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Matthias Koeppe
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src/sage/arith/misc.py: Doctest cosmetics, use more block tags
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Lines changed: 68 additions & 58 deletions

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‎src/sage/arith/misc.py‎

Lines changed: 68 additions & 58 deletions
Original file line numberDiff line numberDiff line change
@@ -107,19 +107,19 @@ def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None,
107107
compute a 200-bit approximation to `sqrt(2)` which is wrong in the
108108
33'rd bit::
109109
110-
sage: # needs sage.rings.real_mpfr
110+
sage: # needs sage.libs.pari sage.rings.real_mpfr
111111
sage: z = sqrt(RealField(200)(2)) + (1/2)^33
112-
sage: p = algdep(z, 4); p # needs sage.libs.pari
112+
sage: p = algdep(z, 4); p
113113
227004321085*x^4 - 216947902586*x^3 - 99411220986*x^2 + 82234881648*x - 211871195088
114-
sage: factor(p) # needs sage.libs.pari
114+
sage: factor(p)
115115
227004321085*x^4 - 216947902586*x^3 - 99411220986*x^2 + 82234881648*x - 211871195088
116-
sage: algdep(z, 4, known_bits=32) # needs sage.libs.pari
116+
sage: algdep(z, 4, known_bits=32)
117117
x^2 - 2
118-
sage: algdep(z, 4, known_digits=10) # needs sage.libs.pari
118+
sage: algdep(z, 4, known_digits=10)
119119
x^2 - 2
120-
sage: algdep(z, 4, use_bits=25) # needs sage.libs.pari
120+
sage: algdep(z, 4, use_bits=25)
121121
x^2 - 2
122-
sage: algdep(z, 4, use_digits=8) # needs sage.libs.pari
122+
sage: algdep(z, 4, use_digits=8)
123123
x^2 - 2
124124
125125
Using the ``height_bound`` and ``proof`` parameters, we can see that
@@ -131,18 +131,18 @@ def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None,
131131
132132
For stronger results, we need more precision::
133133
134-
sage: algdep(pi.n(), 5, height_bound=100, proof=True) is None # needs sage.libs.pari sage.symbolic
134+
sage: # needs sage.libs.pari sage.symbolic
135+
sage: algdep(pi.n(), 5, height_bound=100, proof=True) is None
135136
Traceback (most recent call last):
136137
...
137138
ValueError: insufficient precision for non-existence proof
138-
sage: algdep(pi.n(200), 5, height_bound=100, proof=True) is None # needs sage.libs.pari sage.symbolic
139+
sage: algdep(pi.n(200), 5, height_bound=100, proof=True) is None
139140
True
140-
141-
sage: algdep(pi.n(), 10, height_bound=10, proof=True) is None # needs sage.libs.pari sage.symbolic
141+
sage: algdep(pi.n(), 10, height_bound=10, proof=True) is None
142142
Traceback (most recent call last):
143143
...
144144
ValueError: insufficient precision for non-existence proof
145-
sage: algdep(pi.n(200), 10, height_bound=10, proof=True) is None # needs sage.libs.pari sage.symbolic
145+
sage: algdep(pi.n(200), 10, height_bound=10, proof=True) is None
146146
True
147147
148148
We can also use ``proof=True`` to get positive results::
@@ -594,19 +594,20 @@ def is_pseudoprime(n):
594594
595595
EXAMPLES::
596596
597-
sage: is_pseudoprime(389) # needs sage.libs.pari
597+
sage: # needs sage.libs.pari
598+
sage: is_pseudoprime(389)
598599
True
599-
sage: is_pseudoprime(2000) # needs sage.libs.pari
600+
sage: is_pseudoprime(2000)
600601
False
601-
sage: is_pseudoprime(2) # needs sage.libs.pari
602+
sage: is_pseudoprime(2)
602603
True
603-
sage: is_pseudoprime(-1) # needs sage.libs.pari
604+
sage: is_pseudoprime(-1)
604605
False
605606
sage: factor(-6)
606607
-1 * 2 * 3
607-
sage: is_pseudoprime(1) # needs sage.libs.pari
608+
sage: is_pseudoprime(1)
608609
False
609-
sage: is_pseudoprime(-2) # needs sage.libs.pari
610+
sage: is_pseudoprime(-2)
610611
False
611612
"""
612613
return ZZ(n).is_pseudoprime()
@@ -1635,19 +1636,22 @@ class Sigma:
16351636
16361637
sage: sigma(100,4)
16371638
106811523
1638-
sage: sigma(factorial(100), 3).mod(144169) # needs sage.libs.pari
1639+
1640+
sage: # needs sage.libs.pari
1641+
sage: sigma(factorial(100), 3).mod(144169)
16391642
3672
1640-
sage: sigma(factorial(150), 12).mod(691) # needs sage.libs.pari
1643+
sage: sigma(factorial(150), 12).mod(691)
16411644
176
1642-
sage: RR(sigma(factorial(133),20)) # needs sage.libs.pari sage.rings.real_mpfr
1645+
sage: RR(sigma(factorial(133),20)) # needs sage.rings.real_mpfr
16431646
2.80414775675747e4523
1644-
sage: sigma(factorial(100),0) # needs sage.libs.pari
1647+
sage: sigma(factorial(100),0)
16451648
39001250856960000
1646-
sage: sigma(factorial(41),1) # needs sage.libs.pari
1649+
sage: sigma(factorial(41),1)
16471650
229199532273029988767733858700732906511758707916800
16481651
sage: from numpy import int8 # needs numpy
1649-
sage: sigma(int8(100), int8(4)) # needs numpy sage.libs.pari
1652+
sage: sigma(int8(100), int8(4)) # needs numpy
16501653
106811523
1654+
16511655
sage: from gmpy2 import mpz
16521656
sage: sigma(mpz(100), mpz(4))
16531657
106811523
@@ -1797,13 +1801,13 @@ def gcd(a, b=None, **kwargs):
17971801
The following shows that indeed coercion takes place before computing
17981802
the gcd. This behaviour was introduced in :trac:`10771`::
17991803
1800-
sage: R.<x>=QQ[]
1801-
sage: S.<x>=ZZ[]
1804+
sage: R.<x> = QQ[]
1805+
sage: S.<x> = ZZ[]
18021806
sage: p = S.random_element(degree=(0,10))
18031807
sage: q = R.random_element(degree=(0,10))
1804-
sage: parent(gcd(1/p,q))
1808+
sage: parent(gcd(1/p, q))
18051809
Fraction Field of Univariate Polynomial Ring in x over Rational Field
1806-
sage: parent(gcd([1/p,q]))
1810+
sage: parent(gcd([1/p, q]))
18071811
Fraction Field of Univariate Polynomial Ring in x over Rational Field
18081812
18091813
Make sure we try QQ and not merely ZZ (:trac:`13014`)::
@@ -5068,26 +5072,28 @@ def falling_factorial(x, a):
50685072
50695073
sage: falling_factorial(10, 3)
50705074
720
5071-
sage: falling_factorial(10, RR('3.0')) # needs sage.symbolic
5072-
720.000000000000
5073-
sage: falling_factorial(10, RR('3.3')) # needs sage.symbolic
5074-
1310.11633396601
50755075
sage: falling_factorial(10, 10)
50765076
3628800
50775077
sage: factorial(10)
50785078
3628800
5079-
sage: a = falling_factorial(1 + I, I); a # needs sage.symbolic
5079+
5080+
sage: # needs sage.symbolic
5081+
sage: falling_factorial(10, RR('3.0'))
5082+
720.000000000000
5083+
sage: falling_factorial(10, RR('3.3'))
5084+
1310.11633396601
5085+
sage: a = falling_factorial(1 + I, I); a
50805086
gamma(I + 2)
5081-
sage: CC(a) # needs sage.symbolic
5087+
sage: CC(a)
50825088
0.652965496420167 + 0.343065839816545*I
5083-
sage: falling_factorial(1 + I, 4) # needs sage.symbolic
5089+
sage: falling_factorial(1 + I, 4)
50845090
4*I + 2
5085-
sage: falling_factorial(I, 4) # needs sage.symbolic
5091+
sage: falling_factorial(I, 4)
50865092
-10
50875093
50885094
sage: M = MatrixSpace(ZZ, 4, 4) # needs sage.modules
50895095
sage: A = M([1,0,1,0, 1,0,1,0, 1,0,10,10, 1,0,1,1]) # needs sage.modules
5090-
sage: falling_factorial(A, 2) # A(A - I) # needs sage.modules
5096+
sage: falling_factorial(A, 2) # A(A - I) # needs sage.modules
50915097
[ 1 0 10 10]
50925098
[ 1 0 10 10]
50935099
[ 20 0 101 100]
@@ -5171,18 +5177,16 @@ def rising_factorial(x, a):
51715177
sage: rising_factorial(10,3)
51725178
1320
51735179
5174-
sage: rising_factorial(10, RR('3.0')) # needs sage.symbolic
5180+
sage: # needs sage.symbolic
5181+
sage: rising_factorial(10, RR('3.0'))
51755182
1320.00000000000
5176-
5177-
sage: rising_factorial(10, RR('3.3')) # needs sage.symbolic
5183+
sage: rising_factorial(10, RR('3.3'))
51785184
2826.38895824964
5179-
5180-
sage: a = rising_factorial(1+I, I); a # needs sage.symbolic
5185+
sage: a = rising_factorial(1+I, I); a
51815186
gamma(2*I + 1)/gamma(I + 1)
5182-
sage: CC(a) # needs sage.symbolic
5187+
sage: CC(a)
51835188
0.266816390637832 + 0.122783354006372*I
5184-
5185-
sage: a = rising_factorial(I, 4); a # needs sage.symbolic
5189+
sage: a = rising_factorial(I, 4); a
51865190
-10
51875191
51885192
sage: x = polygen(ZZ)
@@ -5474,7 +5478,8 @@ def three_squares(n):
54745478
sage: three_squares(7^100)
54755479
(0, 0, 1798465042647412146620280340569649349251249)
54765480
sage: three_squares(11^111 - 1) # needs sage.libs.pari
5477-
(616274160655975340150706442680, 901582938385735143295060746161, 6270382387635744140394001363065311967964099981788593947233)
5481+
(616274160655975340150706442680, 901582938385735143295060746161,
5482+
6270382387635744140394001363065311967964099981788593947233)
54785483
sage: three_squares(7 * 2^41) # needs sage.libs.pari
54795484
(1048576, 2097152, 3145728)
54805485
sage: three_squares(7 * 2^42)
@@ -5618,8 +5623,9 @@ def four_squares(n):
56185623
sage: four_squares(1101011011004)
56195624
(90, 102, 1220, 1049290)
56205625
sage: four_squares(10^100 - 1) # needs sage.libs.pari
5621-
(155024616290, 2612183768627, 14142135623730950488016887, 99999999999999999999999999999999999999999999999999)
5622-
sage: for i in range(2^129, 2^129+10000): # long time # needs sage.libs.pari
5626+
(155024616290, 2612183768627, 14142135623730950488016887,
5627+
99999999999999999999999999999999999999999999999999)
5628+
sage: for i in range(2^129, 2^129 + 10000): # long time # needs sage.libs.pari
56235629
....: S = four_squares(i)
56245630
....: assert sum(x^2 for x in S) == i
56255631
@@ -5695,7 +5701,11 @@ def sum_of_k_squares(k, n):
56955701
sage: sum_of_k_squares(5, 9634)
56965702
(0, 1, 2, 5, 98)
56975703
sage: sum_of_k_squares(6, 11^1111 - 1) # needs sage.libs.pari
5698-
(19215400822645944253860920437586326284, 37204645194585992174252915693267578306, 3473654819477394665857484221256136567800161086815834297092488779216863122, 5860191799617673633547572610351797996721850737768032876360978911074629287841061578270832330322236796556721252602860754789786937515870682024273948, 20457423294558182494001919812379023992538802203730791019728543439765347851316366537094696896669915675685581905102118246887673397020172285247862426612188418787649371716686651256443143210952163970564228423098202682066311189439731080552623884051737264415984619097656479060977602722566383385989, 311628095411678159849237738619458396497534696043580912225334269371611836910345930320700816649653412141574887113710604828156159177769285115652741014638785285820578943010943846225597311231847997461959204894255074229895666356909071243390280307709880906261008237873840245959883405303580405277298513108957483306488193844321589356441983980532251051786704380984788999660195252373574924026139168936921591652831237741973242604363696352878914129671292072201700073286987126265965322808664802662993006926302359371379531571194266134916767573373504566621665949840469229781956838744551367172353)
5704+
(19215400822645944253860920437586326284, 37204645194585992174252915693267578306,
5705+
3473654819477394665857484221256136567800161086815834297092488779216863122,
5706+
5860191799617673633547572610351797996721850737768032876360978911074629287841061578270832330322236796556721252602860754789786937515870682024273948,
5707+
20457423294558182494001919812379023992538802203730791019728543439765347851316366537094696896669915675685581905102118246887673397020172285247862426612188418787649371716686651256443143210952163970564228423098202682066311189439731080552623884051737264415984619097656479060977602722566383385989,
5708+
311628095411678159849237738619458396497534696043580912225334269371611836910345930320700816649653412141574887113710604828156159177769285115652741014638785285820578943010943846225597311231847997461959204894255074229895666356909071243390280307709880906261008237873840245959883405303580405277298513108957483306488193844321589356441983980532251051786704380984788999660195252373574924026139168936921591652831237741973242604363696352878914129671292072201700073286987126265965322808664802662993006926302359371379531571194266134916767573373504566621665949840469229781956838744551367172353)
56995709
sage: sum_of_k_squares(7, 0)
57005710
(0, 0, 0, 0, 0, 0, 0)
57015711
sage: sum_of_k_squares(30,999999)
@@ -5954,32 +5964,32 @@ def sort_complex_numbers_for_display(nums):
59545964
59555965
EXAMPLES::
59565966
5967+
sage: # needs sage.rings.complex_double
59575968
sage: import sage.arith.misc
59585969
sage: sort_c = sort_complex_numbers_for_display
5959-
sage: nums = [CDF(i) for i in range(3)] # needs sage.rings.complex_double
5960-
sage: for i in range(3): # needs sage.rings.complex_double
5970+
sage: nums = [CDF(i) for i in range(3)]
5971+
sage: for i in range(3):
59615972
....: nums.append(CDF(i + RDF.random_element(-3e-11, 3e-11),
59625973
....: RDF.random_element()))
59635974
....: nums.append(CDF(i + RDF.random_element(-3e-11, 3e-11),
59645975
....: RDF.random_element()))
5965-
sage: shuffle(nums) # needs sage.rings.complex_double
5966-
sage: nums = sort_c(nums) # needs sage.rings.complex_double
5967-
sage: for i in range(len(nums)): # needs sage.rings.complex_double
5976+
sage: shuffle(nums)
5977+
sage: nums = sort_c(nums)
5978+
sage: for i in range(len(nums)):
59685979
....: if nums[i].imag():
59695980
....: first_non_real = i
59705981
....: break
59715982
....: else:
59725983
....: first_non_real = len(nums)
5973-
sage: assert first_non_real >= 3 # needs sage.rings.complex_double
5974-
sage: for i in range(first_non_real - 1): # needs sage.rings.complex_double
5984+
sage: assert first_non_real >= 3
5985+
sage: for i in range(first_non_real - 1):
59755986
....: assert nums[i].real() <= nums[i + 1].real()
5976-
59775987
sage: def truncate(n):
59785988
....: if n.real() < 1e-10:
59795989
....: return 0
59805990
....: else:
59815991
....: return n.real().n(digits=9)
5982-
sage: for i in range(first_non_real, len(nums)-1): # needs sage.rings.complex_double
5992+
sage: for i in range(first_non_real, len(nums)-1):
59835993
....: assert truncate(nums[i]) <= truncate(nums[i + 1])
59845994
....: if truncate(nums[i]) == truncate(nums[i + 1]):
59855995
....: assert nums[i].imag() <= nums[i+1].imag()

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