Arbitrary precision real intervals using MPFI

AUTHORS:

• Carl Witty (2007-01-21): based on real_mpfr.pyx; changed it to use mpfi rather than mpfr.

• William Stein (2007-01-24): modifications and clean up and docs, etc.

• Niles Johnson (2010-08): Issue #3893: random_element() should pass on *args and **kwds.

• Travis Scrimshaw (2012-10-20): Fixing scientific notation output to fix Issue #13634.

• Travis Scrimshaw (2012-11-02): Added doctests for full coverage

This is a straightforward binding to the MPFI library; it may be useful to refer to its documentation for more details.

An interval is represented as a pair of floating-point numbers \(a\) and \(b\) (where \(a \leq b\)) and is printed as a standard floating-point number with a question mark (for instance, 3.1416?). The question mark indicates that the preceding digit may have an error of \(\pm 1\). These floating-point numbers are implemented using MPFR (the same as the RealNumber elements of RealField_class).

There is also an alternate method of printing, where the interval prints as [a .. b] (for instance, [3.1415 .. 3.1416]).

The interval represents the set \(\{ x : a \leq x \leq b \}\) (so if \(a = b\), then the interval represents that particular floating-point number). The endpoints can include positive and negative infinity, with the obvious meaning. It is also possible to have a NaN (Not-a-Number) interval, which is represented by having either endpoint be NaN.

PRINTING:

There are two styles for printing intervals: ‘brackets’ style and ‘question’ style (the default).

In question style, we print the “known correct” part of the number, followed by a question mark. The question mark indicates that the preceding digit is possibly wrong by \(\pm 1\).

Sage

sage: RIF(sqrt(2))                                                                  # needs sage.symbolic
1.414213562373095?

Python

>>> from sage.all import *
>>> RIF(sqrt(Integer(2)))                                                                  # needs sage.symbolic
1.414213562373095?

However, if the interval is precise (its lower bound is equal to its upper bound) and equal to a not-too-large integer, then we just print that integer.

Sage

sage: RIF(0)
0
sage: RIF(654321)
654321

Python

>>> from sage.all import *
>>> RIF(Integer(0))
0
>>> RIF(Integer(654321))
654321

Sage

sage: RIF(123, 125)
124.?
sage: RIF(123, 126)
1.3?e2

Python

>>> from sage.all import *
>>> RIF(Integer(123), Integer(125))
124.?
>>> RIF(Integer(123), Integer(126))
1.3?e2

As we see in the last example, question style can discard almost a whole digit’s worth of precision. We can reduce this by allowing “error digits”: an error following the question mark, that gives the maximum error of the digit(s) before the question mark. If the error is absent (which it always is in the default printing), then it is taken to be 1.

Sage

sage: RIF(123, 126).str(error_digits=1)
'125.?2'
sage: RIF(123, 127).str(error_digits=1)
'125.?2'
sage: v = RIF(-e, pi); v                                                            # needs sage.symbolic
0.?e1
sage: v.str(error_digits=1)                                                         # needs sage.symbolic
'1.?4'
sage: v.str(error_digits=5)                                                         # needs sage.symbolic
'0.2117?29300'

Python

>>> from sage.all import *
>>> RIF(Integer(123), Integer(126)).str(error_digits=Integer(1))
'125.?2'
>>> RIF(Integer(123), Integer(127)).str(error_digits=Integer(1))
'125.?2'
>>> v = RIF(-e, pi); v                                                            # needs sage.symbolic
0.?e1
>>> v.str(error_digits=Integer(1))                                                         # needs sage.symbolic
'1.?4'
>>> v.str(error_digits=Integer(5))                                                         # needs sage.symbolic
'0.2117?29300'

Error digits also sometimes let us indicate that the interval is actually equal to a single floating-point number:

Sage

sage: RIF(54321/256)
212.19140625000000?
sage: RIF(54321/256).str(error_digits=1)
'212.19140625000000?0'

Python

>>> from sage.all import *
>>> RIF(Integer(54321)/Integer(256))
212.19140625000000?
>>> RIF(Integer(54321)/Integer(256)).str(error_digits=Integer(1))
'212.19140625000000?0'

In brackets style, intervals are printed with the left value rounded down and the right rounded up, which is conservative, but in some ways unsatisfying.

Consider a 3-bit interval containing exactly the floating-point number 1.25. In round-to-nearest or round-down, this prints as 1.2; in round-up, this prints as 1.3. The straightforward options, then, are to print this interval as [1.2 .. 1.2] (which does not even contain the true value, 1.25), or to print it as [1.2 .. 1.3] (which gives the impression that the upper and lower bounds are not equal, even though they really are). Neither of these is very satisfying, but we have chosen the latter.

Sage

sage: R = RealIntervalField(3)
sage: a = R(1.25)
sage: a.str(style='brackets')
'[1.2 .. 1.3]'
sage: a == 5/4
True
sage: a == 2
False

Python

>>> from sage.all import *
>>> R = RealIntervalField(Integer(3))
>>> a = R(RealNumber('1.25'))
>>> a.str(style='brackets')
'[1.2 .. 1.3]'
>>> a == Integer(5)/Integer(4)
True
>>> a == Integer(2)
False

COMPARISONS:

Comparison operations (==, !=, <, <=, >, >=) return True if every value in the first interval has the given relation to every value in the second interval.

This convention for comparison operators has good and bad points. The good:

• Expected transitivity properties hold (if a > b and b == c, then a > c; etc.)

• a == 0 is true if the interval contains only the floating-point number 0; similarly for a == 1

• a > 0 means something useful (that every value in the interval is greater than 0)

The bad:

• Trichotomy fails to hold: there are values (a,b) such that none of a < b, a == b, or a > b are true

• There are values a and b such that a <= b but neither a < b nor a == b hold.

• There are values a and b such that neither a != b nor a == b hold.

Note

Intervals a and b overlap iff not(a != b).

Warning

The cmp(a, b) function should not be used to compare real intervals. Note that cmp will disappear in Python3.

EXAMPLES:

Sage

sage: 0 < RIF(1, 2)
True
sage: 0 == RIF(0)
True
sage: not(0 == RIF(0, 1))
True
sage: not(0 != RIF(0, 1))
True
sage: 0 <= RIF(0, 1)
True
sage: not(0 < RIF(0, 1))
True

Python

>>> from sage.all import *
>>> Integer(0) < RIF(Integer(1), Integer(2))
True
>>> Integer(0) == RIF(Integer(0))
True
>>> not(Integer(0) == RIF(Integer(0), Integer(1)))
True
>>> not(Integer(0) != RIF(Integer(0), Integer(1)))
True
>>> Integer(0) <= RIF(Integer(0), Integer(1))
True
>>> not(Integer(0) < RIF(Integer(0), Integer(1)))
True

Comparison with infinity is defined through coercion to the infinity ring where semi-infinite intervals are sent to their central value (plus or minus infinity); this implements the above convention for inequalities:

Sage

sage: InfinityRing.has_coerce_map_from(RIF)
True
sage: -oo < RIF(-1,1) < oo
True
sage: -oo < RIF(0,oo) <= oo
True
sage: -oo <= RIF(-oo,-1) < oo
True

Python

>>> from sage.all import *
>>> InfinityRing.has_coerce_map_from(RIF)
True
>>> -oo < RIF(-Integer(1),Integer(1)) < oo
True
>>> -oo < RIF(Integer(0),oo) <= oo
True
>>> -oo <= RIF(-oo,-Integer(1)) < oo
True

Comparison by equality shows what the semi-infinite intervals actually coerce to:

Sage

sage: RIF(1,oo) == oo
True
sage: RIF(-oo,-1) == -oo
True

Python

>>> from sage.all import *
>>> RIF(Integer(1),oo) == oo
True
>>> RIF(-oo,-Integer(1)) == -oo
True

For lack of a better value in the infinity ring, the doubly infinite interval coerces to plus infinity:

Sage

sage: RIF(-oo,oo) == oo
True

Python

>>> from sage.all import *
>>> RIF(-oo,oo) == oo
True

If you want to compare two intervals lexicographically, you can use the method lexico_cmp. However, the behavior of this method is not specified if given a non-interval and an interval:

Sage

sage: RIF(0).lexico_cmp(RIF(0, 1))
-1
sage: RIF(0, 1).lexico_cmp(RIF(0))
1
sage: RIF(0, 1).lexico_cmp(RIF(1))
-1
sage: RIF(0, 1).lexico_cmp(RIF(0, 1))
0

Python

>>> from sage.all import *
>>> RIF(Integer(0)).lexico_cmp(RIF(Integer(0), Integer(1)))
-1
>>> RIF(Integer(0), Integer(1)).lexico_cmp(RIF(Integer(0)))
1
>>> RIF(Integer(0), Integer(1)).lexico_cmp(RIF(Integer(1)))
-1
>>> RIF(Integer(0), Integer(1)).lexico_cmp(RIF(Integer(0), Integer(1)))
0

Warning

Mixing symbolic expressions with intervals (in particular, converting constant symbolic expressions to intervals), can lead to incorrect results:

Sage

sage: ref = RealIntervalField(100)(ComplexBallField(100).one().airy_ai().real())
sage: ref
0.135292416312881415524147423515?
sage: val = RIF(airy_ai(1)); val # known bug
0.13529241631288142?
sage: val.overlaps(ref)          # known bug
False

Python

>>> from sage.all import *
>>> ref = RealIntervalField(Integer(100))(ComplexBallField(Integer(100)).one().airy_ai().real())
>>> ref
0.135292416312881415524147423515?
>>> val = RIF(airy_ai(Integer(1))); val # known bug
0.13529241631288142?
>>> val.overlaps(ref)          # known bug
False

sage.rings.real_mpfi.RealInterval(s, upper=None, base=10, pad=0, min_prec=53)

Return the real number defined by the string s as an element of RealIntervalField(prec=n), where n potentially has slightly more (controlled by pad) bits than given by s.

INPUT:

• s – string that defines a real number (or something whose string representation defines a number)

• upper – (default: None) upper endpoint of interval if given, in which case s is the lower endpoint

• base – integer between 2 and 36

• pad – integer (default: 0)

• min_prec – number will have at least this many bits of precision, no matter what

EXAMPLES:

Sage

sage: RealInterval('2.3')
2.300000000000000?
sage: RealInterval(10)
10
sage: RealInterval('1.0000000000000000000000000000000000')
1
sage: RealInterval('1.2345678901234567890123456789012345')
1.23456789012345678901234567890123450?
sage: RealInterval(29308290382930840239842390482, 3^20).str(style='brackets')
'[3.48678440100000000000000000000e9 .. 2.93082903829308402398423904820e28]'

Python

>>> from sage.all import *
>>> RealInterval('2.3')
2.300000000000000?
>>> RealInterval(Integer(10))
10
>>> RealInterval('1.0000000000000000000000000000000000')
1
>>> RealInterval('1.2345678901234567890123456789012345')
1.23456789012345678901234567890123450?
>>> RealInterval(Integer(29308290382930840239842390482), Integer(3)**Integer(20)).str(style='brackets')
'[3.48678440100000000000000000000e9 .. 2.93082903829308402398423904820e28]'

sage.rings.real_mpfi.RealIntervalField(prec=53, sci_not=False)

Construct a RealIntervalField_class, with caching.

INPUT:

• prec – integer (default: 53); precision. The number of bits used to represent the mantissa of a floating-point number. The precision can be any integer between mpfr_prec_min() and mpfr_prec_max(). In the current implementation, mpfr_prec_min() is equal to 2.

• sci_not – boolean (default: False); whether or not to display using scientific notation

EXAMPLES:

Sage

sage: RealIntervalField()
Real Interval Field with 53 bits of precision
sage: RealIntervalField(200, sci_not=True)
Real Interval Field with 200 bits of precision
sage: RealIntervalField(53) is RIF
True
sage: RealIntervalField(200) is RIF
False
sage: RealIntervalField(200) is RealIntervalField(200)
True

Python

>>> from sage.all import *
>>> RealIntervalField()
Real Interval Field with 53 bits of precision
>>> RealIntervalField(Integer(200), sci_not=True)
Real Interval Field with 200 bits of precision
>>> RealIntervalField(Integer(53)) is RIF
True
>>> RealIntervalField(Integer(200)) is RIF
False
>>> RealIntervalField(Integer(200)) is RealIntervalField(Integer(200))
True

See the documentation for RealIntervalField_class for many more examples.

class sage.rings.real_mpfi.RealIntervalFieldElement

Bases: RingElement

A real number interval.

absolute_diameter()

The diameter of this interval (for \([a .. b]\), this is \(b-a\)), rounded upward, as a RealNumber.

EXAMPLES:

Sage

sage: RIF(1, pi).absolute_diameter()                                        # needs sage.symbolic
2.14159265358979

Python

>>> from sage.all import *
>>> RIF(Integer(1), pi).absolute_diameter()                                        # needs sage.symbolic
2.14159265358979

alea()

Return a floating-point number picked at random from the interval.

EXAMPLES:

Sage

sage: RIF(1, 2).alea() # random
1.34696133696137

Python

>>> from sage.all import *
>>> RIF(Integer(1), Integer(2)).alea() # random
1.34696133696137

algdep(n)

Return a polynomial of degree at most \(n\) which is approximately satisfied by self.

Note

The returned polynomial need not be irreducible, and indeed usually won’t be if self is a good approximation to an algebraic number of degree less than \(n\).

Pari needs to know the number of “known good bits” in the number; we automatically get that from the interval width.

ALGORITHM:

Uses the PARI C-library algdep command.

EXAMPLES:

Sage

sage: r = sqrt(RIF(2)); r
1.414213562373095?
sage: r.algdep(5)
x^2 - 2

Python

>>> from sage.all import *
>>> r = sqrt(RIF(Integer(2))); r
1.414213562373095?
>>> r.algdep(Integer(5))
x^2 - 2

If we compute a wrong, but precise, interval, we get a wrong answer:

Sage

sage: r = sqrt(RealIntervalField(200)(2)) + (1/2)^40; r
1.414213562374004543503461652447613117632171875376948073176680?
sage: r.algdep(5)
7266488*x^5 + 22441629*x^4 - 90470501*x^3 + 23297703*x^2 + 45778664*x + 13681026

Python

>>> from sage.all import *
>>> r = sqrt(RealIntervalField(Integer(200))(Integer(2))) + (Integer(1)/Integer(2))**Integer(40); r
1.414213562374004543503461652447613117632171875376948073176680?
>>> r.algdep(Integer(5))
7266488*x^5 + 22441629*x^4 - 90470501*x^3 + 23297703*x^2 + 45778664*x + 13681026

But if we compute an interval that includes the number we mean, we’re much more likely to get the right answer, even if the interval is very imprecise:

Sage

sage: r = r.union(sqrt(2.0))
sage: r.algdep(5)
x^2 - 2

Python

>>> from sage.all import *
>>> r = r.union(sqrt(RealNumber('2.0')))
>>> r.algdep(Integer(5))
x^2 - 2

Even on this extremely imprecise interval we get an answer which is technically correct:

Sage

sage: RIF(-1, 1).algdep(5)
x

Python

>>> from sage.all import *
>>> RIF(-Integer(1), Integer(1)).algdep(Integer(5))
x

arccos()

Return the inverse cosine of self.

EXAMPLES:

Sage

sage: q = RIF.pi()/3; q
1.047197551196598?
sage: i = q.cos(); i
0.500000000000000?
sage: q2 = i.arccos(); q2
1.047197551196598?
sage: q == q2
False
sage: q != q2
False
sage: q2.lower() == q.lower()
False
sage: q - q2
0.?e-15
sage: q in q2
True

Python

>>> from sage.all import *
>>> q = RIF.pi()/Integer(3); q
1.047197551196598?
>>> i = q.cos(); i
0.500000000000000?
>>> q2 = i.arccos(); q2
1.047197551196598?
>>> q == q2
False
>>> q != q2
False
>>> q2.lower() == q.lower()
False
>>> q - q2
0.?e-15
>>> q in q2
True

arccosh()

Return the hyperbolic inverse cosine of self.

EXAMPLES:

Sage

sage: q = RIF.pi()/2
sage: i = q.arccosh() ; i
1.023227478547551?

Python

>>> from sage.all import *
>>> q = RIF.pi()/Integer(2)
>>> i = q.arccosh() ; i
1.023227478547551?

arccoth()

Return the inverse hyperbolic cotangent of self.

EXAMPLES:

Sage

sage: RealIntervalField(100)(2).arccoth()
0.549306144334054845697622618462?
sage: (2.0).arccoth()
0.549306144334055

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(100))(Integer(2)).arccoth()
0.549306144334054845697622618462?
>>> (RealNumber('2.0')).arccoth()
0.549306144334055

arccsch()

Return the inverse hyperbolic cosecant of self.

EXAMPLES:

Sage

sage: RealIntervalField(100)(2).arccsch()
0.481211825059603447497758913425?
sage: (2.0).arccsch()
0.481211825059603

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(100))(Integer(2)).arccsch()
0.481211825059603447497758913425?
>>> (RealNumber('2.0')).arccsch()
0.481211825059603

arcsech()

Return the inverse hyperbolic secant of self.

EXAMPLES:

Sage

sage: RealIntervalField(100)(0.5).arcsech()
1.316957896924816708625046347308?
sage: (0.5).arcsech()
1.31695789692482

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(100))(RealNumber('0.5')).arcsech()
1.316957896924816708625046347308?
>>> (RealNumber('0.5')).arcsech()
1.31695789692482

arcsin()

Return the inverse sine of self.

EXAMPLES:

Sage

sage: q = RIF.pi()/5; q
0.6283185307179587?
sage: i = q.sin(); i
0.587785252292474?
sage: q2 = i.arcsin(); q2
0.628318530717959?
sage: q == q2
False
sage: q != q2
False
sage: q2.lower() == q.lower()
False
sage: q - q2
0.?e-15
sage: q in q2
True

Python

>>> from sage.all import *
>>> q = RIF.pi()/Integer(5); q
0.6283185307179587?
>>> i = q.sin(); i
0.587785252292474?
>>> q2 = i.arcsin(); q2
0.628318530717959?
>>> q == q2
False
>>> q != q2
False
>>> q2.lower() == q.lower()
False
>>> q - q2
0.?e-15
>>> q in q2
True

arcsinh()

Return the hyperbolic inverse sine of self.

EXAMPLES:

Sage

sage: q = RIF.pi()/7
sage: i = q.sinh() ; i
0.464017630492991?
sage: i.arcsinh() - q
0.?e-15

Python

>>> from sage.all import *
>>> q = RIF.pi()/Integer(7)
>>> i = q.sinh() ; i
0.464017630492991?
>>> i.arcsinh() - q
0.?e-15

arctan()

Return the inverse tangent of self.

EXAMPLES:

Sage

sage: q = RIF.pi()/5; q
0.6283185307179587?
sage: i = q.tan(); i
0.726542528005361?
sage: q2 = i.arctan(); q2
0.628318530717959?
sage: q == q2
False
sage: q != q2
False
sage: q2.lower() == q.lower()
False
sage: q - q2
0.?e-15
sage: q in q2
True

Python

>>> from sage.all import *
>>> q = RIF.pi()/Integer(5); q
0.6283185307179587?
>>> i = q.tan(); i
0.726542528005361?
>>> q2 = i.arctan(); q2
0.628318530717959?
>>> q == q2
False
>>> q != q2
False
>>> q2.lower() == q.lower()
False
>>> q - q2
0.?e-15
>>> q in q2
True

arctanh()

Return the hyperbolic inverse tangent of self.

EXAMPLES:

Sage

sage: q = RIF.pi()/7
sage: i = q.tanh() ; i
0.420911241048535?
sage: i.arctanh() - q
0.?e-15

Python

>>> from sage.all import *
>>> q = RIF.pi()/Integer(7)
>>> i = q.tanh() ; i
0.420911241048535?
>>> i.arctanh() - q
0.?e-15

argument()

The argument of this interval, if it is well-defined, in the complex sense. Otherwise raises a ValueError.

OUTPUT:

• an element of the parent of this interval (0 or pi)

EXAMPLES:

Sage

sage: RIF(1).argument()
0
sage: RIF(-1).argument()
3.141592653589794?
sage: RIF(0,1).argument()
0
sage: RIF(-1,0).argument()
3.141592653589794?
sage: RIF(0).argument()
Traceback (most recent call last):
...
ValueError: Can't take the argument of an exact zero
sage: RIF(-1,1).argument()
Traceback (most recent call last):
...
ValueError: Can't take the argument of interval strictly containing zero

Python

>>> from sage.all import *
>>> RIF(Integer(1)).argument()
0
>>> RIF(-Integer(1)).argument()
3.141592653589794?
>>> RIF(Integer(0),Integer(1)).argument()
0
>>> RIF(-Integer(1),Integer(0)).argument()
3.141592653589794?
>>> RIF(Integer(0)).argument()
Traceback (most recent call last):
...
ValueError: Can't take the argument of an exact zero
>>> RIF(-Integer(1),Integer(1)).argument()
Traceback (most recent call last):
...
ValueError: Can't take the argument of interval strictly containing zero

bisection()

Return the bisection of self into two intervals of half the size whose union is self and intersection is center().

EXAMPLES:

Sage

sage: a, b = RIF(1,2).bisection()
sage: a.lower(), a.upper()
(1.00000000000000, 1.50000000000000)
sage: b.lower(), b.upper()
(1.50000000000000, 2.00000000000000)

sage: # needs sage.symbolic
sage: I = RIF(e, pi)
sage: a, b = I.bisection()
sage: a.intersection(b) == RIF(I.center())
True
sage: a.union(b).endpoints() == I.endpoints()
True

Python

>>> from sage.all import *
>>> a, b = RIF(Integer(1),Integer(2)).bisection()
>>> a.lower(), a.upper()
(1.00000000000000, 1.50000000000000)
>>> b.lower(), b.upper()
(1.50000000000000, 2.00000000000000)

>>> # needs sage.symbolic
>>> I = RIF(e, pi)
>>> a, b = I.bisection()
>>> a.intersection(b) == RIF(I.center())
True
>>> a.union(b).endpoints() == I.endpoints()
True

ceil()

Return the ceiling of this interval as an interval.

The ceiling of a real number \(x\) is the smallest integer larger than or equal to \(x\).

See also

• unique_ceil() – return the ceil as an integer if it is unique and raises a ValueError otherwise

• floor() – truncation towards minus infinity

• trunc() – truncation towards zero

• round() – rounding

EXAMPLES:

Sage

sage: (2.99).ceil()
3
sage: (2.00).ceil()
2
sage: (2.01).ceil()
3
sage: R = RealIntervalField(30)
sage: a = R(-9.5, -11.3); a.str(style='brackets')
'[-11.300000012 .. -9.5000000000]'
sage: a.floor().str(style='brackets')
'[-12.000000000 .. -10.000000000]'
sage: a.ceil()
-10.?
sage: ceil(a).str(style='brackets')
'[-11.000000000 .. -9.0000000000]'

Python

>>> from sage.all import *
>>> (RealNumber('2.99')).ceil()
3
>>> (RealNumber('2.00')).ceil()
2
>>> (RealNumber('2.01')).ceil()
3
>>> R = RealIntervalField(Integer(30))
>>> a = R(-RealNumber('9.5'), -RealNumber('11.3')); a.str(style='brackets')
'[-11.300000012 .. -9.5000000000]'
>>> a.floor().str(style='brackets')
'[-12.000000000 .. -10.000000000]'
>>> a.ceil()
-10.?
>>> ceil(a).str(style='brackets')
'[-11.000000000 .. -9.0000000000]'

ceiling()

Return the ceiling of this interval as an interval.

The ceiling of a real number \(x\) is the smallest integer larger than or equal to \(x\).

See also

• unique_ceil() – return the ceil as an integer if it is unique and raises a ValueError otherwise

• floor() – truncation towards minus infinity

• trunc() – truncation towards zero

• round() – rounding

EXAMPLES:

Sage

sage: (2.99).ceil()
3
sage: (2.00).ceil()
2
sage: (2.01).ceil()
3
sage: R = RealIntervalField(30)
sage: a = R(-9.5, -11.3); a.str(style='brackets')
'[-11.300000012 .. -9.5000000000]'
sage: a.floor().str(style='brackets')
'[-12.000000000 .. -10.000000000]'
sage: a.ceil()
-10.?
sage: ceil(a).str(style='brackets')
'[-11.000000000 .. -9.0000000000]'

Python

>>> from sage.all import *
>>> (RealNumber('2.99')).ceil()
3
>>> (RealNumber('2.00')).ceil()
2
>>> (RealNumber('2.01')).ceil()
3
>>> R = RealIntervalField(Integer(30))
>>> a = R(-RealNumber('9.5'), -RealNumber('11.3')); a.str(style='brackets')
'[-11.300000012 .. -9.5000000000]'
>>> a.floor().str(style='brackets')
'[-12.000000000 .. -10.000000000]'
>>> a.ceil()
-10.?
>>> ceil(a).str(style='brackets')
'[-11.000000000 .. -9.0000000000]'

center()

Compute the center of the interval \([a .. b]\) which is \((a+b) / 2\).

EXAMPLES:

Sage

sage: RIF(1, 2).center()
1.50000000000000

Python

>>> from sage.all import *
>>> RIF(Integer(1), Integer(2)).center()
1.50000000000000

contains_zero()

Return True if self is an interval containing zero.

EXAMPLES:

Sage

sage: RIF(0).contains_zero()
True
sage: RIF(1, 2).contains_zero()
False
sage: RIF(-1, 1).contains_zero()
True
sage: RIF(-1, 0).contains_zero()
True

Python

>>> from sage.all import *
>>> RIF(Integer(0)).contains_zero()
True
>>> RIF(Integer(1), Integer(2)).contains_zero()
False
>>> RIF(-Integer(1), Integer(1)).contains_zero()
True
>>> RIF(-Integer(1), Integer(0)).contains_zero()
True

cos()

Return the cosine of self.

EXAMPLES:

Sage

sage: # needs sage.symbolic
sage: t = RIF(pi)/2
sage: t.cos()
0.?e-15
sage: t.cos().str(style='brackets')
'[-1.6081226496766367e-16 .. 6.1232339957367661e-17]'
sage: t.cos().cos()
0.9999999999999999?

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> t = RIF(pi)/Integer(2)
>>> t.cos()
0.?e-15
>>> t.cos().str(style='brackets')
'[-1.6081226496766367e-16 .. 6.1232339957367661e-17]'
>>> t.cos().cos()
0.9999999999999999?

cosh()

Return the hyperbolic cosine of self.

EXAMPLES:

Sage

sage: q = RIF.pi()/12
sage: q.cosh()
1.034465640095511?

Python

>>> from sage.all import *
>>> q = RIF.pi()/Integer(12)
>>> q.cosh()
1.034465640095511?

cot()

Return the cotangent of self.

EXAMPLES:

Sage

sage: RealIntervalField(100)(2).cot()
-0.457657554360285763750277410432?

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(100))(Integer(2)).cot()
-0.457657554360285763750277410432?

coth()

Return the hyperbolic cotangent of self.

EXAMPLES:

Sage

sage: RealIntervalField(100)(2).coth()
1.03731472072754809587780976477?

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(100))(Integer(2)).coth()
1.03731472072754809587780976477?

csc()

Return the cosecant of self.

EXAMPLES:

Sage

sage: RealIntervalField(100)(2).csc()
1.099750170294616466756697397026?

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(100))(Integer(2)).csc()
1.099750170294616466756697397026?

csch()

Return the hyperbolic cosecant of self.

EXAMPLES:

Sage

sage: RealIntervalField(100)(2).csch()
0.275720564771783207758351482163?

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(100))(Integer(2)).csch()
0.275720564771783207758351482163?

diameter()

If 0 is in self, then return absolute_diameter(), otherwise return relative_diameter().

EXAMPLES:

Sage

sage: RIF(1, 2).diameter()
0.666666666666667
sage: RIF(1, 2).absolute_diameter()
1.00000000000000
sage: RIF(1, 2).relative_diameter()
0.666666666666667

sage: # needs sage.symbolic
sage: RIF(pi).diameter()
1.41357985842823e-16
sage: RIF(pi).absolute_diameter()
4.44089209850063e-16
sage: RIF(pi).relative_diameter()
1.41357985842823e-16
sage: (RIF(pi) - RIF(3, 22/7)).diameter()
0.142857142857144
sage: (RIF(pi) - RIF(3, 22/7)).absolute_diameter()
0.142857142857144
sage: (RIF(pi) - RIF(3, 22/7)).relative_diameter()
2.03604377705518

Python

>>> from sage.all import *
>>> RIF(Integer(1), Integer(2)).diameter()
0.666666666666667
>>> RIF(Integer(1), Integer(2)).absolute_diameter()
1.00000000000000
>>> RIF(Integer(1), Integer(2)).relative_diameter()
0.666666666666667

>>> # needs sage.symbolic
>>> RIF(pi).diameter()
1.41357985842823e-16
>>> RIF(pi).absolute_diameter()
4.44089209850063e-16
>>> RIF(pi).relative_diameter()
1.41357985842823e-16
>>> (RIF(pi) - RIF(Integer(3), Integer(22)/Integer(7))).diameter()
0.142857142857144
>>> (RIF(pi) - RIF(Integer(3), Integer(22)/Integer(7))).absolute_diameter()
0.142857142857144
>>> (RIF(pi) - RIF(Integer(3), Integer(22)/Integer(7))).relative_diameter()
2.03604377705518

edges()

Return the lower and upper endpoints of this interval as intervals.

OUTPUT: a 2-tuple of real intervals (lower endpoint, upper endpoint) each containing just one point.

See also

endpoints() which returns the endpoints as real numbers instead of intervals.

EXAMPLES:

Sage

sage: RIF(1,2).edges()
(1, 2)
sage: RIF(pi).edges()                                                       # needs sage.symbolic
(3.1415926535897932?, 3.1415926535897936?)

Python

>>> from sage.all import *
>>> RIF(Integer(1),Integer(2)).edges()
(1, 2)
>>> RIF(pi).edges()                                                       # needs sage.symbolic
(3.1415926535897932?, 3.1415926535897936?)

endpoints(rnd=None)

Return the lower and upper endpoints of this interval.

OUTPUT: a 2-tuple of real numbers (lower endpoint, upper endpoint)

See also

edges() which returns the endpoints as exact intervals instead of real numbers.

EXAMPLES:

Sage

sage: RIF(1,2).endpoints()
(1.00000000000000, 2.00000000000000)
sage: RIF(pi).endpoints()                                                   # needs sage.symbolic
(3.14159265358979, 3.14159265358980)
sage: a = CIF(RIF(1,2), RIF(3,4))
sage: a.real().endpoints()
(1.00000000000000, 2.00000000000000)

Python

>>> from sage.all import *
>>> RIF(Integer(1),Integer(2)).endpoints()
(1.00000000000000, 2.00000000000000)
>>> RIF(pi).endpoints()                                                   # needs sage.symbolic
(3.14159265358979, 3.14159265358980)
>>> a = CIF(RIF(Integer(1),Integer(2)), RIF(Integer(3),Integer(4)))
>>> a.real().endpoints()
(1.00000000000000, 2.00000000000000)

As with lower() and upper(), a rounding mode is accepted:

Sage

sage: RIF(1,2).endpoints('RNDD')[0].parent()
Real Field with 53 bits of precision and rounding RNDD

Python

>>> from sage.all import *
>>> RIF(Integer(1),Integer(2)).endpoints('RNDD')[Integer(0)].parent()
Real Field with 53 bits of precision and rounding RNDD

exp()

Return \(e^\mathtt{self}\).

EXAMPLES:

Sage

sage: r = RIF(0.0)
sage: r.exp()
1

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('0.0'))
>>> r.exp()
1

Sage

sage: r = RIF(32.3)
sage: a = r.exp(); a
1.065888472748645?e14
sage: a.log()
32.30000000000000?

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('32.3'))
>>> a = r.exp(); a
1.065888472748645?e14
>>> a.log()
32.30000000000000?

Sage

sage: r = RIF(-32.3)
sage: r.exp()
9.38184458849869?e-15

Python

>>> from sage.all import *
>>> r = RIF(-RealNumber('32.3'))
>>> r.exp()
9.38184458849869?e-15

exp2()

Return \(2^\mathtt{self}\).

EXAMPLES:

Sage

sage: r = RIF(0.0)
sage: r.exp2()
1

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('0.0'))
>>> r.exp2()
1

Sage

sage: r = RIF(32.0)
sage: r.exp2()
4294967296

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('32.0'))
>>> r.exp2()
4294967296

Sage

sage: r = RIF(-32.3)
sage: r.exp2()
1.891172482530207?e-10

Python

>>> from sage.all import *
>>> r = RIF(-RealNumber('32.3'))
>>> r.exp2()
1.891172482530207?e-10

factorial()

Return the factorial evaluated on self.

EXAMPLES:

Sage

sage: RIF(5).factorial()
120
sage: RIF(2.3,5.7).factorial()
1.?e3
sage: RIF(2.3).factorial()
2.683437381955768?

Python

>>> from sage.all import *
>>> RIF(Integer(5)).factorial()
120
>>> RIF(RealNumber('2.3'),RealNumber('5.7')).factorial()
1.?e3
>>> RIF(RealNumber('2.3')).factorial()
2.683437381955768?

Recover the factorial as integer:

Sage

sage: f = RealIntervalField(200)(50).factorial()
sage: f
3.0414093201713378043612608166064768844377641568960512000000000?e64
sage: f.unique_integer()
30414093201713378043612608166064768844377641568960512000000000000
sage: 50.factorial()
30414093201713378043612608166064768844377641568960512000000000000

Python

>>> from sage.all import *
>>> f = RealIntervalField(Integer(200))(Integer(50)).factorial()
>>> f
3.0414093201713378043612608166064768844377641568960512000000000?e64
>>> f.unique_integer()
30414093201713378043612608166064768844377641568960512000000000000
>>> Integer(50).factorial()
30414093201713378043612608166064768844377641568960512000000000000

floor()

Return the floor of this interval as an interval.

The floor of a real number \(x\) is the largest integer smaller than or equal to \(x\).

See also

• unique_floor() – method which returns the floor as an integer if it is unique or raises a ValueError otherwise

• ceil() – truncation towards plus infinity

• round() – rounding

• trunc() – truncation towards zero

EXAMPLES:

Sage

sage: R = RealIntervalField()
sage: (2.99).floor()
2
sage: (2.00).floor()
2
sage: floor(RR(-5/2))
-3
sage: R = RealIntervalField(100)
sage: a = R(9.5, 11.3); a.str(style='brackets')
'[9.5000000000000000000000000000000 .. 11.300000000000000710542735760101]'
sage: floor(a).str(style='brackets')
'[9.0000000000000000000000000000000 .. 11.000000000000000000000000000000]'
sage: a.floor()
10.?
sage: ceil(a)
11.?
sage: a.ceil().str(style='brackets')
'[10.000000000000000000000000000000 .. 12.000000000000000000000000000000]'

Python

>>> from sage.all import *
>>> R = RealIntervalField()
>>> (RealNumber('2.99')).floor()
2
>>> (RealNumber('2.00')).floor()
2
>>> floor(RR(-Integer(5)/Integer(2)))
-3
>>> R = RealIntervalField(Integer(100))
>>> a = R(RealNumber('9.5'), RealNumber('11.3')); a.str(style='brackets')
'[9.5000000000000000000000000000000 .. 11.300000000000000710542735760101]'
>>> floor(a).str(style='brackets')
'[9.0000000000000000000000000000000 .. 11.000000000000000000000000000000]'
>>> a.floor()
10.?
>>> ceil(a)
11.?
>>> a.ceil().str(style='brackets')
'[10.000000000000000000000000000000 .. 12.000000000000000000000000000000]'

fp_rank_diameter()

Compute the diameter of this interval in terms of the “floating-point rank”.

The floating-point rank is the number of floating-point numbers (of the current precision) contained in the given interval, minus one. An fp_rank_diameter of 0 means that the interval is exact; an fp_rank_diameter of 1 means that the interval is as tight as possible, unless the number you’re trying to represent is actually exactly representable as a floating-point number.

EXAMPLES:

Sage

sage: RIF(12345).fp_rank_diameter()
0
sage: RIF(5/8).fp_rank_diameter()
0
sage: RIF(5/7).fp_rank_diameter()
1

sage: # needs sage.symbolic
sage: RIF(pi).fp_rank_diameter()
1
sage: RIF(-sqrt(2)).fp_rank_diameter()
1
sage: a = RIF(pi)^12345; a
2.06622879260?e6137
sage: a.fp_rank_diameter()
30524
sage: (RIF(sqrt(2)) - RIF(sqrt(2))).fp_rank_diameter()
9671406088542672151117826            # 32-bit
41538374868278620559869609387229186  # 64-bit

Python

>>> from sage.all import *
>>> RIF(Integer(12345)).fp_rank_diameter()
0
>>> RIF(Integer(5)/Integer(8)).fp_rank_diameter()
0
>>> RIF(Integer(5)/Integer(7)).fp_rank_diameter()
1

>>> # needs sage.symbolic
>>> RIF(pi).fp_rank_diameter()
1
>>> RIF(-sqrt(Integer(2))).fp_rank_diameter()
1
>>> a = RIF(pi)**Integer(12345); a
2.06622879260?e6137
>>> a.fp_rank_diameter()
30524
>>> (RIF(sqrt(Integer(2))) - RIF(sqrt(Integer(2)))).fp_rank_diameter()
9671406088542672151117826            # 32-bit
41538374868278620559869609387229186  # 64-bit

Just because we have the best possible interval, doesn’t mean the interval is actually small:

Sage

sage: a = RIF(pi)^12345678901234567890; a                                   # needs sage.symbolic
[2.0985787164673874e323228496 .. +infinity]            # 32-bit
[5.8756537891115869e1388255822130839282 .. +infinity]  # 64-bit
sage: a.fp_rank_diameter()                                                  # needs sage.symbolic
1

Python

>>> from sage.all import *
>>> a = RIF(pi)**Integer(12345678901234567890); a                                   # needs sage.symbolic
[2.0985787164673874e323228496 .. +infinity]            # 32-bit
[5.8756537891115869e1388255822130839282 .. +infinity]  # 64-bit
>>> a.fp_rank_diameter()                                                  # needs sage.symbolic
1

frac()

Return the fractional part of this interval as an interval.

The fractional part \(y\) of a real number \(x\) is the unique element in the interval \((-1,1)\) that has the same sign as \(x\) and such that \(x-y\) is an integer. The integer \(x-y\) can be obtained through the method trunc().

The output of this function is the smallest interval that contains all possible values of \(frac(x)\) for \(x\) in this interval. Note that if it contains an integer then the answer might not be very meaningful. More precisely, if the endpoints are \(a\) and \(b\) then:

• if \(floor(b) > \max(a,0)\) then the interval obtained contains \([0,1]\),

• if \(ceil(a) < \min(b,0)\) then the interval obtained contains \([-1,0]\).

See also

trunc() – return the integer part complement to this fractional part

EXAMPLES:

Sage

sage: RIF(2.37123, 2.372).frac()
0.372?
sage: RIF(-23.12, -23.13).frac()
-0.13?

sage: RIF(.5, 1).frac().endpoints()
(0.000000000000000, 1.00000000000000)
sage: RIF(1, 1.5).frac().endpoints()
(0.000000000000000, 0.500000000000000)

sage: r = RIF(-22.47, -22.468)
sage: r in (r.frac() + r.trunc())
True

sage: r = RIF(18.222, 18.223)
sage: r in (r.frac() + r.trunc())
True

sage: RIF(1.99, 2.025).frac().endpoints()
(0.000000000000000, 1.00000000000000)
sage: RIF(1.99, 2.00).frac().endpoints()
(0.000000000000000, 1.00000000000000)
sage: RIF(2.00, 2.025).frac().endpoints()
(0.000000000000000, 0.0250000000000000)

sage: RIF(-2.1,-0.9).frac().endpoints()
(-1.00000000000000, -0.000000000000000)
sage: RIF(-0.5,0.5).frac().endpoints()
(-0.500000000000000, 0.500000000000000)

Python

>>> from sage.all import *
>>> RIF(RealNumber('2.37123'), RealNumber('2.372')).frac()
0.372?
>>> RIF(-RealNumber('23.12'), -RealNumber('23.13')).frac()
-0.13?

>>> RIF(RealNumber('.5'), Integer(1)).frac().endpoints()
(0.000000000000000, 1.00000000000000)
>>> RIF(Integer(1), RealNumber('1.5')).frac().endpoints()
(0.000000000000000, 0.500000000000000)

>>> r = RIF(-RealNumber('22.47'), -RealNumber('22.468'))
>>> r in (r.frac() + r.trunc())
True

>>> r = RIF(RealNumber('18.222'), RealNumber('18.223'))
>>> r in (r.frac() + r.trunc())
True

>>> RIF(RealNumber('1.99'), RealNumber('2.025')).frac().endpoints()
(0.000000000000000, 1.00000000000000)
>>> RIF(RealNumber('1.99'), RealNumber('2.00')).frac().endpoints()
(0.000000000000000, 1.00000000000000)
>>> RIF(RealNumber('2.00'), RealNumber('2.025')).frac().endpoints()
(0.000000000000000, 0.0250000000000000)

>>> RIF(-RealNumber('2.1'),-RealNumber('0.9')).frac().endpoints()
(-1.00000000000000, -0.000000000000000)
>>> RIF(-RealNumber('0.5'),RealNumber('0.5')).frac().endpoints()
(-0.500000000000000, 0.500000000000000)

gamma()

Return the gamma function evaluated on self.

EXAMPLES:

Sage

sage: RIF(1).gamma()
1
sage: RIF(5).gamma()
24
sage: a = RIF(3,4).gamma(); a
1.?e1
sage: a.lower(), a.upper()
(2.00000000000000, 6.00000000000000)
sage: RIF(-1/2).gamma()
-3.54490770181104?
sage: gamma(-1/2).n(100) in RIF(-1/2).gamma()                               # needs sage.symbolic
True
sage: RIF1000 = RealIntervalField(1000)
sage: 0 in (RIF1000(RealField(2000)(-19/3).gamma()) - RIF1000(-19/3).gamma())
True
sage: gamma(RIF(100))
9.33262154439442?e155
sage: gamma(RIF(-10000/3))
1.31280781451?e-10297

Python

>>> from sage.all import *
>>> RIF(Integer(1)).gamma()
1
>>> RIF(Integer(5)).gamma()
24
>>> a = RIF(Integer(3),Integer(4)).gamma(); a
1.?e1
>>> a.lower(), a.upper()
(2.00000000000000, 6.00000000000000)
>>> RIF(-Integer(1)/Integer(2)).gamma()
-3.54490770181104?
>>> gamma(-Integer(1)/Integer(2)).n(Integer(100)) in RIF(-Integer(1)/Integer(2)).gamma()                               # needs sage.symbolic
True
>>> RIF1000 = RealIntervalField(Integer(1000))
>>> Integer(0) in (RIF1000(RealField(Integer(2000))(-Integer(19)/Integer(3)).gamma()) - RIF1000(-Integer(19)/Integer(3)).gamma())
True
>>> gamma(RIF(Integer(100)))
9.33262154439442?e155
>>> gamma(RIF(-Integer(10000)/Integer(3)))
1.31280781451?e-10297

Verify the result contains the local minima:

Sage

sage: 0.88560319441088 in RIF(1, 2).gamma()
True
sage: 0.88560319441088 in RIF(0.25, 4).gamma()
True
sage: 0.88560319441088 in RIF(1.4616, 1.46164).gamma()
True

sage: (-0.99).gamma()
-100.436954665809
sage: (-0.01).gamma()
-100.587197964411
sage: RIF(-0.99, -0.01).gamma().upper()
-1.60118039970055

Python

>>> from sage.all import *
>>> RealNumber('0.88560319441088') in RIF(Integer(1), Integer(2)).gamma()
True
>>> RealNumber('0.88560319441088') in RIF(RealNumber('0.25'), Integer(4)).gamma()
True
>>> RealNumber('0.88560319441088') in RIF(RealNumber('1.4616'), RealNumber('1.46164')).gamma()
True

>>> (-RealNumber('0.99')).gamma()
-100.436954665809
>>> (-RealNumber('0.01')).gamma()
-100.587197964411
>>> RIF(-RealNumber('0.99'), -RealNumber('0.01')).gamma().upper()
-1.60118039970055

Correctly detects poles:

Sage

sage: gamma(RIF(-3/2,-1/2))
[-infinity .. +infinity]

Python

>>> from sage.all import *
>>> gamma(RIF(-Integer(3)/Integer(2),-Integer(1)/Integer(2)))
[-infinity .. +infinity]

imag()

Return the imaginary part of this real interval.

(Since this is interval is real, this simply returns the zero interval.)

See also

real()

EXAMPLES:

Sage

sage: RIF(2,3).imag()
0

Python

>>> from sage.all import *
>>> RIF(Integer(2),Integer(3)).imag()
0

intersection(other)

Return the intersection of two intervals. If the intervals do not overlap, raises a ValueError.

EXAMPLES:

Sage

sage: RIF(1, 2).intersection(RIF(1.5, 3)).str(style='brackets')
'[1.5000000000000000 .. 2.0000000000000000]'
sage: RIF(1, 2).intersection(RIF(4/3, 5/3)).str(style='brackets')
'[1.3333333333333332 .. 1.6666666666666668]'
sage: RIF(1, 2).intersection(RIF(3, 4))
Traceback (most recent call last):
...
ValueError: intersection of non-overlapping intervals

Python

>>> from sage.all import *
>>> RIF(Integer(1), Integer(2)).intersection(RIF(RealNumber('1.5'), Integer(3))).str(style='brackets')
'[1.5000000000000000 .. 2.0000000000000000]'
>>> RIF(Integer(1), Integer(2)).intersection(RIF(Integer(4)/Integer(3), Integer(5)/Integer(3))).str(style='brackets')
'[1.3333333333333332 .. 1.6666666666666668]'
>>> RIF(Integer(1), Integer(2)).intersection(RIF(Integer(3), Integer(4)))
Traceback (most recent call last):
...
ValueError: intersection of non-overlapping intervals

is_NaN()

Check to see if self is Not-a-Number NaN.

EXAMPLES:

Sage

sage: a = RIF(0) / RIF(0.0,0.00); a
[.. NaN ..]
sage: a.is_NaN()
True

Python

>>> from sage.all import *
>>> a = RIF(Integer(0)) / RIF(RealNumber('0.0'),RealNumber('0.00')); a
[.. NaN ..]
>>> a.is_NaN()
True

is_exact()

Return whether this real interval is exact (i.e. contains exactly one real value).

EXAMPLES:

Sage

sage: RIF(3).is_exact()
True
sage: RIF(2*pi).is_exact()                                                  # needs sage.symbolic
False

Python

>>> from sage.all import *
>>> RIF(Integer(3)).is_exact()
True
>>> RIF(Integer(2)*pi).is_exact()                                                  # needs sage.symbolic
False

is_int()

Check to see whether this interval includes exactly one integer.

OUTPUT:

If this contains exactly one integer, it returns the tuple (True, n), where n is that integer; otherwise, this returns (False, None).

EXAMPLES:

Sage

sage: a = RIF(0.8,1.5)
sage: a.is_int()
(True, 1)
sage: a = RIF(1.1,1.5)
sage: a.is_int()
(False, None)
sage: a = RIF(1,2)
sage: a.is_int()
(False, None)
sage: a = RIF(-1.1, -0.9)
sage: a.is_int()
(True, -1)
sage: a = RIF(0.1, 1.9)
sage: a.is_int()
(True, 1)
sage: RIF(+infinity,+infinity).is_int()
(False, None)

Python

>>> from sage.all import *
>>> a = RIF(RealNumber('0.8'),RealNumber('1.5'))
>>> a.is_int()
(True, 1)
>>> a = RIF(RealNumber('1.1'),RealNumber('1.5'))
>>> a.is_int()
(False, None)
>>> a = RIF(Integer(1),Integer(2))
>>> a.is_int()
(False, None)
>>> a = RIF(-RealNumber('1.1'), -RealNumber('0.9'))
>>> a.is_int()
(True, -1)
>>> a = RIF(RealNumber('0.1'), RealNumber('1.9'))
>>> a.is_int()
(True, 1)
>>> RIF(+infinity,+infinity).is_int()
(False, None)

lexico_cmp(left, right)

Compare two intervals lexicographically.

This means that the left bounds are compared first and then the right bounds are compared if the left bounds coincide.

Return 0 if they are the same interval, -1 if the second is larger, or 1 if the first is larger.

EXAMPLES:

Sage

sage: RIF(0).lexico_cmp(RIF(1))
-1
sage: RIF(0, 1).lexico_cmp(RIF(1))
-1
sage: RIF(0, 1).lexico_cmp(RIF(1, 2))
-1
sage: RIF(0, 0.99999).lexico_cmp(RIF(1, 2))
-1
sage: RIF(1, 2).lexico_cmp(RIF(0, 1))
1
sage: RIF(1, 2).lexico_cmp(RIF(0))
1
sage: RIF(0, 1).lexico_cmp(RIF(0, 2))
-1
sage: RIF(0, 1).lexico_cmp(RIF(0, 1))
0
sage: RIF(0, 1).lexico_cmp(RIF(0, 1/2))
1

Python

>>> from sage.all import *
>>> RIF(Integer(0)).lexico_cmp(RIF(Integer(1)))
-1
>>> RIF(Integer(0), Integer(1)).lexico_cmp(RIF(Integer(1)))
-1
>>> RIF(Integer(0), Integer(1)).lexico_cmp(RIF(Integer(1), Integer(2)))
-1
>>> RIF(Integer(0), RealNumber('0.99999')).lexico_cmp(RIF(Integer(1), Integer(2)))
-1
>>> RIF(Integer(1), Integer(2)).lexico_cmp(RIF(Integer(0), Integer(1)))
1
>>> RIF(Integer(1), Integer(2)).lexico_cmp(RIF(Integer(0)))
1
>>> RIF(Integer(0), Integer(1)).lexico_cmp(RIF(Integer(0), Integer(2)))
-1
>>> RIF(Integer(0), Integer(1)).lexico_cmp(RIF(Integer(0), Integer(1)))
0
>>> RIF(Integer(0), Integer(1)).lexico_cmp(RIF(Integer(0), Integer(1)/Integer(2)))
1

log(base='e')

Return the logarithm of self to the given base.

EXAMPLES:

Sage

sage: R = RealIntervalField()
sage: r = R(2); r.log()
0.6931471805599453?
sage: r = R(-2); r.log()
0.6931471805599453? + 3.141592653589794?*I

Python

>>> from sage.all import *
>>> R = RealIntervalField()
>>> r = R(Integer(2)); r.log()
0.6931471805599453?
>>> r = R(-Integer(2)); r.log()
0.6931471805599453? + 3.141592653589794?*I

log10()

Return log to the base 10 of self.

EXAMPLES:

Sage

sage: r = RIF(16.0); r.log10()
1.204119982655925?
sage: r.log() / RIF(10).log()
1.204119982655925?

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('16.0')); r.log10()
1.204119982655925?
>>> r.log() / RIF(Integer(10)).log()
1.204119982655925?

Sage

sage: r = RIF(39.9); r.log10()
1.600972895686749?

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('39.9')); r.log10()
1.600972895686749?

Sage

sage: r = RIF(0.0)
sage: r.log10()
[-infinity .. -infinity]

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('0.0'))
>>> r.log10()
[-infinity .. -infinity]

Sage

sage: r = RIF(-1.0)
sage: r.log10()
1.364376353841841?*I

Python

>>> from sage.all import *
>>> r = RIF(-RealNumber('1.0'))
>>> r.log10()
1.364376353841841?*I

log2()

Return log to the base 2 of self.

EXAMPLES:

Sage

sage: r = RIF(16.0)
sage: r.log2()
4

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('16.0'))
>>> r.log2()
4

Sage

sage: r = RIF(31.9); r.log2()
4.995484518877507?

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('31.9')); r.log2()
4.995484518877507?

Sage

sage: r = RIF(0.0, 2.0)
sage: r.log2()
[-infinity .. 1.0000000000000000]

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('0.0'), RealNumber('2.0'))
>>> r.log2()
[-infinity .. 1.0000000000000000]

lower(rnd=None)

Return the lower bound of this interval.

INPUT:

• rnd – the rounding mode (default: towards minus infinity, see sage.rings.real_mpfr.RealField for possible values)

The rounding mode does not affect the value returned as a floating-point number, but it does control which variety of RealField the returned number is in, which affects printing and subsequent operations.

EXAMPLES:

Sage

sage: R = RealIntervalField(13)
sage: R.pi().lower().str()
'3.1411'

Python

>>> from sage.all import *
>>> R = RealIntervalField(Integer(13))
>>> R.pi().lower().str()
'3.1411'

Sage

sage: x = R(1.2,1.3); x.str(style='brackets')
'[1.1999 .. 1.3001]'
sage: x.lower()
1.19
sage: x.lower('RNDU')
1.20
sage: x.lower('RNDN')
1.20
sage: x.lower('RNDZ')
1.19
sage: x.lower('RNDA')
1.20
sage: x.lower().parent()
Real Field with 13 bits of precision and rounding RNDD
sage: x.lower('RNDU').parent()
Real Field with 13 bits of precision and rounding RNDU
sage: x.lower('RNDA').parent()
Real Field with 13 bits of precision and rounding RNDA
sage: x.lower() == x.lower('RNDU')
True

Python

>>> from sage.all import *
>>> x = R(RealNumber('1.2'),RealNumber('1.3')); x.str(style='brackets')
'[1.1999 .. 1.3001]'
>>> x.lower()
1.19
>>> x.lower('RNDU')
1.20
>>> x.lower('RNDN')
1.20
>>> x.lower('RNDZ')
1.19
>>> x.lower('RNDA')
1.20
>>> x.lower().parent()
Real Field with 13 bits of precision and rounding RNDD
>>> x.lower('RNDU').parent()
Real Field with 13 bits of precision and rounding RNDU
>>> x.lower('RNDA').parent()
Real Field with 13 bits of precision and rounding RNDA
>>> x.lower() == x.lower('RNDU')
True

magnitude()

The largest absolute value of the elements of the interval.

OUTPUT: a real number with rounding mode RNDU

EXAMPLES:

Sage

sage: RIF(-2, 1).magnitude()
2.00000000000000
sage: RIF(-1, 2).magnitude()
2.00000000000000
sage: parent(RIF(1).magnitude())
Real Field with 53 bits of precision and rounding RNDU

Python

>>> from sage.all import *
>>> RIF(-Integer(2), Integer(1)).magnitude()
2.00000000000000
>>> RIF(-Integer(1), Integer(2)).magnitude()
2.00000000000000
>>> parent(RIF(Integer(1)).magnitude())
Real Field with 53 bits of precision and rounding RNDU

max(*_others)

Return an interval containing the maximum of self and the arguments.

EXAMPLES:

Sage

sage: RIF(-1, 1).max(0).endpoints()
(0.000000000000000, 1.00000000000000)
sage: RIF(-1, 1).max(RIF(2, 3)).endpoints()
(2.00000000000000, 3.00000000000000)
sage: RIF(-1, 1).max(RIF(-100, 100)).endpoints()
(-1.00000000000000, 100.000000000000)
sage: RIF(-1, 1).max(RIF(-100, 100), RIF(5, 10)).endpoints()
(5.00000000000000, 100.000000000000)

Python

>>> from sage.all import *
>>> RIF(-Integer(1), Integer(1)).max(Integer(0)).endpoints()
(0.000000000000000, 1.00000000000000)
>>> RIF(-Integer(1), Integer(1)).max(RIF(Integer(2), Integer(3))).endpoints()
(2.00000000000000, 3.00000000000000)
>>> RIF(-Integer(1), Integer(1)).max(RIF(-Integer(100), Integer(100))).endpoints()
(-1.00000000000000, 100.000000000000)
>>> RIF(-Integer(1), Integer(1)).max(RIF(-Integer(100), Integer(100)), RIF(Integer(5), Integer(10))).endpoints()
(5.00000000000000, 100.000000000000)

Note that if the maximum is one of the given elements, that element will be returned.

Sage

sage: a = RIF(-1, 1)
sage: b = RIF(2, 3)
sage: c = RIF(3, 4)
sage: c.max(a, b) is c
True
sage: b.max(a, c) is c
True
sage: a.max(b, c) is c
True

Python

>>> from sage.all import *
>>> a = RIF(-Integer(1), Integer(1))
>>> b = RIF(Integer(2), Integer(3))
>>> c = RIF(Integer(3), Integer(4))
>>> c.max(a, b) is c
True
>>> b.max(a, c) is c
True
>>> a.max(b, c) is c
True

It might also be convenient to call the method as a function:

Sage

sage: from sage.rings.real_mpfi import RealIntervalFieldElement
sage: RealIntervalFieldElement.max(a, b, c) is c
True
sage: elements = [a, b, c]
sage: RealIntervalFieldElement.max(*elements) is c
True

Python

>>> from sage.all import *
>>> from sage.rings.real_mpfi import RealIntervalFieldElement
>>> RealIntervalFieldElement.max(a, b, c) is c
True
>>> elements = [a, b, c]
>>> RealIntervalFieldElement.max(*elements) is c
True

The generic max does not always do the right thing:

Sage

sage: max(0, RIF(-1, 1))
0
sage: max(RIF(-1, 1), RIF(-100, 100)).endpoints()
(-1.00000000000000, 1.00000000000000)

Python

>>> from sage.all import *
>>> max(Integer(0), RIF(-Integer(1), Integer(1)))
0
>>> max(RIF(-Integer(1), Integer(1)), RIF(-Integer(100), Integer(100))).endpoints()
(-1.00000000000000, 1.00000000000000)

Note that calls involving NaNs try to return a number when possible. This is consistent with IEEE-754-2008 but may be surprising.

Sage

sage: RIF('nan').max(1, 2)
2
sage: RIF(-1/3).max(RIF('nan'))
-0.3333333333333334?
sage: RIF('nan').max(RIF('nan'))
[.. NaN ..]

Python

>>> from sage.all import *
>>> RIF('nan').max(Integer(1), Integer(2))
2
>>> RIF(-Integer(1)/Integer(3)).max(RIF('nan'))
-0.3333333333333334?
>>> RIF('nan').max(RIF('nan'))
[.. NaN ..]

See also

min()

mignitude()

The smallest absolute value of the elements of the interval.

OUTPUT: a real number with rounding mode RNDD

EXAMPLES:

Sage

sage: RIF(-2, 1).mignitude()
0.000000000000000
sage: RIF(-2, -1).mignitude()
1.00000000000000
sage: RIF(3, 4).mignitude()
3.00000000000000
sage: parent(RIF(1).mignitude())
Real Field with 53 bits of precision and rounding RNDD

Python

>>> from sage.all import *
>>> RIF(-Integer(2), Integer(1)).mignitude()
0.000000000000000
>>> RIF(-Integer(2), -Integer(1)).mignitude()
1.00000000000000
>>> RIF(Integer(3), Integer(4)).mignitude()
3.00000000000000
>>> parent(RIF(Integer(1)).mignitude())
Real Field with 53 bits of precision and rounding RNDD

min(*_others)

Return an interval containing the minimum of self and the arguments.

EXAMPLES:

Sage

sage: a = RIF(-1, 1).min(0).endpoints()
sage: a[0] == -1.0 and a[1].abs() == 0.0 # in MPFI, the sign of 0.0 is not specified
True
sage: RIF(-1, 1).min(pi).endpoints()                                        # needs sage.symbolic
(-1.00000000000000, 1.00000000000000)
sage: RIF(-1, 1).min(RIF(-100, 100)).endpoints()
(-100.000000000000, 1.00000000000000)
sage: RIF(-1, 1).min(RIF(-100, 0)).endpoints()
(-100.000000000000, 0.000000000000000)
sage: RIF(-1, 1).min(RIF(-100, 2), RIF(-200, -3)).endpoints()
(-200.000000000000, -3.00000000000000)

Python

>>> from sage.all import *
>>> a = RIF(-Integer(1), Integer(1)).min(Integer(0)).endpoints()
>>> a[Integer(0)] == -RealNumber('1.0') and a[Integer(1)].abs() == RealNumber('0.0') # in MPFI, the sign of 0.0 is not specified
True
>>> RIF(-Integer(1), Integer(1)).min(pi).endpoints()                                        # needs sage.symbolic
(-1.00000000000000, 1.00000000000000)
>>> RIF(-Integer(1), Integer(1)).min(RIF(-Integer(100), Integer(100))).endpoints()
(-100.000000000000, 1.00000000000000)
>>> RIF(-Integer(1), Integer(1)).min(RIF(-Integer(100), Integer(0))).endpoints()
(-100.000000000000, 0.000000000000000)
>>> RIF(-Integer(1), Integer(1)).min(RIF(-Integer(100), Integer(2)), RIF(-Integer(200), -Integer(3))).endpoints()
(-200.000000000000, -3.00000000000000)

Note that if the minimum is one of the given elements, that element will be returned.

Sage

sage: a = RIF(-1, 1)
sage: b = RIF(2, 3)
sage: c = RIF(3, 4)
sage: c.min(a, b) is a
True
sage: b.min(a, c) is a
True
sage: a.min(b, c) is a
True

Python

>>> from sage.all import *
>>> a = RIF(-Integer(1), Integer(1))
>>> b = RIF(Integer(2), Integer(3))
>>> c = RIF(Integer(3), Integer(4))
>>> c.min(a, b) is a
True
>>> b.min(a, c) is a
True
>>> a.min(b, c) is a
True

It might also be convenient to call the method as a function:

Sage

sage: from sage.rings.real_mpfi import RealIntervalFieldElement
sage: RealIntervalFieldElement.min(a, b, c) is a
True
sage: elements = [a, b, c]
sage: RealIntervalFieldElement.min(*elements) is a
True

Python

>>> from sage.all import *
>>> from sage.rings.real_mpfi import RealIntervalFieldElement
>>> RealIntervalFieldElement.min(a, b, c) is a
True
>>> elements = [a, b, c]
>>> RealIntervalFieldElement.min(*elements) is a
True

The generic min does not always do the right thing:

Sage

sage: min(0, RIF(-1, 1))
0
sage: min(RIF(-1, 1), RIF(-100, 100)).endpoints()
(-1.00000000000000, 1.00000000000000)

Python

>>> from sage.all import *
>>> min(Integer(0), RIF(-Integer(1), Integer(1)))
0
>>> min(RIF(-Integer(1), Integer(1)), RIF(-Integer(100), Integer(100))).endpoints()
(-1.00000000000000, 1.00000000000000)

Note that calls involving NaNs try to return a number when possible. This is consistent with IEEE-754-2008 but may be surprising.

Sage

sage: RIF('nan').min(2, 1)
1
sage: RIF(-1/3).min(RIF('nan'))
-0.3333333333333334?
sage: RIF('nan').min(RIF('nan'))
[.. NaN ..]

Python

>>> from sage.all import *
>>> RIF('nan').min(Integer(2), Integer(1))
1
>>> RIF(-Integer(1)/Integer(3)).min(RIF('nan'))
-0.3333333333333334?
>>> RIF('nan').min(RIF('nan'))
[.. NaN ..]

See also

max()

multiplicative_order()

Return \(n\) such that self^n == 1.

Only \(\pm 1\) have finite multiplicative order.

EXAMPLES:

Sage

sage: RIF(1).multiplicative_order()
1
sage: RIF(-1).multiplicative_order()
2
sage: RIF(3).multiplicative_order()
+Infinity

Python

>>> from sage.all import *
>>> RIF(Integer(1)).multiplicative_order()
1
>>> RIF(-Integer(1)).multiplicative_order()
2
>>> RIF(Integer(3)).multiplicative_order()
+Infinity

overlaps(other)

Return True if self and other are intervals with at least one value in common. For intervals a and b, we have a.overlaps(b) iff not(a!=b).

EXAMPLES:

Sage

sage: RIF(0, 1).overlaps(RIF(1, 2))
True
sage: RIF(1, 2).overlaps(RIF(0, 1))
True
sage: RIF(0, 1).overlaps(RIF(2, 3))
False
sage: RIF(2, 3).overlaps(RIF(0, 1))
False
sage: RIF(0, 3).overlaps(RIF(1, 2))
True
sage: RIF(0, 2).overlaps(RIF(1, 3))
True

Python

>>> from sage.all import *
>>> RIF(Integer(0), Integer(1)).overlaps(RIF(Integer(1), Integer(2)))
True
>>> RIF(Integer(1), Integer(2)).overlaps(RIF(Integer(0), Integer(1)))
True
>>> RIF(Integer(0), Integer(1)).overlaps(RIF(Integer(2), Integer(3)))
False
>>> RIF(Integer(2), Integer(3)).overlaps(RIF(Integer(0), Integer(1)))
False
>>> RIF(Integer(0), Integer(3)).overlaps(RIF(Integer(1), Integer(2)))
True
>>> RIF(Integer(0), Integer(2)).overlaps(RIF(Integer(1), Integer(3)))
True

prec()

Return the precision of self.

EXAMPLES:

Sage

sage: RIF(2.1).precision()
53
sage: RealIntervalField(200)(2.1).precision()
200

Python

>>> from sage.all import *
>>> RIF(RealNumber('2.1')).precision()
53
>>> RealIntervalField(Integer(200))(RealNumber('2.1')).precision()
200

precision()

Return the precision of self.

EXAMPLES:

Sage

sage: RIF(2.1).precision()
53
sage: RealIntervalField(200)(2.1).precision()
200

Python

>>> from sage.all import *
>>> RIF(RealNumber('2.1')).precision()
53
>>> RealIntervalField(Integer(200))(RealNumber('2.1')).precision()
200

psi()

Return the digamma function evaluated on self.

OUTPUT: a RealIntervalFieldElement

EXAMPLES:

Sage

sage: psi_1 = RIF(1).psi()
sage: psi_1
-0.577215664901533?
sage: psi_1.overlaps(-RIF.euler_constant())
True

Python

>>> from sage.all import *
>>> psi_1 = RIF(Integer(1)).psi()
>>> psi_1
-0.577215664901533?
>>> psi_1.overlaps(-RIF.euler_constant())
True

real()

Return the real part of this real interval.

(Since this interval is real, this simply returns itself.)

See also

imag()

EXAMPLES:

Sage

sage: RIF(1.2465).real() == RIF(1.2465)
True

Python

>>> from sage.all import *
>>> RIF(RealNumber('1.2465')).real() == RIF(RealNumber('1.2465'))
True

relative_diameter()

The relative diameter of this interval (for \([a .. b]\), this is \((b-a)/((a+b)/2)\)), rounded upward, as a RealNumber.

EXAMPLES:

Sage

sage: RIF(1, pi).relative_diameter()                                        # needs sage.symbolic
1.03418797197910

Python

>>> from sage.all import *
>>> RIF(Integer(1), pi).relative_diameter()                                        # needs sage.symbolic
1.03418797197910

round()

Return the nearest integer of this interval as an interval.

See also

• unique_round() – return the round as an integer if it is unique and raises a ValueError otherwise

• floor() – truncation towards \(-\infty\)

• ceil() – truncation towards \(+\infty\)

• trunc() – truncation towards \(0\)

EXAMPLES:

Sage

sage: RIF(7.2, 7.3).round()
7
sage: RIF(-3.2, -3.1).round()
-3

Python

>>> from sage.all import *
>>> RIF(RealNumber('7.2'), RealNumber('7.3')).round()
7
>>> RIF(-RealNumber('3.2'), -RealNumber('3.1')).round()
-3

Be careful that the answer is not an integer but an interval:

Sage

sage: RIF(2.2, 2.3).round().parent()
Real Interval Field with 53 bits of precision

Python

>>> from sage.all import *
>>> RIF(RealNumber('2.2'), RealNumber('2.3')).round().parent()
Real Interval Field with 53 bits of precision

And in some cases, the lower and upper bounds of this interval do not agree:

Sage

sage: r = RIF(2.5, 3.5).round()
sage: r
4.?
sage: r.lower()
3.00000000000000
sage: r.upper()
4.00000000000000

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('2.5'), RealNumber('3.5')).round()
>>> r
4.?
>>> r.lower()
3.00000000000000
>>> r.upper()
4.00000000000000

sec()

Return the secant of this number.

EXAMPLES:

Sage

sage: RealIntervalField(100)(2).sec()
-2.40299796172238098975460040142?

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(100))(Integer(2)).sec()
-2.40299796172238098975460040142?

sech()

Return the hyperbolic secant of self.

EXAMPLES:

Sage

sage: RealIntervalField(100)(2).sech()
0.265802228834079692120862739820?

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(100))(Integer(2)).sech()
0.265802228834079692120862739820?

simplest_rational(low_open=False, high_open=False)

Return the simplest rational in this interval. Given rationals \(a / b\) and \(c / d\) (both in lowest terms), the former is simpler if \(b<d\) or if \(b = d\) and \(|a| < |c|\).

If optional parameters low_open or high_open are True, then treat this as an open interval on that end.

EXAMPLES:

Sage

sage: RealIntervalField(10)(pi).simplest_rational()                         # needs sage.symbolic
22/7
sage: RealIntervalField(20)(pi).simplest_rational()                         # needs sage.symbolic
355/113
sage: RIF(0.123, 0.567).simplest_rational()
1/2
sage: RIF(RR(1/3).nextabove(), RR(3/7)).simplest_rational()
2/5
sage: RIF(1234/567).simplest_rational()
1234/567
sage: RIF(-8765/432).simplest_rational()
-8765/432
sage: RIF(-1.234, 0.003).simplest_rational()
0
sage: RIF(RR(1/3)).simplest_rational()
6004799503160661/18014398509481984
sage: RIF(RR(1/3)).simplest_rational(high_open=True)
Traceback (most recent call last):
...
ValueError: simplest_rational() on open, empty interval
sage: RIF(1/3, 1/2).simplest_rational()
1/2
sage: RIF(1/3, 1/2).simplest_rational(high_open=True)
1/3
sage: phi = ((RealIntervalField(500)(5).sqrt() + 1)/2)
sage: phi.simplest_rational() == fibonacci(362)/fibonacci(361)
True

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(10))(pi).simplest_rational()                         # needs sage.symbolic
22/7
>>> RealIntervalField(Integer(20))(pi).simplest_rational()                         # needs sage.symbolic
355/113
>>> RIF(RealNumber('0.123'), RealNumber('0.567')).simplest_rational()
1/2
>>> RIF(RR(Integer(1)/Integer(3)).nextabove(), RR(Integer(3)/Integer(7))).simplest_rational()
2/5
>>> RIF(Integer(1234)/Integer(567)).simplest_rational()
1234/567
>>> RIF(-Integer(8765)/Integer(432)).simplest_rational()
-8765/432
>>> RIF(-RealNumber('1.234'), RealNumber('0.003')).simplest_rational()
0
>>> RIF(RR(Integer(1)/Integer(3))).simplest_rational()
6004799503160661/18014398509481984
>>> RIF(RR(Integer(1)/Integer(3))).simplest_rational(high_open=True)
Traceback (most recent call last):
...
ValueError: simplest_rational() on open, empty interval
>>> RIF(Integer(1)/Integer(3), Integer(1)/Integer(2)).simplest_rational()
1/2
>>> RIF(Integer(1)/Integer(3), Integer(1)/Integer(2)).simplest_rational(high_open=True)
1/3
>>> phi = ((RealIntervalField(Integer(500))(Integer(5)).sqrt() + Integer(1))/Integer(2))
>>> phi.simplest_rational() == fibonacci(Integer(362))/fibonacci(Integer(361))
True

sin()

Return the sine of self.

EXAMPLES:

Sage

sage: R = RealIntervalField(100)
sage: R(2).sin()
0.909297426825681695396019865912?

Python

>>> from sage.all import *
>>> R = RealIntervalField(Integer(100))
>>> R(Integer(2)).sin()
0.909297426825681695396019865912?

sinh()

Return the hyperbolic sine of self.

EXAMPLES:

Sage

sage: q = RIF.pi()/12
sage: q.sinh()
0.2648002276022707?

Python

>>> from sage.all import *
>>> q = RIF.pi()/Integer(12)
>>> q.sinh()
0.2648002276022707?

sqrt()

Return a square root of self. Raises an error if self is nonpositive.

If you use square_root() then an interval will always be returned (though it will be NaN if self is nonpositive).

EXAMPLES:

Sage

sage: r = RIF(4.0)
sage: r.sqrt()
2
sage: r.sqrt()^2 == r
True

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('4.0'))
>>> r.sqrt()
2
>>> r.sqrt()**Integer(2) == r
True

Sage

sage: r = RIF(4344)
sage: r.sqrt()
65.90902821313633?
sage: r.sqrt()^2 == r
False
sage: r in r.sqrt()^2
True
sage: r.sqrt()^2 - r
0.?e-11
sage: (r.sqrt()^2 - r).str(style='brackets')
'[-9.0949470177292824e-13 .. 1.8189894035458565e-12]'

Python

>>> from sage.all import *
>>> r = RIF(Integer(4344))
>>> r.sqrt()
65.90902821313633?
>>> r.sqrt()**Integer(2) == r
False
>>> r in r.sqrt()**Integer(2)
True
>>> r.sqrt()**Integer(2) - r
0.?e-11
>>> (r.sqrt()**Integer(2) - r).str(style='brackets')
'[-9.0949470177292824e-13 .. 1.8189894035458565e-12]'

Sage

sage: r = RIF(-2.0)
sage: r.sqrt()
Traceback (most recent call last):
...
ValueError: self (=-2) is not >= 0

Python

>>> from sage.all import *
>>> r = RIF(-RealNumber('2.0'))
>>> r.sqrt()
Traceback (most recent call last):
...
ValueError: self (=-2) is not >= 0

Sage

sage: r = RIF(-2, 2)
sage: r.sqrt()
Traceback (most recent call last):
...
ValueError: self (=0.?e1) is not >= 0

Python

>>> from sage.all import *
>>> r = RIF(-Integer(2), Integer(2))
>>> r.sqrt()
Traceback (most recent call last):
...
ValueError: self (=0.?e1) is not >= 0

square()

Return the square of self.

Note

Squaring an interval is different than multiplying it by itself, because the square can never be negative.

EXAMPLES:

Sage

sage: RIF(1, 2).square().str(style='brackets')
'[1.0000000000000000 .. 4.0000000000000000]'
sage: RIF(-1, 1).square().str(style='brackets')
'[0.0000000000000000 .. 1.0000000000000000]'
sage: (RIF(-1, 1) * RIF(-1, 1)).str(style='brackets')
'[-1.0000000000000000 .. 1.0000000000000000]'

Python

>>> from sage.all import *
>>> RIF(Integer(1), Integer(2)).square().str(style='brackets')
'[1.0000000000000000 .. 4.0000000000000000]'
>>> RIF(-Integer(1), Integer(1)).square().str(style='brackets')
'[0.0000000000000000 .. 1.0000000000000000]'
>>> (RIF(-Integer(1), Integer(1)) * RIF(-Integer(1), Integer(1))).str(style='brackets')
'[-1.0000000000000000 .. 1.0000000000000000]'

square_root()

Return a square root of self. An interval will always be returned (though it will be NaN if self is nonpositive).

EXAMPLES:

Sage

sage: r = RIF(-2.0)
sage: r.square_root()
[.. NaN ..]
sage: r.sqrt()
Traceback (most recent call last):
...
ValueError: self (=-2) is not >= 0

Python

>>> from sage.all import *
>>> r = RIF(-RealNumber('2.0'))
>>> r.square_root()
[.. NaN ..]
>>> r.sqrt()
Traceback (most recent call last):
...
ValueError: self (=-2) is not >= 0

str(base=10, style=None, no_sci=None, e=None, error_digits=None)

Return a string representation of self.

INPUT:

• base – base for output

• style – the printing style; either 'brackets' or 'question' (or None, to use the current default)

• no_sci – if True do not print using scientific notation; if False print with scientific notation; if None (the default), print how the parent prints.

• e – symbol used in scientific notation

• error_digits – the number of digits of error to print, in 'question' style

We support two different styles of printing; 'question' style and 'brackets' style. In question style (the default), we print the “known correct” part of the number, followed by a question mark:

Sage

sage: RIF(pi).str()                                                         # needs sage.symbolic
'3.141592653589794?'
sage: RIF(pi, 22/7).str()                                                   # needs sage.symbolic
'3.142?'
sage: RIF(pi, 22/7).str(style='question')                                   # needs sage.symbolic
'3.142?'

Python

>>> from sage.all import *
>>> RIF(pi).str()                                                         # needs sage.symbolic
'3.141592653589794?'
>>> RIF(pi, Integer(22)/Integer(7)).str()                                                   # needs sage.symbolic
'3.142?'
>>> RIF(pi, Integer(22)/Integer(7)).str(style='question')                                   # needs sage.symbolic
'3.142?'

However, if the interval is precisely equal to some integer that’s not too large, we just return that integer:

Sage

sage: RIF(-42).str()
'-42'
sage: RIF(0).str()
'0'
sage: RIF(12^5).str(base=3)
'110122100000'

Python

>>> from sage.all import *
>>> RIF(-Integer(42)).str()
'-42'
>>> RIF(Integer(0)).str()
'0'
>>> RIF(Integer(12)**Integer(5)).str(base=Integer(3))
'110122100000'

Very large integers, however, revert to the normal question-style printing:

Sage

sage: RIF(3^7).str()
'2187'
sage: RIF(3^7 * 2^256).str()
'2.5323729916201052?e80'

Python

>>> from sage.all import *
>>> RIF(Integer(3)**Integer(7)).str()
'2187'
>>> RIF(Integer(3)**Integer(7) * Integer(2)**Integer(256)).str()
'2.5323729916201052?e80'

In brackets style, we print the lower and upper bounds of the interval within brackets:

Sage

sage: RIF(237/16).str(style='brackets')
'[14.812500000000000 .. 14.812500000000000]'

Python

>>> from sage.all import *
>>> RIF(Integer(237)/Integer(16)).str(style='brackets')
'[14.812500000000000 .. 14.812500000000000]'

Note that the lower bound is rounded down, and the upper bound is rounded up. So even if the lower and upper bounds are equal, they may print differently. (This is done so that the printed representation of the interval contains all the numbers in the internal binary interval.)

For instance, we find the best 10-bit floating point representation of 1/3:

Sage

sage: RR10 = RealField(10)
sage: RR(RR10(1/3))
0.333496093750000

Python

>>> from sage.all import *
>>> RR10 = RealField(Integer(10))
>>> RR(RR10(Integer(1)/Integer(3)))
0.333496093750000

And we see that the point interval containing only this floating-point number prints as a wider decimal interval, that does contain the number:

Sage

sage: RIF10 = RealIntervalField(10)
sage: RIF10(RR10(1/3)).str(style='brackets')
'[0.33349 .. 0.33350]'

Python

>>> from sage.all import *
>>> RIF10 = RealIntervalField(Integer(10))
>>> RIF10(RR10(Integer(1)/Integer(3))).str(style='brackets')
'[0.33349 .. 0.33350]'

We always use brackets style for NaN and infinities:

Sage

sage: RIF(pi, infinity)                                                     # needs sage.symbolic
[3.1415926535897931 .. +infinity]
sage: RIF(NaN)                                                              # needs sage.symbolic
[.. NaN ..]

Python

>>> from sage.all import *
>>> RIF(pi, infinity)                                                     # needs sage.symbolic
[3.1415926535897931 .. +infinity]
>>> RIF(NaN)                                                              # needs sage.symbolic
[.. NaN ..]

Let’s take a closer, formal look at the question style. In its full generality, a number printed in the question style looks like:

MANTISSA ?ERROR eEXPONENT

(without the spaces). The “eEXPONENT” part is optional; if it is missing, then the exponent is 0. (If the base is greater than 10, then the exponent separator is “@” instead of “e”.)

The “ERROR” is optional; if it is missing, then the error is 1.

The mantissa is printed in base \(b\), and always contains a decimal point (also known as a radix point, in bases other than 10). (The error and exponent are always printed in base 10.)

We define the “precision” of a floating-point printed representation to be the positional value of the last digit of the mantissa. For instance, in 2.7?e5, the precision is \(10^4\); in 8.?, the precision is \(10^0\); and in 9.35? the precision is \(10^{-2}\). This precision will always be \(10^k\) for some \(k\) (or, for an arbitrary base \(b\), \(b^k\)).

Then the interval is contained in the interval:

\[\text{mantissa} \cdot b^{\text{exponent}} - \text{error} \cdot b^k .. \text{mantissa} \cdot b^{\text{exponent}} + \text{error} \cdot b^k\]

To control the printing, we can specify a maximum number of error digits. The default is 0, which means that we do not print an error at all (so that the error is always the default, 1).

Now, consider the precisions needed to represent the endpoints (this is the precision that would be produced by v.lower().str(no_sci=False)). Our result is no more precise than the less precise endpoint, and is sufficiently imprecise that the error can be represented with the given number of decimal digits. Our result is the most precise possible result, given these restrictions. When there are two possible results of equal precision and with the same error width, then we pick the one which is farther from zero. (For instance, RIF(0, 123) with two error digits could print as 61.?62 or 62.?62. We prefer the latter because it makes it clear that the interval is known not to be negative.)

EXAMPLES:

Sage

sage: a = RIF(59/27); a
2.185185185185186?
sage: a.str()
'2.185185185185186?'
sage: a.str(style='brackets')
'[2.1851851851851851 .. 2.1851851851851856]'
sage: a.str(16)
'2.2f684bda12f69?'
sage: a.str(no_sci=False)
'2.185185185185186?e0'
sage: pi_appr = RIF(pi, 22/7)
sage: pi_appr.str(style='brackets')
'[3.1415926535897931 .. 3.1428571428571433]'
sage: pi_appr.str()
'3.142?'
sage: pi_appr.str(error_digits=1)
'3.1422?7'
sage: pi_appr.str(error_digits=2)
'3.14223?64'
sage: pi_appr.str(base=36)
'3.6?'
sage: RIF(NaN)                                                              # needs sage.symbolic
[.. NaN ..]
sage: RIF(pi, infinity)                                                     # needs sage.symbolic
[3.1415926535897931 .. +infinity]
sage: RIF(-infinity, pi)                                                    # needs sage.symbolic
[-infinity .. 3.1415926535897936]
sage: RealIntervalField(210)(3).sqrt()
1.732050807568877293527446341505872366942805253810380628055806980?
sage: RealIntervalField(210)(RIF(3).sqrt())
1.732050807568878?
sage: RIF(3).sqrt()
1.732050807568878?
sage: RIF(0, 3^-150)                                                        # needs sage.symbolic
1.?e-71

Python

>>> from sage.all import *
>>> a = RIF(Integer(59)/Integer(27)); a
2.185185185185186?
>>> a.str()
'2.185185185185186?'
>>> a.str(style='brackets')
'[2.1851851851851851 .. 2.1851851851851856]'
>>> a.str(Integer(16))
'2.2f684bda12f69?'
>>> a.str(no_sci=False)
'2.185185185185186?e0'
>>> pi_appr = RIF(pi, Integer(22)/Integer(7))
>>> pi_appr.str(style='brackets')
'[3.1415926535897931 .. 3.1428571428571433]'
>>> pi_appr.str()
'3.142?'
>>> pi_appr.str(error_digits=Integer(1))
'3.1422?7'
>>> pi_appr.str(error_digits=Integer(2))
'3.14223?64'
>>> pi_appr.str(base=Integer(36))
'3.6?'
>>> RIF(NaN)                                                              # needs sage.symbolic
[.. NaN ..]
>>> RIF(pi, infinity)                                                     # needs sage.symbolic
[3.1415926535897931 .. +infinity]
>>> RIF(-infinity, pi)                                                    # needs sage.symbolic
[-infinity .. 3.1415926535897936]
>>> RealIntervalField(Integer(210))(Integer(3)).sqrt()
1.732050807568877293527446341505872366942805253810380628055806980?
>>> RealIntervalField(Integer(210))(RIF(Integer(3)).sqrt())
1.732050807568878?
>>> RIF(Integer(3)).sqrt()
1.732050807568878?
>>> RIF(Integer(0), Integer(3)**-Integer(150))                                                        # needs sage.symbolic
1.?e-71

tan()

Return the tangent of self.

EXAMPLES:

Sage

sage: q = RIF.pi()/3
sage: q.tan()
1.732050807568877?
sage: q = RIF.pi()/6
sage: q.tan()
0.577350269189626?

Python

>>> from sage.all import *
>>> q = RIF.pi()/Integer(3)
>>> q.tan()
1.732050807568877?
>>> q = RIF.pi()/Integer(6)
>>> q.tan()
0.577350269189626?

tanh()

Return the hyperbolic tangent of self.

EXAMPLES:

Sage

sage: q = RIF.pi()/11
sage: q.tanh()
0.2780794292958503?

Python

>>> from sage.all import *
>>> q = RIF.pi()/Integer(11)
>>> q.tanh()
0.2780794292958503?

trunc()

Return the truncation of this interval as an interval.

The truncation of \(x\) is the floor of \(x\) if \(x\) is nonnegative or the ceil of \(x\) if \(x\) is negative.

See also

• unique_trunc() – return the trunc as an integer if it is unique and raises a ValueError otherwise

• floor() – truncation towards \(-\infty\)

• ceil() – truncation towards \(+\infty\)

• round() – rounding

EXAMPLES:

Sage

sage: RIF(2.3, 2.7).trunc()
2
sage: parent(_)
Real Interval Field with 53 bits of precision

sage: RIF(-0.9, 0.9).trunc()
0
sage: RIF(-7.5, -7.3).trunc()
-7

Python

>>> from sage.all import *
>>> RIF(RealNumber('2.3'), RealNumber('2.7')).trunc()
2
>>> parent(_)
Real Interval Field with 53 bits of precision

>>> RIF(-RealNumber('0.9'), RealNumber('0.9')).trunc()
0
>>> RIF(-RealNumber('7.5'), -RealNumber('7.3')).trunc()
-7

In the above example, the obtained interval contains only one element. But on the following it is not the case anymore:

Sage

sage: r = RIF(2.99, 3.01).trunc()
sage: r.upper()
3.00000000000000
sage: r.lower()
2.00000000000000

Python

>>> from sage.all import *
>>> r = RIF(RealNumber('2.99'), RealNumber('3.01')).trunc()
>>> r.upper()
3.00000000000000
>>> r.lower()
2.00000000000000

union(other)

Return the union of two intervals, or of an interval and a real number (more precisely, the convex hull).

EXAMPLES:

Sage

sage: RIF(1, 2).union(RIF(pi, 22/7)).str(style='brackets')
'[1.0000000000000000 .. 3.1428571428571433]'
sage: RIF(1, 2).union(pi).str(style='brackets')
'[1.0000000000000000 .. 3.1415926535897936]'
sage: RIF(1).union(RIF(0, 2)).str(style='brackets')
'[0.0000000000000000 .. 2.0000000000000000]'
sage: RIF(1).union(RIF(-1)).str(style='brackets')
'[-1.0000000000000000 .. 1.0000000000000000]'

Python

>>> from sage.all import *
>>> RIF(Integer(1), Integer(2)).union(RIF(pi, Integer(22)/Integer(7))).str(style='brackets')
'[1.0000000000000000 .. 3.1428571428571433]'
>>> RIF(Integer(1), Integer(2)).union(pi).str(style='brackets')
'[1.0000000000000000 .. 3.1415926535897936]'
>>> RIF(Integer(1)).union(RIF(Integer(0), Integer(2))).str(style='brackets')
'[0.0000000000000000 .. 2.0000000000000000]'
>>> RIF(Integer(1)).union(RIF(-Integer(1))).str(style='brackets')
'[-1.0000000000000000 .. 1.0000000000000000]'

unique_ceil()

Return the unique ceiling of this interval, if it is well defined, otherwise raise a ValueError.

OUTPUT: integer

See also

ceil() – return the ceil as an interval (and never raise error)

EXAMPLES:

Sage

sage: RIF(pi).unique_ceil()                                                 # needs sage.symbolic
4
sage: RIF(100*pi).unique_ceil()                                             # needs sage.symbolic
315
sage: RIF(100, 200).unique_ceil()
Traceback (most recent call last):
...
ValueError: interval does not have a unique ceil

Python

>>> from sage.all import *
>>> RIF(pi).unique_ceil()                                                 # needs sage.symbolic
4
>>> RIF(Integer(100)*pi).unique_ceil()                                             # needs sage.symbolic
315
>>> RIF(Integer(100), Integer(200)).unique_ceil()
Traceback (most recent call last):
...
ValueError: interval does not have a unique ceil

unique_floor()

Return the unique floor of this interval, if it is well defined, otherwise raise a ValueError.

OUTPUT: integer

See also

floor() – return the floor as an interval (and never raise error)

EXAMPLES:

Sage

sage: RIF(pi).unique_floor()                                                # needs sage.symbolic
3
sage: RIF(100*pi).unique_floor()                                            # needs sage.symbolic
314
sage: RIF(100, 200).unique_floor()
Traceback (most recent call last):
...
ValueError: interval does not have a unique floor

Python

>>> from sage.all import *
>>> RIF(pi).unique_floor()                                                # needs sage.symbolic
3
>>> RIF(Integer(100)*pi).unique_floor()                                            # needs sage.symbolic
314
>>> RIF(Integer(100), Integer(200)).unique_floor()
Traceback (most recent call last):
...
ValueError: interval does not have a unique floor

unique_integer()

Return the unique integer in this interval, if there is exactly one, otherwise raise a ValueError.

EXAMPLES:

Sage

sage: RIF(pi).unique_integer()                                              # needs sage.symbolic
Traceback (most recent call last):
...
ValueError: interval contains no integer
sage: RIF(pi, pi+1).unique_integer()                                        # needs sage.symbolic
4
sage: RIF(pi, pi+2).unique_integer()                                        # needs sage.symbolic
Traceback (most recent call last):
...
ValueError: interval contains more than one integer
sage: RIF(100).unique_integer()
100

Python

>>> from sage.all import *
>>> RIF(pi).unique_integer()                                              # needs sage.symbolic
Traceback (most recent call last):
...
ValueError: interval contains no integer
>>> RIF(pi, pi+Integer(1)).unique_integer()                                        # needs sage.symbolic
4
>>> RIF(pi, pi+Integer(2)).unique_integer()                                        # needs sage.symbolic
Traceback (most recent call last):
...
ValueError: interval contains more than one integer
>>> RIF(Integer(100)).unique_integer()
100

unique_round()

Return the unique round (nearest integer) of this interval, if it is well defined, otherwise raise a ValueError.

OUTPUT: integer

See also

round() – return the round as an interval (and never raise error)

EXAMPLES:

Sage

sage: RIF(pi).unique_round()                                                # needs sage.symbolic
3
sage: RIF(1000*pi).unique_round()                                           # needs sage.symbolic
3142
sage: RIF(100, 200).unique_round()
Traceback (most recent call last):
...
ValueError: interval does not have a unique round (nearest integer)
sage: RIF(1.2, 1.7).unique_round()
Traceback (most recent call last):
...
ValueError: interval does not have a unique round (nearest integer)
sage: RIF(0.7, 1.2).unique_round()
1
sage: RIF(-pi).unique_round()                                               # needs sage.symbolic
-3
sage: (RIF(4.5).unique_round(), RIF(-4.5).unique_round())
(5, -5)

Python

>>> from sage.all import *
>>> RIF(pi).unique_round()                                                # needs sage.symbolic
3
>>> RIF(Integer(1000)*pi).unique_round()                                           # needs sage.symbolic
3142
>>> RIF(Integer(100), Integer(200)).unique_round()
Traceback (most recent call last):
...
ValueError: interval does not have a unique round (nearest integer)
>>> RIF(RealNumber('1.2'), RealNumber('1.7')).unique_round()
Traceback (most recent call last):
...
ValueError: interval does not have a unique round (nearest integer)
>>> RIF(RealNumber('0.7'), RealNumber('1.2')).unique_round()
1
>>> RIF(-pi).unique_round()                                               # needs sage.symbolic
-3
>>> (RIF(RealNumber('4.5')).unique_round(), RIF(-RealNumber('4.5')).unique_round())
(5, -5)

unique_sign()

Return the sign of this element if it is well defined.

This method returns \(+1\) if all elements in this interval are positive, \(-1\) if all of them are negative and \(0\) if it contains only zero. Otherwise it raises a ValueError.

EXAMPLES:

Sage

sage: RIF(1.2,5.7).unique_sign()
1
sage: RIF(-3,-2).unique_sign()
-1
sage: RIF(0).unique_sign()
0
sage: RIF(0,1).unique_sign()
Traceback (most recent call last):
...
ValueError: interval does not have a unique sign
sage: RIF(-1,0).unique_sign()
Traceback (most recent call last):
...
ValueError: interval does not have a unique sign
sage: RIF(-0.1, 0.1).unique_sign()
Traceback (most recent call last):
...
ValueError: interval does not have a unique sign

Python

>>> from sage.all import *
>>> RIF(RealNumber('1.2'),RealNumber('5.7')).unique_sign()
1
>>> RIF(-Integer(3),-Integer(2)).unique_sign()
-1
>>> RIF(Integer(0)).unique_sign()
0
>>> RIF(Integer(0),Integer(1)).unique_sign()
Traceback (most recent call last):
...
ValueError: interval does not have a unique sign
>>> RIF(-Integer(1),Integer(0)).unique_sign()
Traceback (most recent call last):
...
ValueError: interval does not have a unique sign
>>> RIF(-RealNumber('0.1'), RealNumber('0.1')).unique_sign()
Traceback (most recent call last):
...
ValueError: interval does not have a unique sign

unique_trunc()

Return the nearest integer toward zero if it is unique, otherwise raise a ValueError.

See also

trunc() – return the truncation as an interval (and never raise error)

EXAMPLES:

Sage

sage: RIF(1.3,1.4).unique_trunc()
1
sage: RIF(-3.3, -3.2).unique_trunc()
-3
sage: RIF(2.9,3.2).unique_trunc()
Traceback (most recent call last):
...
ValueError: interval does not have a unique trunc (nearest integer toward zero)
sage: RIF(-3.1,-2.9).unique_trunc()
Traceback (most recent call last):
...
ValueError: interval does not have a unique trunc (nearest integer toward zero)

Python

>>> from sage.all import *
>>> RIF(RealNumber('1.3'),RealNumber('1.4')).unique_trunc()
1
>>> RIF(-RealNumber('3.3'), -RealNumber('3.2')).unique_trunc()
-3
>>> RIF(RealNumber('2.9'),RealNumber('3.2')).unique_trunc()
Traceback (most recent call last):
...
ValueError: interval does not have a unique trunc (nearest integer toward zero)
>>> RIF(-RealNumber('3.1'),-RealNumber('2.9')).unique_trunc()
Traceback (most recent call last):
...
ValueError: interval does not have a unique trunc (nearest integer toward zero)

upper(rnd=None)

Return the upper bound of self.

INPUT:

• rnd – the rounding mode (default: towards plus infinity, see sage.rings.real_mpfr.RealField for possible values)

The rounding mode does not affect the value returned as a floating-point number, but it does control which variety of RealField the returned number is in, which affects printing and subsequent operations.

EXAMPLES:

Sage

sage: R = RealIntervalField(13)
sage: R.pi().upper().str()
'3.1417'

Python

>>> from sage.all import *
>>> R = RealIntervalField(Integer(13))
>>> R.pi().upper().str()
'3.1417'

Sage

sage: R = RealIntervalField(13)
sage: x = R(1.2,1.3); x.str(style='brackets')
'[1.1999 .. 1.3001]'
sage: x.upper()
1.31
sage: x.upper('RNDU')
1.31
sage: x.upper('RNDN')
1.30
sage: x.upper('RNDD')
1.30
sage: x.upper('RNDZ')
1.30
sage: x.upper('RNDA')
1.31
sage: x.upper().parent()
Real Field with 13 bits of precision and rounding RNDU
sage: x.upper('RNDD').parent()
Real Field with 13 bits of precision and rounding RNDD
sage: x.upper() == x.upper('RNDD')
True

Python

>>> from sage.all import *
>>> R = RealIntervalField(Integer(13))
>>> x = R(RealNumber('1.2'),RealNumber('1.3')); x.str(style='brackets')
'[1.1999 .. 1.3001]'
>>> x.upper()
1.31
>>> x.upper('RNDU')
1.31
>>> x.upper('RNDN')
1.30
>>> x.upper('RNDD')
1.30
>>> x.upper('RNDZ')
1.30
>>> x.upper('RNDA')
1.31
>>> x.upper().parent()
Real Field with 13 bits of precision and rounding RNDU
>>> x.upper('RNDD').parent()
Real Field with 13 bits of precision and rounding RNDD
>>> x.upper() == x.upper('RNDD')
True

zeta(a=None)

Return the image of this interval by the Hurwitz zeta function.

For a = 1 (or a = None), this computes the Riemann zeta function.

EXAMPLES:

Sage

sage: zeta(RIF(3))
1.202056903159594?
sage: _.parent()
Real Interval Field with 53 bits of precision
sage: RIF(3).zeta(1/2)
8.41439832211716?

Python

>>> from sage.all import *
>>> zeta(RIF(Integer(3)))
1.202056903159594?
>>> _.parent()
Real Interval Field with 53 bits of precision
>>> RIF(Integer(3)).zeta(Integer(1)/Integer(2))
8.41439832211716?

class sage.rings.real_mpfi.RealIntervalField_class

Bases: RealIntervalField

Class of the real interval field.

INPUT:

• prec – integer (default: 53); precision prec is the number of bits used to represent the mantissa of a floating-point number. The precision can be any integer between mpfr_prec_min() and mpfr_prec_max(). In the current implementation, mpfr_prec_min() is equal to 2.

• sci_not – boolean (default: False); whether or not to display using scientific notation

EXAMPLES:

Sage

sage: RealIntervalField(10)
Real Interval Field with 10 bits of precision
sage: RealIntervalField()
Real Interval Field with 53 bits of precision
sage: RealIntervalField(100000)
Real Interval Field with 100000 bits of precision

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(10))
Real Interval Field with 10 bits of precision
>>> RealIntervalField()
Real Interval Field with 53 bits of precision
>>> RealIntervalField(Integer(100000))
Real Interval Field with 100000 bits of precision

Note

The default precision is 53, since according to the GMP manual: ‘mpfr should be able to exactly reproduce all computations with double-precision machine floating-point numbers (double type in C), except the default exponent range is much wider and subnormal numbers are not implemented.’

EXAMPLES:

Creation of elements.

First with default precision. First we coerce elements of various types, then we coerce intervals:

Sage

sage: RIF = RealIntervalField(); RIF
Real Interval Field with 53 bits of precision
sage: RIF(3)
3
sage: RIF(RIF(3))
3
sage: RIF(pi)                                                                   # needs sage.symbolic
3.141592653589794?
sage: RIF(RealField(53)('1.5'))
1.5000000000000000?
sage: RIF(-2/19)
-0.1052631578947369?
sage: RIF(-3939)
-3939
sage: RIF(-3939r)
-3939
sage: RIF('1.5')
1.5000000000000000?
sage: R200 = RealField(200)
sage: RIF(R200.pi())
3.141592653589794?
sage: RIF(10^100)
1.000000000000000?e100

Python

>>> from sage.all import *
>>> RIF = RealIntervalField(); RIF
Real Interval Field with 53 bits of precision
>>> RIF(Integer(3))
3
>>> RIF(RIF(Integer(3)))
3
>>> RIF(pi)                                                                   # needs sage.symbolic
3.141592653589794?
>>> RIF(RealField(Integer(53))('1.5'))
1.5000000000000000?
>>> RIF(-Integer(2)/Integer(19))
-0.1052631578947369?
>>> RIF(-Integer(3939))
-3939
>>> RIF(-3939)
-3939
>>> RIF('1.5')
1.5000000000000000?
>>> R200 = RealField(Integer(200))
>>> RIF(R200.pi())
3.141592653589794?
>>> RIF(Integer(10)**Integer(100))
1.000000000000000?e100

The base must be explicitly specified as a named parameter:

Sage

sage: RIF('101101', base=2)
45
sage: RIF('+infinity')
[+infinity .. +infinity]
sage: RIF('[1..3]').str(style='brackets')
'[1.0000000000000000 .. 3.0000000000000000]'

Python

>>> from sage.all import *
>>> RIF('101101', base=Integer(2))
45
>>> RIF('+infinity')
[+infinity .. +infinity]
>>> RIF('[1..3]').str(style='brackets')
'[1.0000000000000000 .. 3.0000000000000000]'

All string-like types are accepted:

Sage

sage: RIF(b"100", u"100")
100

Python

>>> from sage.all import *
>>> RIF(b"100", u"100")
100

Next we coerce some 2-tuples, which define intervals:

Sage

sage: RIF((-1.5, -1.3))
-1.4?
sage: RIF((RDF('-1.5'), RDF('-1.3')))
-1.4?
sage: RIF((1/3,2/3)).str(style='brackets')
'[0.33333333333333331 .. 0.66666666666666675]'

Python

>>> from sage.all import *
>>> RIF((-RealNumber('1.5'), -RealNumber('1.3')))
-1.4?
>>> RIF((RDF('-1.5'), RDF('-1.3')))
-1.4?
>>> RIF((Integer(1)/Integer(3),Integer(2)/Integer(3))).str(style='brackets')
'[0.33333333333333331 .. 0.66666666666666675]'

The extra parentheses aren’t needed:

Sage

sage: RIF(1/3,2/3).str(style='brackets')
'[0.33333333333333331 .. 0.66666666666666675]'
sage: RIF((1,2)).str(style='brackets')
'[1.0000000000000000 .. 2.0000000000000000]'
sage: RIF((1r,2r)).str(style='brackets')
'[1.0000000000000000 .. 2.0000000000000000]'
sage: RIF((pi, e)).str(style='brackets')
'[2.7182818284590450 .. 3.1415926535897936]'

Python

>>> from sage.all import *
>>> RIF(Integer(1)/Integer(3),Integer(2)/Integer(3)).str(style='brackets')
'[0.33333333333333331 .. 0.66666666666666675]'
>>> RIF((Integer(1),Integer(2))).str(style='brackets')
'[1.0000000000000000 .. 2.0000000000000000]'
>>> RIF((1,2)).str(style='brackets')
'[1.0000000000000000 .. 2.0000000000000000]'
>>> RIF((pi, e)).str(style='brackets')
'[2.7182818284590450 .. 3.1415926535897936]'

Values which can be represented as an exact floating-point number (of the precision of this RealIntervalField) result in a precise interval, where the lower bound is equal to the upper bound (even if they print differently). Other values typically result in an interval where the lower and upper bounds are adjacent floating-point numbers.

Sage

sage: def check(x):
....:     return (x, x.lower() == x.upper())
sage: check(RIF(pi))                                                            # needs sage.symbolic
(3.141592653589794?, False)
sage: check(RIF(RR(pi)))                                                        # needs sage.symbolic
(3.1415926535897932?, True)
sage: check(RIF(1.5))
(1.5000000000000000?, True)
sage: check(RIF('1.5'))
(1.5000000000000000?, True)
sage: check(RIF(0.1))
(0.10000000000000001?, True)
sage: check(RIF(1/10))
(0.10000000000000000?, False)
sage: check(RIF('0.1'))
(0.10000000000000000?, False)

Python

>>> from sage.all import *
>>> def check(x):
...     return (x, x.lower() == x.upper())
>>> check(RIF(pi))                                                            # needs sage.symbolic
(3.141592653589794?, False)
>>> check(RIF(RR(pi)))                                                        # needs sage.symbolic
(3.1415926535897932?, True)
>>> check(RIF(RealNumber('1.5')))
(1.5000000000000000?, True)
>>> check(RIF('1.5'))
(1.5000000000000000?, True)
>>> check(RIF(RealNumber('0.1')))
(0.10000000000000001?, True)
>>> check(RIF(Integer(1)/Integer(10)))
(0.10000000000000000?, False)
>>> check(RIF('0.1'))
(0.10000000000000000?, False)

Similarly, when specifying both ends of an interval, the lower end is rounded down and the upper end is rounded up:

Sage

sage: outward = RIF(1/10, 7/10); outward.str(style='brackets')
'[0.099999999999999991 .. 0.70000000000000007]'
sage: nearest = RIF(RR(1/10), RR(7/10)); nearest.str(style='brackets')
'[0.10000000000000000 .. 0.69999999999999996]'
sage: nearest.lower() - outward.lower()
1.38777878078144e-17
sage: outward.upper() - nearest.upper()
1.11022302462516e-16

Python

>>> from sage.all import *
>>> outward = RIF(Integer(1)/Integer(10), Integer(7)/Integer(10)); outward.str(style='brackets')
'[0.099999999999999991 .. 0.70000000000000007]'
>>> nearest = RIF(RR(Integer(1)/Integer(10)), RR(Integer(7)/Integer(10))); nearest.str(style='brackets')
'[0.10000000000000000 .. 0.69999999999999996]'
>>> nearest.lower() - outward.lower()
1.38777878078144e-17
>>> outward.upper() - nearest.upper()
1.11022302462516e-16

Some examples with a real interval field of higher precision:

Sage

sage: R = RealIntervalField(100)
sage: R(3)
3
sage: R(R(3))
3
sage: R(pi)                                                                     # needs sage.symbolic
3.14159265358979323846264338328?
sage: R(-2/19)
-0.1052631578947368421052631578948?
sage: R(e,pi).str(style='brackets')                                             # needs sage.symbolic
'[2.7182818284590452353602874713512 .. 3.1415926535897932384626433832825]'

Python

>>> from sage.all import *
>>> R = RealIntervalField(Integer(100))
>>> R(Integer(3))
3
>>> R(R(Integer(3)))
3
>>> R(pi)                                                                     # needs sage.symbolic
3.14159265358979323846264338328?
>>> R(-Integer(2)/Integer(19))
-0.1052631578947368421052631578948?
>>> R(e,pi).str(style='brackets')                                             # needs sage.symbolic
'[2.7182818284590452353602874713512 .. 3.1415926535897932384626433832825]'

See also

• sage.rings.real_mpfi

• sage.rings.complex_interval_field

• sage.rings.real_arb.RealBallField (alternative implementation of real intervals, with more features)

Element

alias of RealIntervalFieldElement

algebraic_closure()

Return the algebraic closure of this interval field, i.e., the complex interval field with the same precision.

EXAMPLES:

Sage

sage: RIF.algebraic_closure()
Complex Interval Field with 53 bits of precision
sage: RIF.algebraic_closure() is CIF
True
sage: RealIntervalField(100).algebraic_closure()
Complex Interval Field with 100 bits of precision

Python

>>> from sage.all import *
>>> RIF.algebraic_closure()
Complex Interval Field with 53 bits of precision
>>> RIF.algebraic_closure() is CIF
True
>>> RealIntervalField(Integer(100)).algebraic_closure()
Complex Interval Field with 100 bits of precision

characteristic()

Return 0, since the field of real numbers has characteristic 0.

EXAMPLES:

Sage

sage: RealIntervalField(10).characteristic()
0

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(10)).characteristic()
0

complex_field()

Return complex field of the same precision.

EXAMPLES:

Sage

sage: RIF.complex_field()
Complex Interval Field with 53 bits of precision

Python

>>> from sage.all import *
>>> RIF.complex_field()
Complex Interval Field with 53 bits of precision

construction()

Return the functorial construction of self, namely, completion of the rational numbers with respect to the prime at \(\infty\), and the note that this is an interval field.

Also preserves other information that makes this field unique (e.g. precision, print mode).

EXAMPLES:

Sage

sage: R = RealIntervalField(123)
sage: c, S = R.construction(); S
Rational Field
sage: R == c(S)
True

Python

>>> from sage.all import *
>>> R = RealIntervalField(Integer(123))
>>> c, S = R.construction(); S
Rational Field
>>> R == c(S)
True

euler_constant()

Return Euler’s gamma constant to the precision of this field.

EXAMPLES:

Sage

sage: RealIntervalField(100).euler_constant()
0.577215664901532860606512090083?

Python

>>> from sage.all import *
>>> RealIntervalField(Integer(100)).euler_constant()
0.577215664901532860606512090083?

gen(i=0)

Return the i-th generator of self.

EXAMPLES:

Sage

sage: RIF.gen(0)
1
sage: RIF.gen(1)
Traceback (most recent call last):
...
IndexError: self has only one generator

Python

>>> from sage.all import *
>>> RIF.gen(Integer(0))
1
>>> RIF.gen(Integer(1))
Traceback (most recent call last):
...
IndexError: self has only one generator

gens()

Return a list of generators.

EXAMPLES:

Sage

sage: RIF.gens()
[1]

Python

>>> from sage.all import *
>>> RIF.gens()
[1]

is_exact()

Return whether or not this field is exact, which is always False.

EXAMPLES:

Sage

sage: RIF.is_exact()
False

Python

>>> from sage.all import *
>>> RIF.is_exact()
False

log2()

Return \(\log(2)\) to the precision of this field.

EXAMPLES:

Sage

sage: R=RealIntervalField(100)
sage: R.log2()
0.693147180559945309417232121458?
sage: R(2).log()
0.693147180559945309417232121458?

Python

>>> from sage.all import *
>>> R=RealIntervalField(Integer(100))
>>> R.log2()
0.693147180559945309417232121458?
>>> R(Integer(2)).log()
0.693147180559945309417232121458?

lower_field()

Return the RealField_class with rounding mode 'RNDD' (rounding towards minus infinity).

EXAMPLES:

Sage

sage: RIF.lower_field()
Real Field with 53 bits of precision and rounding RNDD
sage: RealIntervalField(200).lower_field()
Real Field with 200 bits of precision and rounding RNDD

Python

>>> from sage.all import *
>>> RIF.lower_field()
Real Field with 53 bits of precision and rounding RNDD
>>> RealIntervalField(Integer(200)).lower_field()
Real Field with 200 bits of precision and rounding RNDD

middle_field()

Return the RealField_class with rounding mode 'RNDN' (rounding towards nearest).

EXAMPLES:

Sage

sage: RIF.middle_field()
Real Field with 53 bits of precision
sage: RealIntervalField(200).middle_field()
Real Field with 200 bits of precision

Python

>>> from sage.all import *
>>> RIF.middle_field()
Real Field with 53 bits of precision
>>> RealIntervalField(Integer(200)).middle_field()
Real Field with 200 bits of precision

name()

Return the name of self.

EXAMPLES:

Sage

sage: RIF.name()
'IntervalRealIntervalField53'
sage: RealIntervalField(200).name()
'IntervalRealIntervalField200'

Python

>>> from sage.all import *
>>> RIF.name()
'IntervalRealIntervalField53'
>>> RealIntervalField(Integer(200)).name()
'IntervalRealIntervalField200'

ngens()

Return the number of generators of self, which is 1.

EXAMPLES:

Sage

sage: RIF.ngens()
1

Python

>>> from sage.all import *
>>> RIF.ngens()
1

pi()

Return \(\pi\) to the precision of this field.

EXAMPLES:

Sage

sage: R = RealIntervalField(100)
sage: R.pi()
3.14159265358979323846264338328?
sage: R.pi().sqrt()/2
0.88622692545275801364908374167?
sage: R = RealIntervalField(150)
sage: R.pi().sqrt()/2
0.886226925452758013649083741670572591398774728?

Python

>>> from sage.all import *
>>> R = RealIntervalField(Integer(100))
>>> R.pi()
3.14159265358979323846264338328?
>>> R.pi().sqrt()/Integer(2)
0.88622692545275801364908374167?
>>> R = RealIntervalField(Integer(150))
>>> R.pi().sqrt()/Integer(2)
0.886226925452758013649083741670572591398774728?

prec()

Return the precision of this field (in bits).

EXAMPLES:

Sage

sage: RIF.precision()
53
sage: RealIntervalField(200).precision()
200

Python

>>> from sage.all import *
>>> RIF.precision()
53
>>> RealIntervalField(Integer(200)).precision()
200

precision()

Return the precision of this field (in bits).

EXAMPLES:

Sage

sage: RIF.precision()
53
sage: RealIntervalField(200).precision()
200

Python

>>> from sage.all import *
>>> RIF.precision()
53
>>> RealIntervalField(Integer(200)).precision()
200

random_element(*args, **kwds)

Return a random element of self. Any arguments or keywords are passed onto the random element function in real field.

By default, this is uniformly distributed in \([-1, 1]\).

EXAMPLES:

Sage

sage: RIF.random_element().parent() is RIF
True
sage: -100 <= RIF.random_element(-100, 100) <= 100
True

Python

>>> from sage.all import *
>>> RIF.random_element().parent() is RIF
True
>>> -Integer(100) <= RIF.random_element(-Integer(100), Integer(100)) <= Integer(100)
True

Passes extra positional or keyword arguments through:

Sage

sage: 0 <= RIF.random_element(min=0, max=100) <= 100
True
sage: -100 <= RIF.random_element(min=-100, max=0) <= 0
True

Python

>>> from sage.all import *
>>> Integer(0) <= RIF.random_element(min=Integer(0), max=Integer(100)) <= Integer(100)
True
>>> -Integer(100) <= RIF.random_element(min=-Integer(100), max=Integer(0)) <= Integer(0)
True

scientific_notation(status=None)

Set or return the scientific notation printing flag.

If this flag is True then real numbers with this space as parent print using scientific notation.

INPUT:

• status – boolean optional flag

EXAMPLES:

Sage

sage: RIF(0.025)
0.025000000000000002?
sage: RIF.scientific_notation(True)
sage: RIF(0.025)
2.5000000000000002?e-2
sage: RIF.scientific_notation(False)
sage: RIF(0.025)
0.025000000000000002?

Python

>>> from sage.all import *
>>> RIF(RealNumber('0.025'))
0.025000000000000002?
>>> RIF.scientific_notation(True)
>>> RIF(RealNumber('0.025'))
2.5000000000000002?e-2
>>> RIF.scientific_notation(False)
>>> RIF(RealNumber('0.025'))
0.025000000000000002?

to_prec(prec)

Return a real interval field to the given precision.

EXAMPLES:

Sage

sage: RIF.to_prec(200)
Real Interval Field with 200 bits of precision
sage: RIF.to_prec(20)
Real Interval Field with 20 bits of precision
sage: RIF.to_prec(53) is RIF
True

Python

>>> from sage.all import *
>>> RIF.to_prec(Integer(200))
Real Interval Field with 200 bits of precision
>>> RIF.to_prec(Integer(20))
Real Interval Field with 20 bits of precision
>>> RIF.to_prec(Integer(53)) is RIF
True

upper_field()

Return the RealField_class with rounding mode 'RNDU' (rounding towards plus infinity).

EXAMPLES:

Sage

sage: RIF.upper_field()
Real Field with 53 bits of precision and rounding RNDU
sage: RealIntervalField(200).upper_field()
Real Field with 200 bits of precision and rounding RNDU

Python

>>> from sage.all import *
>>> RIF.upper_field()
Real Field with 53 bits of precision and rounding RNDU
>>> RealIntervalField(Integer(200)).upper_field()
Real Field with 200 bits of precision and rounding RNDU

zeta(n=2)

Return an \(n\)-th root of unity in the real field, if one exists, or raise a ValueError otherwise.

EXAMPLES:

Sage

sage: R = RealIntervalField()
sage: R.zeta()
-1
sage: R.zeta(1)
1
sage: R.zeta(5)
Traceback (most recent call last):
...
ValueError: No 5th root of unity in self

Python

>>> from sage.all import *
>>> R = RealIntervalField()
>>> R.zeta()
-1
>>> R.zeta(Integer(1))
1
>>> R.zeta(Integer(5))
Traceback (most recent call last):
...
ValueError: No 5th root of unity in self

sage.rings.real_mpfi.is_RealIntervalField(x)

Check if x is a RealIntervalField_class.

EXAMPLES:

Sage

sage: sage.rings.real_mpfi.is_RealIntervalField(RIF)
doctest:warning...
DeprecationWarning: The function is_RealIntervalField is deprecated;
use 'isinstance(..., RealIntervalField_class)' instead.
See https://github.com/sagemath/sage/issues/38128 for details.
True
sage: sage.rings.real_mpfi.is_RealIntervalField(RealIntervalField(200))
True

Python

>>> from sage.all import *
>>> sage.rings.real_mpfi.is_RealIntervalField(RIF)
doctest:warning...
DeprecationWarning: The function is_RealIntervalField is deprecated;
use 'isinstance(..., RealIntervalField_class)' instead.
See https://github.com/sagemath/sage/issues/38128 for details.
True
>>> sage.rings.real_mpfi.is_RealIntervalField(RealIntervalField(Integer(200)))
True

sage.rings.real_mpfi.is_RealIntervalFieldElement(x)

Check if x is a RealIntervalFieldElement.

EXAMPLES:

Sage

sage: sage.rings.real_mpfi.is_RealIntervalFieldElement(RIF(2.2))
doctest:warning...
DeprecationWarning: The function is_RealIntervalFieldElement is deprecated;
use 'isinstance(..., RealIntervalFieldElement)' instead.
See https://github.com/sagemath/sage/issues/38128 for details.
True
sage: sage.rings.real_mpfi.is_RealIntervalFieldElement(RealIntervalField(200)(2.2))
True

Python

>>> from sage.all import *
>>> sage.rings.real_mpfi.is_RealIntervalFieldElement(RIF(RealNumber('2.2')))
doctest:warning...
DeprecationWarning: The function is_RealIntervalFieldElement is deprecated;
use 'isinstance(..., RealIntervalFieldElement)' instead.
See https://github.com/sagemath/sage/issues/38128 for details.
True
>>> sage.rings.real_mpfi.is_RealIntervalFieldElement(RealIntervalField(Integer(200))(RealNumber('2.2')))
True