gcd in generic polynomial rings with many generators
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Martin R
Nov 3, 2024, 6:31:44 AM11/3/24
to sage-devel
Are the following timings to be expected?
sage: R = SymmetricFunctions(QQ).h().fraction_field()
sage: P = PolynomialRing(R, 200, "x")
sage: e = P.gen(199); e
x199
sage: %time e.gcd(e)
CPU times: user 41.1 s, sys: 2.24 s, total: 43.4 s
Wall time: 43.5 s
x199
sage: P = PolynomialRing(R, 200, "x")
sage: e = P.gen(199); e
x199
sage: %time e.gcd(e)
CPU times: user 41.1 s, sys: 2.24 s, total: 43.4 s
Wall time: 43.5 s
x199
sage: R = QQ
sage: P = PolynomialRing(R, 200, "x")
sage: e = P.gen(199); e
x199
sage: %time e.gcd(e)
CPU times: user 41 µs, sys: 0 ns, total: 41 µs
Wall time: 47 µs
x199
sage: P = PolynomialRing(R, 200, "x")
sage: e = P.gen(199); e
x199
sage: %time e.gcd(e)
CPU times: user 41 µs, sys: 0 ns, total: 41 µs
Wall time: 47 µs
x199
sage: R = QQbar
sage: P = PolynomialRing(R, 200, "x")
sage: e = P.gen(199); e
x199
sage: %time e.gcd(e)
CPU times: user 3.02 s, sys: 183 ms, total: 3.2 s
Wall time: 3.21 s
x199
sage: P = PolynomialRing(R, 200, "x")
sage: e = P.gen(199); e
x199
sage: %time e.gcd(e)
CPU times: user 3.02 s, sys: 183 ms, total: 3.2 s
Wall time: 3.21 s
x199
Martin R
Nov 3, 2024, 11:11:23 AM11/3/24
to sage-devel
Possibly the problem is that MPolynomial.gcd calls UniqueFactorizationDomains.ParentMethods._gcd_univariate_polynomial, which recurses on MPolynomial.gcd, but the setup done by MPolynomial.gcd is possibly not cheap.
Martin R
Nov 3, 2024, 3:57:05 PM11/3/24
to sage-devel
I think I see things slightly clearer now. One problem is, that, by design, we successively reduce to polynomial rings with fewer generators. More precisely, MPolynomial.gcd converts the input to a univariate polynomial in one of the generators, and UniqueFactorizationDomains.ParentMethods._gcd_univariate_polynomial then relies on the existence of a gcd for the new base ring.
To do this, the new base ring, i.e., the ring with one fewer generator, must be constructed, and this is rather expensive. If I understand correctly, almost all the time is spent there. This is done in `MPolynomial.polynomial` using `S = PolynomialRing(R.base_ring(), Z)`
So, the questions are:
1. is there a (much) faster way to construct the univariate polynomial ring? I tried to do S = R.__class__(R.base_ring(), len(Z), [str(z) for z in Z], R.term_order()) when Z has at least two variables (because of the difference between univariate and multivariate polynomial rings) but to my surprise this was actually slower.
2. can we bypass the construction of polynomial rings entirely?
Martin R
Nov 4, 2024, 5:45:45 AM11/4/24
to sage-devel
Ah, S = R.__class__(R.base_ring(), len(Z), [str(z) for z in Z], R.term_order()) is silly, because it bypasses the caching mechanism.
In fact, I'm not sure anymore whether I am on the right track: the basic problem is described at the bottom of https://github.com/sagemath/sage/pull/38108. Looking at %prun (see below), I thought that a lot of time is wasted in constructing polynomial rings, especially the number of calls to PolynomialRing seems very large to me. However, looking at sage.rings.polynomial.polynomial_ring_constructor._cache.keys(), these are actually all have very few generators (at most 7), so the title of this post ("many generators") is completely wrong. It may still be problematic that `_gcd_univariate_polynomial` constructs them so often.
I'll investigate further. Help would be very much appreciated, of course.
Martin
%prun -s cumulative repr(D)
92714635 function calls (92104545 primitive calls) in 71.609 seconds
Ordered by: cumulative time
ncalls tottime percall cumtime percall filename:lineno(function)
1 0.000 0.000 71.783 71.783 {built-in method builtins.exec}
1 0.000 0.000 71.783 71.783 <string>:1(<module>)
790/1 0.002 0.000 71.783 71.783 {built-in method builtins.repr}
141/1 0.000 0.000 71.783 71.783 lazy_series.py:1550(_repr_)
29/1 0.001 0.000 71.783 71.783 lazy_series.py:5930(_format_series)
29/1 0.000 0.000 71.776 71.776 lazy_series.py:5959(<listcomp>)
100/11 0.000 0.000 71.775 6.525 lazy_series.py:320(__getitem__)
62/11 0.000 0.000 71.775 6.525 stream.py:1782(__getitem__)
7 0.000 0.000 71.775 10.254 stream.py:2096(_compute)
996 0.006 0.000 67.615 0.068 infinite_polynomial_element.py:1266(gcd)
17828/996 0.851 0.000 67.579 0.068 {method 'gcd' of 'sage.rings.polynomial.multi_polynomial.MPolynomial' objects}
18124/990 0.204 0.000 62.978 0.064 unique_factorization_domains.py:132(_gcd_univariate_polynomial)
1788/1062 0.026 0.000 55.622 0.052 fraction_field.py:560(_element_constructor_)
7 0.001 0.000 51.446 7.349 stream.py:2046(_solve_linear_equations_and_subs)
28 0.008 0.000 51.324 1.833 stream.py:1900(_subs_in_caches)
402 0.004 0.000 51.111 0.127 {method 'map_coefficients' of 'sage.rings.polynomial.multi_polynomial.MPolynomial' objects}
713 0.001 0.000 50.436 0.071 stream.py:1964(<lambda>)
713 0.019 0.000 50.435 0.071 stream.py:1922(subs)
233237/233209 0.949 0.000 46.016 0.000 multi_polynomial_element.py:457(_repr_)
233237/233209 4.064 0.000 44.502 0.000 {method 'poly_repr' of 'sage.rings.polynomial.polydict.PolyDict' objects}
1517930 2.132 0.000 43.219 0.000 magmatic_algebras.py:208(_product_from_product_on_basis_multiply)
1517930 1.505 0.000 40.697 0.000 free_module.py:1054(linear_combination)
1517930 4.470 0.000 38.232 0.000 {built-in method sage.data_structures.blas_dict.linear_combination}
183026/182240 1.749 0.000 34.106 0.000 polynomial_ring_constructor.py:60(PolynomialRing)
3034737 1.263 0.000 33.762 0.000 free_module.py:1078(<genexpr>)
3034737 2.818 0.000 32.499 0.000 magmatic_algebras.py:222(<genexpr>)
183026/182666 0.943 0.000 30.224 0.000 {built-in method sage.structure.category_object.normalize_names}
1516807 4.375 0.000 29.414 0.000 multiplicative.py:41(product_on_basis)
7 0.000 0.000 20.329 2.904 stream.py:1983(_collect_equations)
71/7 0.000 0.000 20.321 2.903 stream.py:396(__getitem__)
28/7 0.000 0.000 20.321 2.903 stream.py:2741(get_coefficient)
226182/164092 0.072 0.000 17.707 0.000 {built-in method builtins.sum}
24/19 0.000 0.000 17.654 0.929 stream.py:2808(get_coefficient)
24/19 0.000 0.000 17.654 0.929 additive_monoids.py:52(sum)
52/42 0.000 0.000 17.582 0.419 stream.py:2827(<genexpr>)
17861 0.111 0.000 15.377 0.001 polynomial_singular_interface.py:181(_singular_)
17861 0.366 0.000 15.266 0.001 polynomial_singular_interface.py:348(_singular_init_)
1516811 1.379 0.000 12.021 0.000 poor_man_map.py:241(__call__)
103578/87895 0.835 0.000 11.820 0.000 multi_polynomial_ring.py:177(__call__)
10284/8981 0.165 0.000 10.894 0.001 multi_polynomial_element.py:321(_mul_)
1516807 1.980 0.000 10.686 0.000 partition.py:462(__classcall_private__)
1516811 8.360 0.000 10.642 0.000 free_module.py:1106(_monomial)
12991 0.154 0.000 9.839 0.001 multi_polynomial_element.py:1849(_floordiv_)
109401 0.631 0.000 9.452 0.000 sfa.py:3241(gcd)
1626805 2.513 0.000 8.706 0.000 partition.py:6350(_element_constructor_)
35668/35580 0.190 0.000 8.503 0.000 rings.py:1148(__getitem__)
1454 0.017 0.000 6.957 0.005 multi_polynomial_element.py:121(__call__)
111193 0.404 0.000 6.924 0.000 sfa.py:6841(_to_polynomials)
7 0.000 0.000 6.363 0.909 lazy_series.py:5265(coefficient)
82/63 0.001 0.000 5.441 0.086 lazy_series_ring.py:2648(_element_constructor_)
1738587 1.196 0.000 5.187 0.000 partition.py:518(__init__)
443 0.005 0.000 4.027 0.009 fraction_field.py:712(resolve_fractions)
12497 0.297 0.000 3.871 0.000 multi_polynomial_ring.py:565(monomial_quotient)
14242 0.031 0.000 3.856 0.000 multi_polynomial_element.py:356(_rmul_)
14242 0.177 0.000 3.775 0.000 {method 'scalar_rmult' of 'sage.rings.polynomial.polydict.PolyDict' objects}
1738587 2.717 0.000 3.752 0.000 combinat.py:1526(__init__)
1326 0.003 0.000 3.042 0.002 multi_polynomial_element.py:335(_lmul_)
1326 0.016 0.000 3.033 0.002 {method 'scalar_lmult' of 'sage.rings.polynomial.polydict.PolyDict' objects}
28 0.001 0.000 2.644 0.094 multi_polynomial_element.py:307(_sub_)
1426 0.091 0.000 2.622 0.002 multi_polynomial_element.py:1432(subs)
107192 0.399 0.000 2.374 0.000 sfa.py:6873(<listcomp>)
3089637 1.873 0.000 2.165 0.000 free_module.py:1204(_from_dict)
15888578 2.153 0.000 2.153 0.000 {built-in method builtins.isinstance}
29 0.000 0.000 2.138 0.074 fraction_field.py:351(wrapper)
111193 0.328 0.000 2.015 0.000 sfa.py:6879(_from_polynomial)
456555/234169 0.361 0.000 1.912 0.000 {built-in method builtins.max}
110451 0.183 0.000 1.898 0.000 sfa.py:1797(is_unit)
103141/103022 0.288 0.000 1.863 0.000 {method 'coerce' of 'sage.structure.parent.Parent' objects}
5014 0.093 0.000 1.765 0.000 {method 'pseudo_quo_rem' of 'sage.rings.polynomial.polynomial_element.Polynomial' objects}
333579 0.259 0.000 1.745 0.000 sfa.py:6866(<genexpr>)
109994 0.802 0.000 1.657 0.000 modules_with_basis.py:1520(coefficient)
242646 1.101 0.000 1.587 0.000 polynomial_ring.py:336(_element_constructor_)
296 0.013 0.000 1.431 0.005 fraction_field.py:942(_gcd_univariate_polynomial)
107192 0.700 0.000 1.427 0.000 sfa.py:6905(<dictcomp>)
1737437 0.895 0.000 1.287 0.000 partition.py:6394(__contains__)
3234864 0.817 0.000 1.194 0.000 combinat.py:1447(__iter__)
708 0.002 0.000 1.090 0.002 infinite_polynomial_element.py:698(_lmul_)
137507/133349 0.151 0.000 1.065 0.000 multi_polynomial_element.py:1820(__eq__)
408 0.003 0.000 1.042 0.003 {method 'numerator' of 'sage.rings.polynomial.multi_polynomial.MPolynomial' objects}
592 0.010 0.000 1.026 0.002 {method 'numerator' of 'sage.rings.polynomial.polynomial_element.Polynomial' objects}
3234741 0.692 0.000 1.021 0.000 combinat.py:1421(__len__)
150040/149686 0.147 0.000 1.009 0.000 polynomial_ring_constructor.py:750(_single_variate)
1738587 0.680 0.000 1.002 0.000 partition.py:550(__hash__)
32986 0.152 0.000 0.967 0.000 polynomial_ring_constructor.py:831(_multi_variate)
7241407/7040280 0.848 0.000 0.908 0.000 {built-in method builtins.len}
112 0.000 0.000 0.904 0.008 lazy_series.py:2976(_mul_)
9 0.000 0.000 0.897 0.100 lazy_series.py:3091(<listcomp>)
18 0.000 0.000 0.897 0.050 lazy_series.py:3091(<genexpr>)
12647 0.012 0.000 0.882 0.000 {method 'is_one' of 'sage.structure.element.RingElement' objects}
1738587 0.841 0.000 0.841 0.000 combinat.py:1111(__init__)
222386 0.722 0.000 0.822 0.000 modules_with_basis.py:1651(support)
111782 0.265 0.000 0.688 0.000 partition.py:6615(from_exp)
214384 0.171 0.000 0.671 0.000 sfa.py:6873(<dictcomp>)
226135 0.338 0.000 0.636 0.000 term_order.py:999(sortkey_degrevlex)
183026 0.111 0.000 0.615 0.000 polynomial_ring_constructor.py:740(_get_from_cache)
20353/20206 0.019 0.000 0.598 0.000 multi_polynomial_ring.py:502(<dictcomp>)
32986 0.274 0.000 0.594 0.000 term_order.py:570(__init__)
144787 0.264 0.000 0.576 0.000 multi_polynomial_element.py:425(__init__)
201127 0.300 0.000 0.568 0.000 partition.py:3770(to_exp)
377/365 0.007 0.000 0.547 0.001 infinite_polynomial_ring.py:858(_element_constructor_)
7 0.000 0.000 0.537 0.077 lazy_series_ring.py:2833(<listcomp>)
183026 0.227 0.000 0.504 0.000 {method 'get' of 'sage.misc.weak_dict.WeakValueDictionary' objects}
873167 0.397 0.000 0.493 0.000 combinat.py:1202(__eq__)
1208 0.004 0.000 0.439 0.000 polynomial_ring.py:1928(__init__)
51473 0.172 0.000 0.432 0.000 multi_polynomial_element.py:827(__iter__)
1210 0.009 0.000 0.432 0.000 polynomial_ring.py:1801(__init__)
1210 0.051 0.000 0.420 0.000 polynomial_ring.py:246(__init__)
1894151/1836583 0.378 0.000 0.418 0.000 {built-in method builtins.hash}
3267511 0.409 0.000 0.409 0.000 {method 'monomial_coefficients' of 'sage.modules.with_basis.indexed_element.IndexedFreeModuleElement' objects}
638 0.024 0.000 0.392 0.001 classical.py:90(_element_constructor_)
3272864 0.386 0.000 0.386 0.000 {built-in method builtins.iter}
3491586 0.337 0.000 0.337 0.000 {method 'items' of 'dict' objects}
28565 0.086 0.000 0.301 0.000 multi_polynomial_element.py:571(degree)
1210 0.005 0.000 0.298 0.000 unital_algebras.py:74(__init_extra__)
1210 0.037 0.000 0.286 0.000 polynomial_ring.py:706(_coerce_map_from_base_ring)
259200 0.135 0.000 0.284 0.000 term_order.py:888(sortkey)
1516807 0.281 0.000 0.281 0.000 {method 'sort' of 'list' objects}
37976 0.010 0.000 0.264 0.000 {built-in method builtins.next}
1111 0.003 0.000 0.257 0.000 polynomial_ring.py:2159(__init__)
868 0.006 0.000 0.239 0.000 multi_polynomial_element.py:940(coefficient)
423513 0.214 0.000 0.238 0.000 support_view.py:102(__iter__)
2245496 0.230 0.000 0.230 0.000 {method 'base_ring' of 'sage.structure.category_object.CategoryObject' objects}
1792 0.019 0.000 0.226 0.000 sfa.py:3202(_floordiv_)
776 0.007 0.000 0.221 0.000 {method 'format' of 'str' objects}
21915 0.151 0.000 0.219 0.000 {method '_mpoly_dict_recursive' of 'sage.rings.polynomial.polynomial_element.Polynomial' objects}
184753 0.186 0.000 0.212 0.000 {method 'remove_zeros' of 'sage.rings.polynomial.polydict.PolyDict' objects}
54371 0.038 0.000 0.210 0.000 unital_algebras.py:354(from_base_ring_from_one_basis)
868 0.017 0.000 0.209 0.000 multi_polynomial_element.py:1373(is_monomial)
272 0.001 0.000 0.201 0.001 stream.py:1936(fix_cache)
4448 0.015 0.000 0.191 0.000 infinite_polynomial_element.py:1628(_richcmp_)
32785 0.030 0.000 0.186 0.000 re.py:232(findall)
25879 0.053 0.000 0.186 0.000 multi_polynomial_element.py:1211(_exponents)
183090 0.104 0.000 0.184 0.000 {built-in method builtins.any}
17855 0.082 0.000 0.176 0.000 polynomial_singular_interface.py:52(_do_singular_init_)
1550 0.007 0.000 0.174 0.000 {sage.misc.misc_c.prod}
1210 0.013 0.000 0.173 0.000 polynomial_ring.py:1240(gen)
54896 0.055 0.000 0.173 0.000 free_module.py:1083(term)
1546536 0.173 0.000 0.173 0.000 {method 'one' of 'sage.rings.ring.Ring' objects}
290 0.011 0.000 0.172 0.001 {method 'monic' of 'sage.rings.polynomial.polynomial_element.Polynomial' objects}
1536888 0.166 0.000 0.166 0.000 {method 'parent' of 'sage.structure.element.Element' objects}
262784 0.151 0.000 0.151 0.000 {built-in method builtins.getattr}
28784 0.052 0.000 0.146 0.000 multi_polynomial_ring.py:166(__hash__)
15174 0.034 0.000 0.143 0.000 {method 'coefficients' of 'sage.rings.polynomial.polynomial_element.Polynomial' objects}
4001 0.059 0.000 0.138 0.000 sfa.py:6875(<listcomp>)
82221 0.103 0.000 0.138 0.000 monoids.py:277(is_one)
1312/1310 0.008 0.000 0.138 0.000 polynomial_ring.py:734(_coerce_map_from_)
19069 0.089 0.000 0.133 0.000 polynomial_singular_interface.py:382(can_convert_to_singular)
631 0.000 0.000 0.129 0.000 infinite_polynomial_element.py:271(_repr_)
3939/2632 0.042 0.000 0.126 0.000 homset.py:87(Hom)
402 0.001 0.000 0.125 0.000 infinite_polynomial_element.py:555(numerator)
473 0.002 0.000 0.122 0.000 infinite_polynomial_element.py:1683(_mul_)
1507/1505 0.006 0.000 0.121 0.000 {method 'coerce_map_from' of 'sage.structure.parent.Parent' objects}
116841 0.076 0.000 0.121 0.000 multi_polynomial_element.py:805(monomial_coefficients)
107192 0.117 0.000 0.117 0.000 {method 'monomial_coefficients' of 'sage.rings.polynomial.polynomial_element.Polynomial' objects}
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22917 0.066 0.000 0.109 0.000 multi_polynomial_element.py:1352(is_generator)
10704/4397 0.071 0.000 0.104 0.000 {method 'has_coerce_map_from' of 'sage.structure.parent.Parent' objects}
32785 0.104 0.000 0.104 0.000 {method 'findall' of 're.Pattern' objects}
35668 0.102 0.000 0.102 0.000 rings.py:1365(<listcomp>)
7 0.007 0.001 0.100 0.014 {method 'solve_right' of 'sage.matrix.matrix2.Matrix' objects}
12497 0.040 0.000 0.100 0.000 multi_polynomial_ring.py:771(monomial_divides)
100617 0.068 0.000 0.096 0.000 <frozen importlib._bootstrap>:404(parent)
4001 0.057 0.000 0.095 0.000 sfa.py:6908(<dictcomp>)
99/97 0.001 0.000 0.094 0.001 fraction_field.py:219(_coerce_map_from_)
486632 0.091 0.000 0.091 0.000 multi_polynomial_element.py:416(element)
25886 0.026 0.000 0.090 0.000 {built-in method builtins.sorted}
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32986 0.049 0.000 0.061 0.000 term_order.py:1907(__eq__)
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3244 0.011 0.000 0.031 0.000 infinite_polynomial_element.py:180(__classcall_private__)
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312/192 0.000 0.000 0.029 0.000 repr.py:7(coeff_repr)
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92714635 function calls (92104545 primitive calls) in 71.609 seconds
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1 0.000 0.000 0.000 0.000 {method 'set_immutable' of 'sage.structure.element.ModuleElementWithMutability' objects}
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