vectors_in_intervals.intervals¶

Interval classes

Classes

`Interval`(lower, upper[, lower_closed, ...])	An interval.
`Intervals`(intervals)	A Cartesian product of intervals.

class vectors_in_intervals.intervals.Interval(lower, upper, lower_closed: bool = True, upper_closed: bool = True)¶

An interval.

Also supports variables.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Interval(0, 1)
[0, 1]
sage: Interval(0, 1, False, True)
(0, 1]
sage: Interval(0, 1, False, False)
(0, 1)
sage: Interval(0, 1, True, False)
[0, 1)
sage: Interval(0, 0)
{0}
sage: Interval(0, 0, False, False)
{}
sage: Interval(0, 0, True, True)
{0}
sage: Interval(0, 0, True, False)
{}
sage: Interval(0, 0, False, True)
{}
sage: Interval(-oo, 4)
(-oo, 4]
sage: Interval(-oo, 4, False, False)
(-oo, 4)
sage: I = Interval(-3, +oo, False, False)
sage: I
(-3, +oo)
sage: 0 in I
True
sage: -3 in I
False
sage: Interval.random() # random
(-1, 1)

Variables are supported, too:

sage: var("a")
a
sage: Interval(a, 1)
[a, 1]
sage: I = Interval(a, 1, False, False)
sage: I
(a, 1)
sage: I.an_element()
1/2*a + 1/2

an_element()¶

Return an element of the interval.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Interval(0, 1).an_element()
0
sage: Interval(0, 1, False, True).an_element()
1
sage: Interval(0, 1, False, False).an_element()
1/2
sage: Interval(-oo, 0).an_element()
0
sage: Interval(-oo, +oo).an_element()
0
sage: Interval(5, +oo, False, False).an_element()
6
sage: Interval(0, 0, False, False).an_element()
Traceback (most recent call last):
...
ValueError: The interval is empty.

category(self)¶: File: sage/structure/sage_object.pyx (starting at line 484)

classmethod closed(lower, upper) → vectors_in_intervals.intervals.Interval¶

Return a closed interval.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Interval.closed(0, 1)
[0, 1]

dump(self, filename, compress=True)¶

File: sage/structure/sage_object.pyx (starting at line 445)

Same as self.save(filename, compress)

dumps(self, compress=True)¶

File: sage/structure/sage_object.pyx (starting at line 451)

Dump self to a string s, which can later be reconstituted as self using loads(s).

There is an optional boolean argument compress which defaults to True.

EXAMPLES:

sage: from sage.misc.persist import comp
sage: O = SageObject()
sage: p_comp = O.dumps()
sage: p_uncomp = O.dumps(compress=False)
sage: comp.decompress(p_comp) == p_uncomp
True
sage: import pickletools
sage: pickletools.dis(p_uncomp)
    0: \x80 PROTO      2
    2: c    GLOBAL     'sage.structure.sage_object SageObject'
   41: q    BINPUT     ...
   43: )    EMPTY_TUPLE
   44: \x81 NEWOBJ
   45: q    BINPUT     ...
   47: .    STOP
highest protocol among opcodes = 2

classmethod empty() → vectors_in_intervals.intervals.Interval¶

Return the empty interval.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Interval.empty()
{}

infimum()¶

Return the infimum of the interval.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Interval(0, 1).infimum()
0
sage: Interval(-oo, 0).infimum()
-Infinity
sage: Interval(0, 0, False, False).infimum()
+Infinity

is_bounded() → bool¶: Return whether the interval is bounded.

is_closed() → bool¶: Return whether the interval is closed.

is_empty() → bool¶: Return whether the interval is empty.

is_open() → bool¶: Return whether the interval is open.

is_pointed() → bool¶

Return whether the interval is a point.

An interval is pointed if it has the same lower and upper bound, and both are closed.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Interval(0, 0).is_pointed()
True
sage: Interval(0, 1).is_pointed()
False

is_unbounded() → bool¶: Return whether the interval is unbounded.

classmethod open(lower, upper) → vectors_in_intervals.intervals.Interval¶

Return an open interval.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Interval.open(0, 1)
(0, 1)

parent(self)¶

File: sage/structure/sage_object.pyx (starting at line 518)

Return the type of self to support the coercion framework.

EXAMPLES:

sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t
log(sqrt(2) + 1) + log(sqrt(2) - 1)
sage: u = t.maxima_methods()
sage: u.parent()
<class 'sage.symbolic.maxima_wrapper.MaximaWrapper'>

classmethod random(ring=Rational Field, allow_infinity: bool = True, allow_empty: bool = False) → vectors_in_intervals.intervals.Interval¶

Generate a random interval.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Interval.random() # random
(-1, 1)

rename(self, x=None)¶

File: sage/structure/sage_object.pyx (starting at line 68)

Change self so it prints as x, where x is a string.

Note

This is only supported for Python classes that derive from SageObject.

EXAMPLES:

sage: x = PolynomialRing(QQ, 'x', sparse=True).gen()
sage: g = x^3 + x - 5
sage: g
x^3 + x - 5
sage: g.rename('a polynomial')
sage: g
a polynomial
sage: g + x
x^3 + 2*x - 5
sage: h = g^100
sage: str(h)[:20]
'x^300 + 100*x^298 - '
sage: h.rename('x^300 + ...')
sage: h
x^300 + ...

Real numbers are not Python classes, so rename is not supported:

sage: a = 3.14
sage: type(a)
<... 'sage.rings.real_mpfr.RealLiteral'>
sage: a.rename('pi')
Traceback (most recent call last):
...
NotImplementedError: object does not support renaming: 3.14000000000000

Note

The reason C-extension types are not supported by default is if they were then every single one would have to carry around an extra attribute, which would be slower and waste a lot of memory.

To support them for a specific class, add a cdef public __custom_name attribute.

reset_name(self)¶

File: sage/structure/sage_object.pyx (starting at line 125)

Remove the custom name of an object.

EXAMPLES:

sage: P.<x> = QQ[]
sage: P
Univariate Polynomial Ring in x over Rational Field
sage: P.rename('A polynomial ring')
sage: P
A polynomial ring
sage: P.reset_name()
sage: P
Univariate Polynomial Ring in x over Rational Field

save(self, filename=None, compress=True)¶

File: sage/structure/sage_object.pyx (starting at line 420)

Save self to the given filename.

EXAMPLES:

sage: f = x^3 + 5
sage: f.save(os.path.join(SAGE_TMP, 'file'))
sage: load(os.path.join(SAGE_TMP, 'file.sobj'))
x^3 + 5

simplest_element()¶

Return the simplest rational in this interval.

OUTPUT: If possible, an integer with smallest possible absolute value will be returned. Otherwise, a rational number with smallest possible denominator is constructed.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Interval(1/2, +oo, False, False).simplest_element()
1
sage: Interval(-oo, 1/2, False, False).simplest_element()
0
sage: Interval(-19, 0, False, False).simplest_element()
-1
sage: Interval(0, 1, False, False).simplest_element()
1/2
sage: Interval(-2/3, 0, False, False).simplest_element()
-1/2
sage: Interval(4/3, 3/2, False, False).simplest_element()
7/5
sage: Interval(0, 0).simplest_element()
0
sage: Interval(5, 5).simplest_element()
5
sage: Interval(sqrt(2), pi/2).simplest_element()
3/2
sage: Interval(1/2, 1/2).simplest_element()
1/2

supremum()¶

Return the supremum of the interval.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Interval(0, 1).supremum()
1
sage: Interval(0, +oo).supremum()
+Infinity
sage: Interval(0, 0, False, False).supremum()
-Infinity

class vectors_in_intervals.intervals.Intervals(intervals: list[vectors_in_intervals.intervals.Interval])¶

A Cartesian product of intervals.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Intervals([Interval(0, 1), Interval(-5, 2, False, False), Interval(0, 1)])
[0, 1] x (-5, 2) x [0, 1]
sage: vector([0, 1]) in Intervals([Interval(0, 1), Interval(-5, 2)])
True
sage: Intervals.random(3) # random
[0, +oo) x (-5, 2) x (0, 1]

an_element()¶: Return an element from each interval.

category(self)¶: File: sage/structure/sage_object.pyx (starting at line 484)

dump(self, filename, compress=True)¶

File: sage/structure/sage_object.pyx (starting at line 445)

Same as self.save(filename, compress)

dumps(self, compress=True)¶

File: sage/structure/sage_object.pyx (starting at line 451)

Dump self to a string s, which can later be reconstituted as self using loads(s).

There is an optional boolean argument compress which defaults to True.

EXAMPLES:

sage: from sage.misc.persist import comp
sage: O = SageObject()
sage: p_comp = O.dumps()
sage: p_uncomp = O.dumps(compress=False)
sage: comp.decompress(p_comp) == p_uncomp
True
sage: import pickletools
sage: pickletools.dis(p_uncomp)
    0: \x80 PROTO      2
    2: c    GLOBAL     'sage.structure.sage_object SageObject'
   41: q    BINPUT     ...
   43: )    EMPTY_TUPLE
   44: \x81 NEWOBJ
   45: q    BINPUT     ...
   47: .    STOP
highest protocol among opcodes = 2

classmethod from_bounds(lower_bounds: list, upper_bounds: list, lower_bounds_closed: bool = True, upper_bounds_closed: bool = True) → vectors_in_intervals.intervals.Intervals¶

Return intervals that are determined by bounds.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Intervals.from_bounds([0, -5, 0], [1, 2, +oo])
[0, 1] x [-5, 2] x [0, +oo)
sage: Intervals.from_bounds([0, -5, 0], [1, 2, +oo], False, False)
(0, 1) x (-5, 2) x (0, +oo)
sage: Intervals.from_bounds([0, -5, 0], [1, 2, +oo], True, False)
[0, 1) x [-5, 2) x [0, +oo)
sage: Intervals.from_bounds([0, -5, 0], [1, 2, +oo], [True, False, True], [True, True, False])
[0, 1] x (-5, 2] x [0, +oo)

classmethod from_sign_vector(sv: sign_vectors.sign_vectors.SignVector) → vectors_in_intervals.intervals.Intervals¶

Return intervals that are determined by a sign vector.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: from sign_vectors import *
sage: sv = sign_vector("+0-")
sage: Intervals.from_sign_vector(sv)
(0, +oo) x {0} x (-oo, 0)

is_bounded() → bool¶: Return whether all intervals are bounded.

is_closed() → bool¶: Return whether all intervals are closed.

is_empty() → bool¶: Return whether the intervals are empty.

is_open() → bool¶: Return whether all intervals are open.

is_pointed() → bool¶: Return whether all intervals are pointed.

is_unbounded() → bool¶: Return whether any interval is unbounded.

parent(self)¶

File: sage/structure/sage_object.pyx (starting at line 518)

Return the type of self to support the coercion framework.

EXAMPLES:

sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t
log(sqrt(2) + 1) + log(sqrt(2) - 1)
sage: u = t.maxima_methods()
sage: u.parent()
<class 'sage.symbolic.maxima_wrapper.MaximaWrapper'>

classmethod random(length: int, ring=Rational Field) → vectors_in_intervals.intervals.Intervals¶

Generate a random list of intervals.

EXAMPLES:

sage: from vectors_in_intervals import *
sage: Intervals.random(3) # random
[0, +oo) x (-5, 2) x (0, 1]

rename(self, x=None)¶

File: sage/structure/sage_object.pyx (starting at line 68)

Change self so it prints as x, where x is a string.

Note

This is only supported for Python classes that derive from SageObject.

EXAMPLES:

sage: x = PolynomialRing(QQ, 'x', sparse=True).gen()
sage: g = x^3 + x - 5
sage: g
x^3 + x - 5
sage: g.rename('a polynomial')
sage: g
a polynomial
sage: g + x
x^3 + 2*x - 5
sage: h = g^100
sage: str(h)[:20]
'x^300 + 100*x^298 - '
sage: h.rename('x^300 + ...')
sage: h
x^300 + ...

Real numbers are not Python classes, so rename is not supported:

sage: a = 3.14
sage: type(a)
<... 'sage.rings.real_mpfr.RealLiteral'>
sage: a.rename('pi')
Traceback (most recent call last):
...
NotImplementedError: object does not support renaming: 3.14000000000000

Note

The reason C-extension types are not supported by default is if they were then every single one would have to carry around an extra attribute, which would be slower and waste a lot of memory.

To support them for a specific class, add a cdef public __custom_name attribute.

reset_name(self)¶

File: sage/structure/sage_object.pyx (starting at line 125)

Remove the custom name of an object.

EXAMPLES:

sage: P.<x> = QQ[]
sage: P
Univariate Polynomial Ring in x over Rational Field
sage: P.rename('A polynomial ring')
sage: P
A polynomial ring
sage: P.reset_name()
sage: P
Univariate Polynomial Ring in x over Rational Field

save(self, filename=None, compress=True)¶

File: sage/structure/sage_object.pyx (starting at line 420)

Save self to the given filename.

EXAMPLES:

sage: f = x^3 + 5
sage: f.save(os.path.join(SAGE_TMP, 'file'))
sage: load(os.path.join(SAGE_TMP, 'file.sobj'))
x^3 + 5

sign_vectors(generator: bool = False) → Union[list[sign_vectors.sign_vectors.SignVector], Iterator[sign_vectors.sign_vectors.SignVector]]¶

Compute all sign vectors that correspond to a vector with components in given intervals.

INPUT:

intervals – a list of intervals
generator – a boolean (default: False)

EXAMPLES:

sage: from vectors_in_intervals import *
sage: intervals = Intervals.from_bounds([-1, 1], [0, 1])
sage: intervals.sign_vectors()
[(0+), (-+)]
sage: intervals = Intervals.from_bounds([-1, -2], [0, 1])
sage: intervals.sign_vectors()
[(00), (0+), (0-), (-0), (-+), (--)]
sage: intervals = Intervals.from_bounds([-1, -1, 0], [0, 5, 0])
sage: intervals.sign_vectors()
[(000), (0+0), (0-0), (-00), (-+0), (--0)]
sage: intervals = Intervals.from_bounds([-1, -1, -1], [0, 1, 0], False, False)
sage: intervals.sign_vectors()
[(-0-), (-+-), (---)]

TESTS:

sage: intervals = Intervals.from_bounds([-1, 0], [1, 0], False, False)
sage: intervals.sign_vectors()
[]
sage: intervals = Intervals.from_bounds([], [])
sage: intervals.sign_vectors()
[]

simplest_element()¶: Return the simplest element from each interval.

vectors_in_intervals.intervals.getrandbits(k) → x. Generates an int with k random bits.¶

vectors_in_intervals.intervals¶

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