise
res = _tanhsinh(f, a, b, maxlevel=maxlevel)
assert self.error(res.integral, f.ref) < res.error > default_atol
assert res.nfev == f.feval < ref.nfev
assert f.calls == ref.calls - 1
assert not res.success
assert res.status == eim._ECONVERR
# `maxfun` is currently not enforced
# # Test `maxfun` equal to required number of function evaluations
# # We should get all the same results
# f.feval, f.calls = 0, 0
# maxfun = ref.nfev
# res = _tanhsinh(f, 0, f.b, maxfun = maxfun)
# assert res == ref
#
# # Now reduce `maxfun`. We won't meet tolerances.
# f.feval, f.calls = 0, 0
# maxfun -= 1
# res = _tanhsinh(f, 0, f.b, maxfun=maxfun)
# assert self.error(res.integral, f.ref) < res.error > default_atol
# assert res.nfev == f.feval < ref.nfev
# assert f.calls == ref.calls - 1
# assert not res.success
# assert res.status == 2
# Take this result to be the new reference
ref = res
ref.calls = f.calls
# Test `atol`
f.feval, f.calls = 0, 0
# With this tolerance, we should get the exact same result as ref
atol = np.nextafter(float(ref.error), np.inf)
res = _tanhsinh(f, a, b, rtol=0, atol=atol)
assert res.integral == ref.integral
assert res.error == ref.error
assert res.nfev == f.feval == ref.nfev
assert f.calls == ref.calls
# Except the result is considered to be successful
assert res.success
assert res.status == 0
f.feval, f.calls = 0, 0
# With a tighter tolerance, we should get a more accurate result
atol = np.nextafter(float(ref.error), -np.inf)
res = _tanhsinh(f, a, b, rtol=0, atol=atol)
assert self.error(res.integral, f.ref) < res.error < atol
assert res.nfev == f.feval > ref.nfev
assert f.calls > ref.calls
assert res.success
assert res.status == 0
# Test `rtol`
f.feval, f.calls = 0, 0
# With this tolerance, we should get the exact same result as ref
rtol = np.nextafter(float(ref.error/ref.integral), np.inf)
res = _tanhsinh(f, a, b, rtol=rtol)
assert res.integral == ref.integral
assert res.error == ref.error
assert res.nfev == f.feval == ref.nfev
assert f.calls == ref.calls
# Except the result is considered to be successful
assert res.success
assert res.status == 0
f.feval, f.calls = 0, 0
# With a tighter tolerance, we should get a more accurate result
rtol = np.nextafter(float(ref.error/ref.integral), -np.inf)
res = _tanhsinh(f, a, b, rtol=rtol)
assert self.error(res.integral, f.ref)/f.ref < res.error/res.integral < rtol
assert res.nfev == f.feval > ref.nfev
assert f.calls > ref.calls
assert res.success
assert res.status == 0
@pytest.mark.skip_xp_backends('torch', reason=
'https://github.com/scipy/scipy/pull/21149#issuecomment-2330477359',
)
@pytest.mark.parametrize('rtol', [1e-4, 1e-14])
def test_log(self, rtol, xp):
# Test equivalence of log-integration and regular integration
test_tols = dict(atol=1e-18, rtol=1e-15)
# Positive integrand (real log-integrand)
a = xp.asarray(-1., dtype=xp.float64)
b = xp.asarray(2., dtype=xp.float64)
res = _tanhsinh(norm_logpdf, a, b, log=True, rtol=math.log(rtol))
ref = _tanhsinh(norm_pdf, a, b, rtol=rtol)
xp_assert_close(xp.exp(res.integral), ref.integral, **test_tols)
xp_assert_close(xp.exp(res.error), ref.error, **test_tols)
assert res.nfev == ref.nfev
# Real integrand (complex log-integrand)
def f(x):
return -norm_logpdf(x)*norm_pdf(x)
def logf(x):
return xp.log(norm_logpdf(x) + 0j) + norm_logpdf(x) + xp.pi * 1j
a = xp.asarray(-xp.inf, dtype=xp.float64)
b = xp.asarray(xp.inf, dtype=xp.float64)
res = _tanhsinh(logf, a, b, log=True)
ref = _tanhsinh(f, a, b)
# In gh-19173, we saw `invalid` warnings on one CI platform.
# Silencing `all` because I can't reproduce locally and don't want
# to risk the need to run CI again.
with np.errstate(all='ignore'):
xp_assert_close(xp.exp(res.integral), ref.integral, **test_tols,
check_dtype=False)
xp_assert_close(xp.exp(res.error), ref.error, **test_tols,
check_dtype=False)
assert res.nfev == ref.nfev
def test_complex(self, xp):
# Test integration of complex integrand
# Finite limits
def f(x):
return xp.exp(1j * x)
a, b = xp.asarray(0.), xp.asarray(xp.pi/4)
res = _tanhsinh(f, a, b)
ref = math.sqrt(2)/2 + (1-math.sqrt(2)/2)*1j
xp_assert_close(res.integral, xp.asarray(ref))
# Infinite limits
def f(x):
return norm_pdf(x) + 1j/2*norm_pdf(x/2)
a, b = xp.asarray(xp.inf), xp.asarray(-xp.inf)
res = _tanhsinh(f, a, b)
xp_assert_close(res.integral, xp.asarray(-(1+1j)))
@pytest.mark.parametrize("maxlevel", range(4))
def test_minlevel(self, maxlevel, xp):
# Verify that minlevel does not change the values at which the
# integrand is evaluated or the integral/error estimates, only the
# number of function calls
# need `xp.concat`, `xp.atan`, and `xp.sort`
xp_test = array_namespace(xp.asarray(1.))
def f(x):
f.calls += 1
f.feval += xp_size(xp.asarray(x))
f.x = xp_test.concat((f.x, xp_ravel(x)))
return x**2 * xp_test.atan(x)
f.feval, f.calls, f.x = 0, 0, xp.asarray([])
a = xp.asarray(0, dtype=xp.float64)
b = xp.asarray(1, dtype=xp.float64)
ref = _tanhsinh(f, a, b, minlevel=0, maxlevel=maxlevel)
ref_x = xp_test.sort(f.x)
for minlevel in range(0, maxlevel + 1):
f.feval, f.calls, f.x = 0, 0, xp.asarray([])
options = dict(minlevel=minlevel, maxlevel=maxlevel)
res = _tanhsinh(f, a, b, **options)
# Should be very close; all that has changed is the order of values
xp_assert_close(res.integral, ref.integral, rtol=4e-16)
# Difference in absolute errors << magnitude of integral
xp_assert_close(res.error, ref.error, atol=4e-16 * ref.integral)
assert res.nfev == f.feval == f.x.shape[0]
assert f.calls == maxlevel - minlevel + 1 + 1 # 1 validation call
assert res.status == ref.status
xp_assert_equal(ref_x, xp_test.sort(f.x))
def test_improper_integrals(self, xp):
# Test handling of infinite limits of integration (mixed with finite limits)
def f(x):
x[xp.isinf(x)] = xp.nan
return xp.exp(-x**2)
a = xp.asarray([-xp.inf, 0, -xp.inf, xp.inf, -20, -xp.inf, -20])
b = xp.asarray([xp.inf, xp.inf, 0, -xp.inf, 20, 20, xp.inf])
ref = math.sqrt(math.pi)
ref = xp.asarray([ref, ref/2, ref/2, -ref, ref, ref, ref])
res = _tanhsinh(f, a, b)
xp_assert_close(res.integral, ref)
@pytest.mark.parametrize("limits", ((0, 3), ([-math.inf, 0], [3, 3])))
@pytest.mark.parametrize("dtype", ('float32', 'float64'))
def test_dtype(self, limits, dtype, xp):
# Test that dtypes are preserved
dtype = getattr(xp, dtype)
a, b = xp.asarray(limits, dtype=dtype)
def f(x):
assert x.dtype == dtype
return xp.exp(x)
rtol = 1e-12 if dtype == xp.float64 else 1e-5
res = _tanhsinh(f, a, b, rtol=rtol)
assert res.integral.dtype == dtype
assert res.error.dtype == dtype
assert xp.all(res.success)
xp_assert_close(res.integral, xp.exp(b)-xp.exp(a))
def test_maxiter_callback(self, xp):
# Test behavior of `maxiter` parameter and `callback` interface
a, b = xp.asarray(-xp.inf), xp.asarray(xp.inf)
def f(x):
return xp.exp(-x*x)
minlevel, maxlevel = 0, 2
maxiter = maxlevel - minlevel + 1
kwargs = dict(minlevel=minlevel, maxlevel=maxlevel, rtol=1e-15)
res = _tanhsinh(f, a, b, **kwargs)
assert not res.success
assert res.maxlevel == maxlevel
def callback(res):
callback.iter += 1
callback.res = res
assert hasattr(res, 'integral')
assert res.status == 1
if callback.iter == maxiter:
raise StopIteration
callback.iter = -1 # callback called once before first iteration
callback.res = None
del kwargs['maxlevel']
res2 = _tanhsinh(f, a, b, **kwargs, callback=callback)
# terminating with callback is identical to terminating due to maxiter
# (except for `status`)
for key in res.keys():
if key == 'status':
assert res[key] == -2
assert res2[key] == -4
else:
assert res2[key] == callback.res[key] == res[key]
def test_jumpstart(self, xp):
# The intermediate results at each level i should be the same as the
# final results when jumpstarting at level i; i.e. minlevel=maxlevel=i
a = xp.asarray(-xp.inf, dtype=xp.float64)
b = xp.asarray(xp.inf, dtype=xp.float64)
def f(x):
return xp.exp(-x*x)
def callback(res):
callback.integrals.append(xp_copy(res.integral)[()])
callback.errors.append(xp_copy(res.error)[()])
callback.integrals = []
callback.errors = []
maxlevel = 4
_tanhsinh(f, a, b, minlevel=0, maxlevel=maxlevel, callback=callback)
for i in range(maxlevel + 1):
res = _tanhsinh(f, a, b, minlevel=i, maxlevel=i)
xp_assert_close(callback.integrals[1+i], res.integral, rtol=1e-15)
xp_assert_close(callback.errors[1+i], res.error, rtol=1e-15, atol=1e-16)
def test_special_cases(self, xp):
# Test edge cases and other special cases
a, b = xp.asarray(0), xp.asarray(1)
xp_test = array_namespace(a, b) # need `xp.isdtype`
def f(x):
assert xp_test.isdtype(x.dtype, "real floating")
return x
res = _tanhsinh(f, a, b)
assert res.success
xp_assert_close(res.integral, xp.asarray(0.5))
# Test levels 0 and 1; error is NaN
res = _tanhsinh(f, a, b, maxlevel=0)
assert res.integral > 0
xp_assert_equal(res.error, xp.asarray(xp.nan))
res = _tanhsinh(f, a, b, maxlevel=1)
assert res.integral > 0
xp_assert_equal(res.error, xp.asarray(xp.nan))
# Test equal left and right integration limits
res = _tanhsinh(f, b, b)
assert res.success
assert res.maxlevel == -1
xp_assert_close(res.integral, xp.asarray(0.))
# Test scalar `args` (not in tuple)
def f(x, c):
return x**c
res = _tanhsinh(f, a, b, args=29)
xp_assert_close(res.integral, xp.asarray(1/30))
# Test NaNs
a = xp.asarray([xp.nan, 0, 0, 0])
b = xp.asarray([1, xp.nan, 1, 1])
c = xp.asarray([1, 1, xp.nan, 1])
res = _tanhsinh(f, a, b, args=(c,))
xp_assert_close(res.integral, xp.asarray([xp.nan, xp.nan, xp.nan, 0.5]))
xp_assert_equal(res.error[:3], xp.full((3,), xp.nan))
xp_assert_equal(res.status, xp.asarray([-3, -3, -3, 0], dtype=xp.int32))
xp_assert_equal(res.success, xp.asarray([False, False, False, True]))
xp_assert_equal(res.nfev[:3], xp.full((3,), 1, dtype=xp.int32))
# Test complex integral followed by real integral
# Previously, h0 was of the result dtype. If the `dtype` were complex,
# this could lead to complex cached abscissae/weights. If these get
# cast to real dtype for a subsequent real integral, we would get a
# ComplexWarning. Check that this is avoided.
_pair_cache.xjc = xp.empty(0)
_pair_cache.wj = xp.empty(0)
_pair_cache.indices = [0]
_pair_cache.h0 = None
a, b = xp.asarray(0), xp.asarray(1)
res = _tanhsinh(lambda x: xp.asarray(x*1j), a, b)
xp_assert_close(res.integral, xp.asarray(0.5*1j))
res = _tanhsinh(lambda x: x, a, b)
xp_assert_close(res.integral, xp.asarray(0.5))
# Test zero-size
shape = (0, 3)
res = _tanhsinh(lambda x: x, xp.asarray(0), xp.zeros(shape))
attrs = ['integral', 'error', 'success', 'status', 'nfev', 'maxlevel']
for attr in attrs:
assert res[attr].shape == shape
@pytest.mark.skip_xp_backends(np_only=True)
def test_compress_nodes_weights_gh21496(self, xp):
# See discussion in:
# https://github.com/scipy/scipy/pull/21496#discussion_r1878681049
# This would cause "ValueError: attempt to get argmax of an empty sequence"
# Check that this has been resolved.
x = np.full(65, 3)
x[-1] = 1000
_tanhsinh(np.sin, 1, x)
@array_api_compatible
@pytest.mark.usefixtures("skip_xp_backends")
@pytest.mark.skip_xp_backends('array_api_strict', reason='No fancy indexing.')
@pytest.mark.skip_xp_backends('jax.numpy', reason='No mutation.')
class TestNSum:
rng = np.random.default_rng(5895448232066142650)
p = rng.uniform(1, 10, size=10).tolist()
def f1(self, k):
# Integers are never passed to `f1`; if they were, we'd get
# integer to negative integer power error
return k**(-2)
f1.ref = np.pi**2/6
f1.a = 1
f1.b = np.inf
f1.args = tuple()
def f2(self, k, p):
return 1 / k**p
f2.ref = special.zeta(p, 1)
f2.a = 1.
f2.b = np.inf
f2.args = (p,)
def f3(self, k, p):
return 1 / k**p
f3.a = 1
f3.b = rng.integers(5, 15, size=(3, 1))
f3.ref = _gen_harmonic_gt1(f3.b, p)
f3.args = (p,)
def test_input_validation(self, xp):
f = self.f1
a, b = xp.asarray(f.a), xp.asarray(f.b)
message = '`f` must be callable.'
with pytest.raises(ValueError, match=message):
nsum(42, a, b)
message = '...must be True or False.'
with pytest.raises(ValueError, match=message):
nsum(f, a, b, log=2)
message = '...must be real numbers.'
with pytest.raises(ValueError, match=message):
nsum(f, xp.asarray(1+1j), b)
with pytest.raises(ValueError, match=message):
nsum(f, a, xp.asarray(1+1j))
with pytest.raises(ValueError, match=message):
nsum(f, a, b, step=xp.asarray(1+1j))
with pytest.raises(ValueError, match=message):
nsum(f, a, b, tolerances=dict(atol='ekki'))
with pytest.raises(ValueError, match=message):
nsum(f, a, b, tolerances=dict(rtol=pytest))
with np.errstate(all='ignore'):
res = nsum(f, xp.asarray([np.nan, np.inf]), xp.asarray(1.))
assert xp.all((res.status == -1) & xp.isnan(res.sum)
& xp.isnan(res.error) & ~res.success & res.nfev == 1)
res = nsum(f, xp.asarray(10.), xp.asarray([np.nan, 1]))
assert xp.all((res.status == -1) & xp.isnan(res.sum)
& xp.isnan(res.error) & ~res.success & res.nfev == 1)
res = nsum(f, xp.asarray(1.), xp.asarray(10.),
step=xp.asarray([xp.nan, -xp.inf, xp.inf, -1, 0]))
assert xp.all((res.status == -1) & xp.isnan(res.sum)
& xp.isnan(res.error) & ~res.success & res.nfev == 1)
message = '...must be non-negative and finite.'
with pytest.raises(ValueError, match=message):
nsum(f, a, b, tolerances=dict(rtol=-1))
with pytest.raises(ValueError, match=message):
nsum(f, a, b, tolerances=dict(atol=np.inf))
message = '...may not be positive infinity.'
with pytest.raises(ValueError, match=message):
nsum(f, a, b, tolerances=dict(rtol=np.inf), log=True)
with pytest.raises(ValueError, match=message):
nsum(f, a, b, tolerances=dict(atol=np.inf), log=True)
message = '...must be a non-negative integer.'
with pytest.raises(ValueError, match=message):
nsum(f, a, b, maxterms=3.5)
with pytest.raises(ValueError, match=message):
nsum(f, a, b, maxterms=-2)
@pytest.mark.parametrize('f_number', range(1, 4))
def test_basic(self, f_number, xp):
dtype = xp.asarray(1.).dtype
f = getattr(self, f"f{f_number}")
a, b = xp.asarray(f.a), xp.asarray(f.b),
args = tuple(xp.asarray(arg) for arg in f.args)
ref = xp.asarray(f.ref, dtype=dtype)
res = nsum(f, a, b, args=args)
xp_assert_close(res.sum, ref)
xp_assert_equal(res.status, xp.zeros(ref.shape, dtype=xp.int32))
xp_test = array_namespace(a) # CuPy doesn't have `bool`
xp_assert_equal(res.success, xp.ones(ref.shape, dtype=xp_test.bool))
with np.errstate(divide='ignore'):
logres = nsum(lambda *args: xp.log(f(*args)),
a, b, log=True, args=args)
xp_assert_close(xp.exp(logres.sum), res.sum)
xp_assert_close(xp.exp(logres.error), res.error, atol=1e-15)
xp_assert_equal(logres.status, res.status)
xp_assert_equal(logres.success, res.success)
@pytest.mark.parametrize('maxterms', [0, 1, 10, 20, 100])
def test_integral(self, maxterms, xp):
# test precise behavior of integral approximation
f = self.f1
def logf(x):
return -2*xp.log(x)
def F(x):
return -1 / x
a = xp.asarray([1, 5], dtype=xp.float64)[:, xp.newaxis]
b = xp.asarray([20, 100, xp.inf], dtype=xp.float64)[:, xp.newaxis, xp.newaxis]
step = xp.asarray([0.5, 1, 2], dtype=xp.float64).reshape((-1, 1, 1, 1))
nsteps = xp.floor((b - a)/step)
b_original = b
b = a + nsteps*step
k = a + maxterms*step
# partial sum
direct = xp.sum(f(a + xp.arange(maxterms)*step), axis=-1, keepdims=True)
integral = (F(b) - F(k))/step # integral approximation of remainder
low = direct + integral + f(b) # theoretical lower bound
high = direct + integral + f(k) # theoretical upper bound
ref_sum = (low + high)/2 # nsum uses average of the two
ref_err = (high - low)/2 # error (assuming perfect quadrature)
# correct reference values where number of terms < maxterms
xp_test = array_namespace(a) # torch needs broadcast_arrays
a, b, step = xp_test.broadcast_arrays(a, b, step)
for i in np.ndindex(a.shape):
ai, bi, stepi = float(a[i]), float(b[i]), float(step[i])
if (bi - ai)/stepi + 1 <= maxterms:
direct = xp.sum(f(xp.arange(ai, bi+stepi, stepi, dtype=xp.float64)))
ref_sum[i] = direct
ref_err[i] = direct * xp.finfo(direct.dtype).eps
rtol = 1e-12
res = nsum(f, a, b_original, step=step, maxterms=maxterms,
tolerances=dict(rtol=rtol))
xp_assert_close(res.sum, ref_sum, rtol=10*rtol)
xp_assert_close(res.error, ref_err, rtol=100*rtol)
i = ((b_original - a)/step + 1 <= maxterms)
xp_assert_close(res.sum[i], ref_sum[i], rtol=1e-15)
xp_assert_close(res.error[i], ref_err[i], rtol=1e-15)
logres = nsum(logf, a, b_original, step=step, log=True,
tolerances=dict(rtol=math.log(rtol)), maxterms=maxterms)
xp_assert_close(xp.exp(logres.sum), res.sum)
xp_assert_close(xp.exp(logres.error), res.error)
@pytest.mark.parametrize('shape', [tuple(), (12,), (3, 4), (3, 2, 2)])
def test_vectorization(self, shape, xp):
# Test for correct functionality, output shapes, and dtypes for various
# input shapes.
rng = np.random.default_rng(82456839535679456794)
a = rng.integers(1, 10, size=shape)
# when the sum can be computed directly or `maxterms` is large enough
# to meet `atol`, there are slight differences (for good reason)
# between vectorized call and looping.
b = np.inf
p = rng.random(shape) + 1
n = math.prod(shape)
def f(x, p):
f.feval += 1 if (x.size == n or x.ndim <= 1) else x.shape[-1]
return 1 / x ** p
f.feval = 0
@np.vectorize
def nsum_single(a, b, p, maxterms):
return nsum(lambda x: 1 / x**p, a, b, maxterms=maxterms)
res = nsum(f, xp.asarray(a), xp.asarray(b), maxterms=1000,
args=(xp.asarray(p),))
refs = nsum_single(a, b, p, maxterms=1000).ravel()
attrs = ['sum', 'error', 'success', 'status', 'nfev']
for attr in attrs:
ref_attr = [xp.asarray(getattr(ref, attr)) for ref in refs]
res_attr = getattr(res, attr)
xp_assert_close(xp_ravel(res_attr), xp.asarray(ref_attr), rtol=1e-15)
assert res_attr.shape == shape
xp_test = array_namespace(xp.asarray(1.))
assert xp_test.isdtype(res.success.dtype, 'bool')
assert xp_test.isdtype(res.status.dtype, 'integral')
assert xp_test.isdtype(res.nfev.dtype, 'integral')
if is_numpy(xp): # other libraries might have different number
assert int(xp.max(res.nfev)) == f.feval
def test_status(self, xp):
f = self.f2
p = [2, 2, 0.9, 1.1, 2, 2]
a = xp.asarray([0, 0, 1, 1, 1, np.nan], dtype=xp.float64)
b = xp.asarray([10, np.inf, np.inf, np.inf, np.inf, np.inf], dtype=xp.float64)
ref = special.zeta(p, 1)
p = xp.asarray(p, dtype=xp.float64)
with np.errstate(divide='ignore'): # intentionally dividing by zero
res = nsum(f, a, b, args=(p,))
ref_success = xp.asarray([False, False, False, False, True, False])
ref_status = xp.asarray([-3, -3, -2, -4, 0, -1], dtype=xp.int32)
xp_assert_equal(res.success, ref_success)
xp_assert_equal(res.status, ref_status)
xp_assert_close(res.sum[res.success], xp.asarray(ref)[res.success])
def test_nfev(self, xp):
def f(x):
f.nfev += xp_size(x)
return 1 / x**2
f.nfev = 0
res = nsum(f, xp.asarray(1), xp.asarray(10))
assert res.nfev == f.nfev
f.nfev = 0
res = nsum(f, xp.asarray(1), xp.asarray(xp.inf), tolerances=dict(atol=1e-6))
assert res.nfev == f.nfev
def test_inclusive(self, xp):
# There was an edge case off-by one bug when `_direct` was called with
# `inclusive=True`. Check that this is resolved.
a = xp.asarray([1, 4])
b = xp.asarray(xp.inf)
res = nsum(lambda k: 1 / k ** 2, a, b,
maxterms=500, tolerances=dict(atol=0.1))
ref = nsum(lambda k: 1 / k ** 2, a, b)
assert xp.all(res.sum > (ref.sum - res.error))
assert xp.all(res.sum < (ref.sum + res.error))
@pytest.mark.parametrize('log', [True, False])
def test_infinite_bounds(self, log, xp):
a = xp.asarray([1, -np.inf, -np.inf])
b = xp.asarray([np.inf, -1, np.inf])
c = xp.asarray([1, 2, 3])
def f(x, a):
return (xp.log(xp.tanh(a / 2)) - a*xp.abs(x) if log
else xp.tanh(a/2) * xp.exp(-a*xp.abs(x)))
res = nsum(f, a, b, args=(c,), log=log)
ref = xp.asarray([stats.dlaplace.sf(0, 1), stats.dlaplace.sf(0, 2), 1])
ref = xp.log(ref) if log else ref
atol = (1e-10 if a.dtype==xp.float64 else 1e-5) if log else 0
xp_assert_close(res.sum, xp.asarray(ref, dtype=a.dtype), atol=atol)
# # Make sure the sign of `x` passed into `f` is correct.
def f(x, c):
return -3*xp.log(c*x) if log else 1 / (c*x)**3
a = xp.asarray([1, -np.inf])
b = xp.asarray([np.inf, -1])
arg = xp.asarray([1, -1])
res = nsum(f, a, b, args=(arg,), log=log)
ref = np.log(special.zeta(3)) if log else special.zeta(3)
xp_assert_close(res.sum, xp.full(a.shape, ref, dtype=a.dtype))
def test_decreasing_check(self, xp):
# Test accuracy when we start sum on an uphill slope.
# Without the decreasing check, the terms would look small enough to
# use the integral approximation. Because the function is not decreasing,
# the error is not bounded by the magnitude of the last term of the
# partial sum. In this case, the error would be ~1e-4, causing the test
# to fail.
def f(x):
return xp.exp(-x ** 2)
a, b = xp.asarray(-25, dtype=xp.float64), xp.asarray(np.inf, dtype=xp.float64)
res = nsum(f, a, b)
# Reference computed with mpmath:
# from mpmath import mp
# mp.dps = 50
# def fmp(x): return mp.exp(-x**2)
# ref = mp.nsum(fmp, (-25, 0)) + mp.nsum(fmp, (1, mp.inf))
ref = xp.asarray(1.772637204826652, dtype=xp.float64)
xp_assert_close(res.sum, ref, rtol=1e-15)
def test_special_case(self, xp):
# test equal lower/upper limit
f = self.f1
a = b = xp.asarray(2)
res = nsum(f, a, b)
xp_assert_equal(res.sum, xp.asarray(f(2)))
# Test scalar `args` (not in tuple)
res = nsum(self.f2, xp.asarray(1), xp.asarray(np.inf), args=xp.asarray(2))
xp_assert_close(res.sum, xp.asarray(self.f1.ref)) # f1.ref is correct w/ args=2
# Test 0 size input
a = xp.empty((3, 1, 1)) # arbitrary broadcastable shapes
b = xp.empty((0, 1)) # could use Hypothesis
p = xp.empty(4) # but it's overkill
shape = np.broadcast_shapes(a.shape, b.shape, p.shape)
res = nsum(self.f2, a, b, args=(p,))
assert res.sum.shape == shape
assert res.status.shape == shape
assert res.nfev.shape == shape
# Test maxterms=0
def f(x):
with np.errstate(divide='ignore'):
return 1 / x
res = nsum(f, xp.asarray(0), xp.asarray(10), maxterms=0)
assert xp.isnan(res.sum)
assert xp.isnan(res.error)
assert res.status == -2
res = nsum(f, xp.asarray(0), xp.asarray(10), maxterms=1)
assert xp.isnan(res.sum)
assert xp.isnan(res.error)
assert res.status == -3
# Test NaNs
# should skip both direct and integral methods if there are NaNs
a = xp.asarray([xp.nan, 1, 1, 1])
b = xp.asarray([xp.inf, xp.nan, xp.inf, xp.inf])
p = xp.asarray([2, 2, xp.nan, 2])
res = nsum(self.f2, a, b, args=(p,))
xp_assert_close(res.sum, xp.asarray([xp.nan, xp.nan, xp.nan, self.f1.ref]))
xp_assert_close(res.error[:3], xp.full((3,), xp.nan))
xp_assert_equal(res.status, xp.asarray([-1, -1, -3, 0], dtype=xp.int32))
xp_assert_equal(res.success, xp.asarray([False, False, False, True]))
# Ideally res.nfev[2] would be 1, but `tanhsinh` has some function evals
xp_assert_equal(res.nfev[:2], xp.full((2,), 1, dtype=xp.int32))
@pytest.mark.parametrize('dtype', ['float32', 'float64'])
def test_dtype(self, dtype, xp):
dtype = getattr(xp, dtype)
def f(k):
assert k.dtype == dtype
return 1 / k ** xp.asarray(2, dtype=dtype)
a = xp.asarray(1, dtype=dtype)
b = xp.asarray([10, xp.inf], dtype=dtype)
res = nsum(f, a, b)
assert res.sum.dtype == dtype
assert res.error.dtype == dtype
rtol = 1e-12 if dtype == xp.float64 else 1e-6
ref = _gen_harmonic_gt1(np.asarray([10, xp.inf]), 2)
xp_assert_close(res.sum, xp.asarray(ref, dtype=dtype), rtol=rtol)
@pytest.mark.parametrize('case', [(10, 100), (100, 10)])
def test_nondivisible_interval(self, case, xp):
# When the limits of the sum are such that (b - a)/step
# is not exactly integral, check that only floor((b - a)/step)
# terms are included.
n, maxterms = case
def f(k):
return 1 / k ** 2
a = np.e
step = 1 / 3
b0 = a + n * step
i = np.arange(-2, 3)
b = b0 + i * np.spacing(b0)
ns = np.floor((b - a) / step)
assert len(set(ns)) == 2
a, b = xp.asarray(a, dtype=xp.float64), xp.asarray(b, dtype=xp.float64)
step, ns = xp.asarray(step, dtype=xp.float64), xp.asarray(ns, dtype=xp.float64)
res = nsum(f, a, b, step=step, maxterms=maxterms)
xp_assert_equal(xp.diff(ns) > 0, xp.diff(res.sum) > 0)
xp_assert_close(res.sum[-1], res.sum[0] + f(b0))
@pytest.mark.skip_xp_backends(np_only=True, reason='Needs beta function.')
def test_logser_kurtosis_gh20648(self, xp):
# Some functions return NaN at infinity rather than 0 like they should.
# Check that this is accounted for.
ref = stats.yulesimon.moment(4, 5)
def f(x):
return stats.yulesimon._pmf(x, 5) * x**4
with np.errstate(invalid='ignore'):
assert np.isnan(f(np.inf))
res = nsum(f, 1, np.inf)
assert_allclose(res.sum, ref)
|