Miscellaneous functions¶
AUTHORS:
- William Stein
- William Stein (2006-04-26): added workaround for Windows where most users’ home directory has a space in it.
- Robert Bradshaw (2007-09-20): Ellipsis range/iterator.
TESTS:
The following test, verifying that trac ticket #16181 has been resolved, needs to stay at the beginning of this file so that its context is not poisoned by other tests:
sage: sage.misc.misc.inject_variable('a', 0)
sage: a
0
Check the fix from trac ticket #8323:
sage: 'name' in globals()
False
sage: 'func' in globals()
False
Test deprecation:
sage: sage.misc.misc.srange(5)
doctest:...: DeprecationWarning:
Importing srange from here is deprecated. If you need to use it, please import it directly from sage.arith.srange
See http://trac.sagemath.org/20094 for details.
[0, 1, 2, 3, 4]
sage: sage.misc.all.srange(5)
doctest:...: DeprecationWarning:
Importing srange from here is deprecated. If you need to use it, please import it directly from sage.arith.srange
See http://trac.sagemath.org/20334 for details.
[0, 1, 2, 3, 4]
sage: sage.misc.misc.sxrange(5)
doctest:...: DeprecationWarning:
Importing sxrange from here is deprecated. If you need to use it, please import it directly from sage.arith.srange
See http://trac.sagemath.org/20094 for details.
<generator object at 0x...>
sage: sage.misc.misc.cancel_alarm()
doctest:...: DeprecationWarning:
Importing cancel_alarm from here is deprecated. If you need to use it, please import it directly from cysignals.alarm
See http://trac.sagemath.org/20002 for details.
-
class
sage.misc.misc.
AttrCallObject
(name, args, kwds)¶ Bases:
object
TESTS:
sage: f = attrcall('core', 3); f *.core(3) sage: TestSuite(f).run()
-
class
sage.misc.misc.
BackslashOperator
¶ Implements Matlab-style backslash operator for solving systems:
A \ b
The preparser converts this to multiplications using
BackslashOperator()
.EXAMPLES:
sage: preparse("A \ matrix(QQ,2,1,[1/3,'2/3'])") "A * BackslashOperator() * matrix(QQ,Integer(2),Integer(1),[Integer(1)/Integer(3),'2/3'])" sage: preparse("A \ matrix(QQ,2,1,[1/3,2*3])") 'A * BackslashOperator() * matrix(QQ,Integer(2),Integer(1),[Integer(1)/Integer(3),Integer(2)*Integer(3)])' sage: preparse("A \ B + C") 'A * BackslashOperator() * B + C' sage: preparse("A \ eval('C+D')") "A * BackslashOperator() * eval('C+D')" sage: preparse("A \ x / 5") 'A * BackslashOperator() * x / Integer(5)' sage: preparse("A^3 \ b") 'A**Integer(3) * BackslashOperator() * b'
-
class
sage.misc.misc.
GlobalCputime
(t)¶ Container for CPU times of subprocesses.
AUTHOR:
- Martin Albrecht - (2008-12): initial version
EXAMPLE:
Objects of this type are returned if
subprocesses=True
is passed tocputime()
:sage: cputime(subprocesses=True) # indirect doctest, output random 0.2347431
We can use it to keep track of the CPU time spent in Singular for example:
sage: t = cputime(subprocesses=True) sage: P = PolynomialRing(QQ,7,'x') sage: I = sage.rings.ideal.Katsura(P) sage: gb = I.groebner_basis() # calls Singular sage: cputime(subprocesses=True) - t # output random 0.462987
For further processing we can then convert this container to a float:
sage: t = cputime(subprocesses=True) sage: float(t) #output somewhat random 2.1088339999999999
See also
-
sage.misc.misc.
assert_attribute
(x, attr, init=None)¶ If the object x has the attribute attr, do nothing. If not, set x.attr to init.
-
sage.misc.misc.
attrcall
(name, *args, **kwds)¶ Returns a callable which takes in an object, gets the method named name from that object, and calls it with the specified arguments and keywords.
INPUT:
name
- a string of the name of the method you want to callargs, kwds
- arguments and keywords to be passed to the method
EXAMPLES:
sage: f = attrcall('core', 3); f *.core(3) sage: [f(p) for p in Partitions(5)] [[2], [1, 1], [1, 1], [3, 1, 1], [2], [2], [1, 1]]
-
class
sage.misc.misc.
cached_attribute
(method, name=None)¶ Bases:
object
Computes attribute value and caches it in the instance.
-
sage.misc.misc.
call_method
(obj, name, *args, **kwds)¶ Call the method
name
onobj
.This has to exist somewhere in Python!!!
See also
operator.methodcaller()
attrcal()
EXAMPLES:
sage: from sage.misc.misc import call_method sage: call_method(1, "__add__", 2) 3
-
sage.misc.misc.
cmp_props
(left, right, props)¶
-
sage.misc.misc.
coeff_repr
(c, is_latex=False)¶
-
sage.misc.misc.
compose
(f, g)¶ Return the composition of one-variable functions: \(f \circ g\)
See also
self_compose()
andnest()
- INPUT:
- \(f\) – a function of one variable
- \(g\) – another function of one variable
- OUTPUT:
- A function, such that compose(f,g)(x) = f(g(x))
EXAMPLES:
sage: def g(x): return 3*x sage: def f(x): return x + 1 sage: h1 = compose(f,g) sage: h2 = compose(g,f) sage: _ = var ('x') sage: h1(x) 3*x + 1 sage: h2(x) 3*x + 3
sage: _ = function('f g') sage: _ = var ('x') sage: compose(f,g)(x) f(g(x))
-
sage.misc.misc.
cputime
(t=0, subprocesses=False)¶ Return the time in CPU seconds since Sage started, or with optional argument
t
, return the time sincet
. This is how much time Sage has spent using the CPU. Ifsubprocesses=False
this does not count time spent in subprocesses spawned by Sage (e.g., Gap, Singular, etc.). Ifsubprocesses=True
this function tries to take all subprocesses with a workingcputime()
implementation into account.The measurement for the main Sage process is done via a call to
resource.getrusage()
, so it avoids the wraparound problems intime.clock()
on Cygwin.INPUT:
t
- (optional) time in CPU seconds, ift
is a result from an earlier call withsubprocesses=True
, thensubprocesses=True
is assumed.- subprocesses – (optional), include subprocesses (default:
False
)
OUTPUT:
float
- time in CPU seconds ifsubprocesses=False
GlobalCputime
- object which holds CPU times of subprocesses otherwise
EXAMPLES:
sage: t = cputime() sage: F = gp.factor(2^199-1) sage: cputime(t) # somewhat random 0.010999000000000092 sage: t = cputime(subprocesses=True) sage: F = gp.factor(2^199-1) sage: cputime(t) # somewhat random 0.091999 sage: w = walltime() sage: F = gp.factor(2^199-1) sage: walltime(w) # somewhat random 0.58425593376159668
Note
Even with
subprocesses=True
there is no guarantee that the CPU time is reported correctly because subprocesses can be started and terminated at any given time.
-
sage.misc.misc.
embedded
()¶ Return True if this copy of Sage is running embedded in the Sage notebook.
EXAMPLES:
sage: sage.misc.misc.embedded() # output True if in the notebook False
-
sage.misc.misc.
exists
(S, P)¶ If S contains an element x such that P(x) is True, this function returns True and the element x. Otherwise it returns False and None.
Note that this function is NOT suitable to be used in an if-statement or in any place where a boolean expression is expected. For those situations, use the Python built-in
any(P(x) for x in S)
INPUT:
S
- object (that supports enumeration)P
- function that returns True or False
OUTPUT:
bool
- whether or not P is True for some element x of Sobject
- x
EXAMPLES: lambda functions are very useful when using the exists function:
sage: exists([1,2,5], lambda x : x > 7) (False, None) sage: exists([1,2,5], lambda x : x > 3) (True, 5)
The following example is similar to one in the MAGMA handbook. We check whether certain integers are a sum of two (small) cubes:
sage: cubes = [t**3 for t in range(-10,11)] sage: exists([(x,y) for x in cubes for y in cubes], lambda v : v[0]+v[1] == 218) (True, (-125, 343)) sage: exists([(x,y) for x in cubes for y in cubes], lambda v : v[0]+v[1] == 219) (False, None)
-
sage.misc.misc.
forall
(S, P)¶ If P(x) is true every x in S, return True and None. If there is some element x in S such that P is not True, return False and x.
Note that this function is NOT suitable to be used in an if-statement or in any place where a boolean expression is expected. For those situations, use the Python built-in
all(P(x) for x in S)
INPUT:
S
- object (that supports enumeration)P
- function that returns True or False
OUTPUT:
bool
- whether or not P is True for all elements of Sobject
- x
EXAMPLES: lambda functions are very useful when using the forall function. As a toy example we test whether certain integers are greater than 3.
sage: forall([1,2,5], lambda x : x > 3) (False, 1) sage: forall([1,2,5], lambda x : x > 0) (True, None)
Next we ask whether every positive integer less than 100 is a product of at most 2 prime factors:
sage: forall(range(1,100), lambda n : len(factor(n)) <= 2) (False, 30)
The answer is no, and 30 is a counterexample. However, every positive integer 100 is a product of at most 3 primes.
sage: forall(range(1,100), lambda n : len(factor(n)) <= 3) (True, None)
-
sage.misc.misc.
generic_cmp
(x, y)¶ Compare x and y and return -1, 0, or 1.
This is similar to x.__cmp__(y), but works even in some cases when a .__cmp__ method isn’t defined.
-
sage.misc.misc.
get_main_globals
()¶ Return the main global namespace.
EXAMPLES:
sage: from sage.misc.misc import get_main_globals sage: G = get_main_globals() sage: bla = 1 sage: G['bla'] 1 sage: bla = 2 sage: G['bla'] 2 sage: G['ble'] = 5 sage: ble 5
This is analogous to
globals()
, except that it can be called from any function, even if it is in a Python module:sage: def f(): ....: G = get_main_globals() ....: assert G['bli'] == 14 ....: G['blo'] = 42 sage: bli = 14 sage: f() sage: blo 42
ALGORITHM:
The main global namespace is discovered by going up the frame stack until the frame for the
__main__
module is found. Should this frame not be found (this should not occur in normal operation), an exception “ValueError: call stack is not deep enough” will be raised by_getframe
.See
inject_variable_test()
for a real test that this works within deeply nested calls in a function defined in a Python module.
-
sage.misc.misc.
get_verbose
()¶ Return the global Sage verbosity level.
INPUT: int level: an integer between 0 and 2, inclusive.
OUTPUT: changes the state of the verbosity flag.
EXAMPLES:
sage: get_verbose() 0 sage: set_verbose(2) sage: get_verbose() 2 sage: set_verbose(0)
-
sage.misc.misc.
get_verbose_files
()¶
-
sage.misc.misc.
getitem
(v, n)¶ Variant of getitem that coerces to an int if a TypeError is raised.
(This is not needed anymore - classes should define an __index__ method.)
Thus, e.g.,
getitem(v,n)
will work even if \(v\) is a Python list and \(n\) is a Sage integer.EXAMPLES:
sage: v = [1,2,3]
The following used to fail in Sage <= 1.3.7. Now it works fine:
sage: v[ZZ(1)] 2
This always worked.
sage: getitem(v, ZZ(1)) 2
-
sage.misc.misc.
inject_variable
(name, value)¶ Inject a variable into the main global namespace.
INPUT:
name
– a stringvalue
– anything
EXAMPLES:
sage: from sage.misc.misc import inject_variable sage: inject_variable("a", 314) sage: a 314
A warning is issued the first time an existing value is overwritten:
sage: inject_variable("a", 271) doctest:...: RuntimeWarning: redefining global value `a` sage: a 271 sage: inject_variable("a", 272) sage: a 272
That’s because warn seem to not reissue twice the same warning:
sage: from warnings import warn sage: warn(“blah”) doctest:...: UserWarning: blah sage: warn(“blah”)Use with care!
-
sage.misc.misc.
inject_variable_test
(name, value, depth)¶ A function for testing deep calls to inject_variable
TESTS:
sage: from sage.misc.misc import inject_variable_test sage: inject_variable_test("a0", 314, 0) sage: a0 314 sage: inject_variable_test("a1", 314, 1) sage: a1 314 sage: inject_variable_test("a2", 314, 2) sage: a2 314 sage: inject_variable_test("a2", 271, 2) doctest:...: RuntimeWarning: redefining global value `a2` sage: a2 271
-
sage.misc.misc.
is_in_string
(line, pos)¶ Returns True if the character at position pos in line occurs within a string.
EXAMPLES:
sage: from sage.misc.misc import is_in_string sage: line = 'test(\'#\')' sage: is_in_string(line, line.rfind('#')) True sage: is_in_string(line, line.rfind(')')) False
-
sage.misc.misc.
is_iterator
(it)¶ Tests if it is an iterator.
The mantra
if hasattr(it, 'next')
was used to tests ifit
is an iterator. This is not quite correct sinceit
could have anext
methods with a different semantic.EXAMPLES:
sage: it = iter([1,2,3]) sage: is_iterator(it) True sage: class wrong(): ... def __init__(self): self.n = 5 ... def next(self): ... self.n -= 1 ... if self.n == 0: raise StopIteration ... return self.n sage: x = wrong() sage: is_iterator(x) False sage: list(x) Traceback (most recent call last): ... TypeError: iteration over non-sequence sage: class good(wrong): ... def __iter__(self): return self sage: x = good() sage: is_iterator(x) True sage: list(x) [4, 3, 2, 1] sage: P = Partitions(3) sage: is_iterator(P) False sage: is_iterator(iter(P)) True
-
class
sage.misc.misc.
lazy_prop
(calculate_function)¶ Bases:
object
-
sage.misc.misc.
nest
(f, n, x)¶ Return \(f(f(...f(x)...))\), where the composition occurs n times.
See also
compose()
andself_compose()
- INPUT:
- \(f\) – a function of one variable
- \(n\) – a nonnegative integer
- \(x\) – any input for \(f\)
- OUTPUT:
- \(f(f(...f(x)...))\), where the composition occurs n times
EXAMPLES:
sage: def f(x): return x^2 + 1 sage: x = var('x') sage: nest(f, 3, x) ((x^2 + 1)^2 + 1)^2 + 1
sage: _ = function('f') sage: _ = var('x') sage: nest(f, 10, x) f(f(f(f(f(f(f(f(f(f(x))))))))))
sage: _ = function('f') sage: _ = var('x') sage: nest(f, 0, x) x
-
sage.misc.misc.
newton_method_sizes
(N)¶ Returns a sequence of integers \(1 = a_1 \leq a_2 \leq \cdots \leq a_n = N\) such that \(a_j = \lceil a_{j+1} / 2 \rceil\) for all \(j\).
This is useful for Newton-style algorithms that double the precision at each stage. For example if you start at precision 1 and want an answer to precision 17, then it’s better to use the intermediate stages 1, 2, 3, 5, 9, 17 than to use 1, 2, 4, 8, 16, 17.
INPUT:
N
- positive integer
EXAMPLES:
sage: newton_method_sizes(17) [1, 2, 3, 5, 9, 17] sage: newton_method_sizes(16) [1, 2, 4, 8, 16] sage: newton_method_sizes(1) [1]
AUTHORS:
- David Harvey (2006-09-09)
-
sage.misc.misc.
pad_zeros
(s, size=3)¶ EXAMPLES:
sage: pad_zeros(100) '100' sage: pad_zeros(10) '010' sage: pad_zeros(10, 5) '00010' sage: pad_zeros(389, 5) '00389' sage: pad_zeros(389, 10) '0000000389'
-
sage.misc.misc.
powerset
(X)¶ Iterator over the list of all subsets of the iterable X, in no particular order. Each list appears exactly once, up to order.
INPUT:
X
- an iterable
OUTPUT: iterator of lists
EXAMPLES:
sage: list(powerset([1,2,3])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] sage: [z for z in powerset([0,[1,2]])] [[], [0], [[1, 2]], [0, [1, 2]]]
Iterating over the power set of an infinite set is also allowed:
sage: i = 0 sage: L = [] sage: for x in powerset(ZZ): ....: if i > 10: ....: break ....: else: ....: i += 1 ....: L.append(x) sage: print(" ".join(str(x) for x in L)) [] [0] [1] [0, 1] [-1] [0, -1] [1, -1] [0, 1, -1] [2] [0, 2] [1, 2]
You may also use subsets as an alias for powerset:
sage: subsets([1,2,3]) <generator object powerset at 0x...> sage: list(subsets([1,2,3])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] The reason we return lists instead of sets is that the elements of sets must be hashable and many structures on which one wants the powerset consist of non-hashable objects.
AUTHORS:
- William Stein
- Nils Bruin (2006-12-19): rewrite to work for not-necessarily finite objects X.
-
sage.misc.misc.
prop
(f)¶
-
sage.misc.misc.
random_sublist
(X, s)¶ Return a pseudo-random sublist of the list X where the probability of including a particular element is s.
INPUT:
X
- lists
- floating point number between 0 and 1
OUTPUT: list
EXAMPLES:
sage: S = [1,7,3,4,18] sage: random_sublist(S, 0.5) [1, 3, 4] sage: random_sublist(S, 0.5) [1, 3]
-
sage.misc.misc.
repr_lincomb
(terms, is_latex=False, scalar_mult='*', strip_one=False, repr_monomial=None, latex_scalar_mult=None)¶ Compute a string representation of a linear combination of some formal symbols.
INPUT:
terms
– list of terms, as pairs (support, coefficient)is_latex
– whether to produce latex (default:False
)scalar_mult
– string representing the multiplication (default:'*'
)latex_scalar_mult
– latex string representing the multiplication (default:''
ifscalar_mult
is'*'
; otherwisescalar_mult
)coeffs
– for backward compatibility
OUTPUT:
str
- a string
EXAMPLES:
sage: repr_lincomb([('a',1), ('b',-2), ('c',3)]) 'a - 2*b + 3*c' sage: repr_lincomb([('a',0), ('b',-2), ('c',3)]) '-2*b + 3*c' sage: repr_lincomb([('a',0), ('b',2), ('c',3)]) '2*b + 3*c' sage: repr_lincomb([('a',1), ('b',0), ('c',3)]) 'a + 3*c' sage: repr_lincomb([('a',-1), ('b','2+3*x'), ('c',3)]) '-a + (2+3*x)*b + 3*c' sage: repr_lincomb([('a', '1+x^2'), ('b', '2+3*x'), ('c', 3)]) '(1+x^2)*a + (2+3*x)*b + 3*c' sage: repr_lincomb([('a', '1+x^2'), ('b', '-2+3*x'), ('c', 3)]) '(1+x^2)*a + (-2+3*x)*b + 3*c' sage: repr_lincomb([('a', 1), ('b', -2), ('c', -3)]) 'a - 2*b - 3*c' sage: t = PolynomialRing(RationalField(),'t').gen() sage: repr_lincomb([('a', -t), ('s', t - 2), ('', t^2 + 2)]) '-t*a + (t-2)*s + (t^2+2)'
Examples for
scalar_mult
:sage: repr_lincomb([('a',1), ('b',2), ('c',3)], scalar_mult='*') 'a + 2*b + 3*c' sage: repr_lincomb([('a',2), ('b',0), ('c',-3)], scalar_mult='**') '2**a - 3**c' sage: repr_lincomb([('a',-1), ('b',2), ('c',3)], scalar_mult='**') '-a + 2**b + 3**c'
Examples for
scalar_mult
andis_latex
:sage: repr_lincomb([('a',-1), ('b',2), ('c',3)], is_latex=True) '-a + 2b + 3c' sage: repr_lincomb([('a',-1), ('b',-1), ('c',3)], is_latex=True, scalar_mult='*') '-a - b + 3c' sage: repr_lincomb([('a',-1), ('b',2), ('c',-3)], is_latex=True, scalar_mult='**') '-a + 2**b - 3**c' sage: repr_lincomb([('a',-2), ('b',-1), ('c',-3)], is_latex=True, latex_scalar_mult='*') '-2*a - b - 3*c'
Examples for
strip_one
:sage: repr_lincomb([ ('a',1), (1,-2), ('3',3) ]) 'a - 2*1 + 3*3' sage: repr_lincomb([ ('a',-1), (1,1), ('3',3) ]) '-a + 1 + 3*3' sage: repr_lincomb([ ('a',1), (1,-2), ('3',3) ], strip_one = True) 'a - 2 + 3*3' sage: repr_lincomb([ ('a',-1), (1,1), ('3',3) ], strip_one = True) '-a + 1 + 3*3' sage: repr_lincomb([ ('a',1), (1,-1), ('3',3) ], strip_one = True) 'a - 1 + 3*3'
Examples for
repr_monomial
:sage: repr_lincomb([('a',1), ('b',2), ('c',3)], repr_monomial = lambda s: s+"1") 'a1 + 2*b1 + 3*c1'
-
sage.misc.misc.
sage_makedirs
(dir)¶ Python version of
mkdir -p
: try to create a directory, and also create all intermediate directories as necessary. Succeed silently if the directory already exists (unlikeos.makedirs()
). Raise other errors (like permission errors) normally.EXAMPLES:
sage: from sage.misc.misc import sage_makedirs sage: sage_makedirs(DOT_SAGE) # no output
The following fails because we are trying to create a directory in place of an ordinary file (the main Sage executable):
sage: sage_executable = os.path.join(SAGE_ROOT, 'sage') sage: sage_makedirs(sage_executable) Traceback (most recent call last): ... OSError: ...
-
sage.misc.misc.
self_compose
(f, n)¶ Return the function \(f\) composed with itself \(n\) times.
See
nest()
if you want \(f(f(...(f(x))...))\) for known \(x\).- INPUT:
- \(f\) – a function of one variable
- \(n\) – a nonnegative integer
- OUTPUT:
- A function, the result of composing \(f\) with itself \(n\) times
EXAMPLES:
sage: def f(x): return x^2 + 1 sage: g = self_compose(f, 3) sage: x = var('x') sage: g(x) ((x^2 + 1)^2 + 1)^2 + 1
sage: def f(x): return x + 1 sage: g = self_compose(f, 10000) sage: g(0) 10000
sage: x = var('x') sage: self_compose(sin, 0)(x) x
-
sage.misc.misc.
set_verbose
(level, files='all')¶ Set the global Sage verbosity level.
INPUT:
level
- an integer between 0 and 2, inclusive.files
(default: ‘all’): list of files to make verbose, or- ‘all’ to make ALL files verbose (the default).
OUTPUT: changes the state of the verbosity flag and possibly appends to the list of files that are verbose.
EXAMPLES:
sage: set_verbose(2) sage: verbose("This is Sage.", level=1) # not tested VERBOSE1 (?): This is Sage. sage: verbose("This is Sage.", level=2) # not tested VERBOSE2 (?): This is Sage. sage: verbose("This is Sage.", level=3) # not tested [no output] sage: set_verbose(0)
-
sage.misc.misc.
set_verbose_files
(file_name)¶
-
sage.misc.misc.
some_tuples
(elements, repeat, bound)¶ Return an iterator over at most
bound
number ofrepeat
-tuples ofelements
.TESTS:
sage: from sage.misc.misc import some_tuples sage: l = some_tuples([0,1,2,3], 2, 3) sage: l <itertools.islice object at ...> sage: len(list(l)) 3 sage: l = some_tuples(range(50), 3, 10) sage: len(list(l)) 10
Todo
Currently, this only return an iterator over the first element of the Cartesian product. It would be smarter to return something more “random like” as it is used in tests. However, this should remain deterministic.
-
sage.misc.misc.
sourcefile
(object)¶ Work out which source or compiled file an object was defined in.
-
sage.misc.misc.
strunc
(s, n=60)¶ Truncate at first space after position n, adding ‘...’ if nontrivial truncation.
-
sage.misc.misc.
subsets
(X)¶ Iterator over the list of all subsets of the iterable X, in no particular order. Each list appears exactly once, up to order.
INPUT:
X
- an iterable
OUTPUT: iterator of lists
EXAMPLES:
sage: list(powerset([1,2,3])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] sage: [z for z in powerset([0,[1,2]])] [[], [0], [[1, 2]], [0, [1, 2]]]
Iterating over the power set of an infinite set is also allowed:
sage: i = 0 sage: L = [] sage: for x in powerset(ZZ): ....: if i > 10: ....: break ....: else: ....: i += 1 ....: L.append(x) sage: print(" ".join(str(x) for x in L)) [] [0] [1] [0, 1] [-1] [0, -1] [1, -1] [0, 1, -1] [2] [0, 2] [1, 2]
You may also use subsets as an alias for powerset:
sage: subsets([1,2,3]) <generator object powerset at 0x...> sage: list(subsets([1,2,3])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] The reason we return lists instead of sets is that the elements of sets must be hashable and many structures on which one wants the powerset consist of non-hashable objects.
AUTHORS:
- William Stein
- Nils Bruin (2006-12-19): rewrite to work for not-necessarily finite objects X.
-
sage.misc.misc.
to_gmp_hex
(n)¶
-
sage.misc.misc.
todo
(mesg='')¶
-
sage.misc.misc.
typecheck
(x, C, var='x')¶ Check that x is of instance C. If not raise a TypeError with an error message.
-
sage.misc.misc.
union
(x, y=None)¶ Return the union of x and y, as a list. The resulting list need not be sorted and can change from call to call.
INPUT:
x
- iterabley
- iterable (may optionally omitted)
OUTPUT: list
EXAMPLES:
sage: answer = union([1,2,3,4], [5,6]); answer [1, 2, 3, 4, 5, 6] sage: union([1,2,3,4,5,6], [5,6]) == answer True sage: union((1,2,3,4,5,6), [5,6]) == answer True sage: union((1,2,3,4,5,6), set([5,6])) == answer True
-
sage.misc.misc.
uniq
(x)¶ Return the sublist of all elements in the list x that is sorted and is such that the entries in the sublist are unique.
EXAMPLES:
sage: v = uniq([1,1,8,-5,3,-5,'a','x','a']) sage: v # potentially random ordering of output ['a', 'x', -5, 1, 3, 8] sage: set(v) == set(['a', 'x', -5, 1, 3, 8]) True
-
sage.misc.misc.
unset_verbose_files
(file_name)¶
-
sage.misc.misc.
verbose
(mesg='', t=0, level=1, caller_name=None)¶ Print a message if the current verbosity is at least level.
INPUT:
mesg
- str, a message to printt
- int, optional, if included, will also print cputime(t), - which is the time since time t. Thus t should have been obtained with t=cputime()level
- int, (default: 1) the verbosity level of what we are printingcaller_name
- string (default: None), the name of the calling function; in most cases Python can deduce this, so it need not be provided.
OUTPUT: possibly prints a message to stdout; also returns cputime()
EXAMPLE:
sage: set_verbose(1) sage: t = cputime() sage: t = verbose("This is Sage.", t, level=1, caller_name="william") # not tested VERBOSE1 (william): This is Sage. (time = 0.0) sage: set_verbose(0)
-
sage.misc.misc.
walltime
(t=0)¶ Return the wall time in second, or with optional argument t, return the wall time since time t. “Wall time” means the time on a wall clock, i.e., the actual time.
INPUT:
t
- (optional) float, time in CPU seconds
OUTPUT:
float
- time in seconds
EXAMPLES:
sage: w = walltime() sage: F = factor(2^199-1) sage: walltime(w) # somewhat random 0.8823847770690918
-
sage.misc.misc.
word_wrap
(s, ncols=85)¶