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.
class sage.misc.misc.AttrCallObject(name, args, kwds)

Bases: object

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

EXAMPLES:

Objects of this type are returned if subprocesses=True is passed to cputime():

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

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 call
• args, 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]]

sage.misc.misc.call_method(obj, name, *args, **kwds)

Call the method name on obj.

This has to exist somewhere in Python!!!

operator.methodcaller() attrcal()

EXAMPLES:

sage: from sage.misc.misc import call_method
3

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 nest()

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 since t. This is how much time Sage has spent using the CPU. If subprocesses=False this does not count time spent in subprocesses spawned by Sage (e.g., Gap, Singular, etc.). If subprocesses=True this function tries to take all subprocesses with a working cputime() implementation into account.

The measurement for the main Sage process is done via a call to resource.getrusage(), so it avoids the wraparound problems in time.clock() on Cygwin.

INPUT:

• t - (optional) time in CPU seconds, if t is a result from an earlier call with subprocesses=True, then subprocesses=True is assumed.
• subprocesses – (optional), include subprocesses (default: False)

OUTPUT:

• float - time in CPU seconds if subprocesses=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 S
• object - 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 S
• object - 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.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.inject_variable(name, value, warn=True)

Inject a variable into the main global namespace.

INPUT:

• name – a string
• value – anything
• warn – a boolean (default: False)

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”)

Warnings can be disabled:

sage: b = 3
sage: inject_variable("b", 42, warn=False)
sage: b
42


Use with care!

sage.misc.misc.inject_variable_test(name, value, depth)

A function for testing deep calls to inject_variable

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 if it is an iterator. This is not quite correct since it could have a next methods with a different semantic.

EXAMPLES:

sage: it = iter([1,2,3])
sage: is_iterator(it)
True

sage: class wrong():  # py2
....:    def __init__(self): self.n = 5
....:    def next(self):
....:        self.n -= 1
....:        if self.n == 0: raise StopIteration
....:        return self.n
sage: class wrong():  # py3
....:    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)  # py2
Traceback (most recent call last):
...
TypeError: iteration over non-sequence
sage: list(x)  # py3
Traceback (most recent call last):
...
TypeError: 'wrong' object is not iterable

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

sage.misc.misc.nest(f, n, x)

Return $$f(f(...f(x)...))$$, where the composition occurs n times.

See also compose() and self_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'
'010'
'00010'
'00389'
'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.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 - list
• s - 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: '' if scalar_mult is '*'; otherwise scalar_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 and is_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(dirname)

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 (unlike os.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:

sage: filename = tmp_filename()
sage: sage_makedirs(filename)
Traceback (most recent call last):
...
OSError: [Errno ...] File exists: ...

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, max_samples=None)

Return an iterator over at most bound number of repeat-tuples of elements.

INPUT:

• elements – an iterable
• repeat – integer (default None), the length of the tuples to be returned. If None, just returns entries from elements.
• bound – the maximum number of tuples returned (ignored if max_samples given)
• max_samples – non-negative integer (default None). If given, then a sample of the possible tuples will be returned, instead of the first few in the standard order.

OUTPUT:

If max_samples is not provided, an iterator over the first bound tuples of length repeat, in the standard nested-for-loop order.

If max_samples is provided, a list of at most max_samples tuples, sampled uniformly from the possibilities. In this case, elements must be finite.

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.try_read(obj, splitlines=False)

Determine if a given object is a readable file-like object and if so read and return its contents.

That is, the object has a callable method named read() which takes no arguments (except self) then the method is executed and the contents are returned.

Alternatively, if the splitlines=True is given, first splitlines() is tried, then if that fails read().splitlines().

If either method fails, None is returned.

INPUT:

• obj – typically a $$file$$ or $$io.BaseIO$$ object, but any other object with a read() method is accepted.
• splitlines$$bool$$, optional; if True, return a list of lines instead of a string.

EXAMPLES:

sage: import io
sage: filename = tmp_filename()
sage: with open(filename, 'w') as fobj:
....:     _ = fobj.write('a\nb\nc')
sage: with open(filename) as fobj:
a
b
c
sage: with open(filename) as fobj:
['a\n', 'b\n', 'c']


The following example is identical to the above example on Python 3, but different on Python 2 where open != io.open:

sage: with io.open(filename) as fobj:
a
b
c


I/O buffers:

sage: buf = io.StringIO(u'a\nb\nc')
a
b
c
sage: _ = buf.seek(0); try_read(buf, splitlines=True)
[u'a\n', u'b\n', u'c']
sage: buf = io.BytesIO(b'a\nb\nc')
True
sage: _ = buf.seek(0)
sage: try_read(buf, splitlines=True) == [b'a\n', b'b\n', b'c']
True


sage: class MyFile(object):
....:     def read(self): return 'Hello world!'
'Hello world!'
['Hello world!']


sage: try_read(1) is None
True

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 - iterable
• y - iterable (may optionally omitted)

OUTPUT: list

EXAMPLES:

sage: answer = union([1,2,3,4], [5,6]); answer
[1, 2, 3, 4, 5, 6]
True
True
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: uniq([1, 1, 8, -5, 3, -5, -13, 13, -13])
doctest:...: DeprecationWarning: the output of uniq(X) being sorted is deprecated; use sorted(set(X)) instead if you want sorted output
See https://trac.sagemath.org/27014 for details.
[-13, -5, 1, 3, 8, 13]

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 print
• t - 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 printing
• caller_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()

EXAMPLES:

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)