Arbitrary precision complex intervals

This is a simple complex interval package, using intervals which are axis-aligned rectangles in the complex plane. It has very few special functions, and it does not use any special tricks to keep the size of the intervals down.

AUTHORS:

These authors wrote complex_mpfr.pyx (renamed from complex_number.pyx):

- William Stein (2006-01-26): complete rewrite
- Joel B. Mohler (2006-12-16): naive rewrite into pyrex
- William Stein(2007-01): rewrite of Mohler's rewrite

Then complex_number.pyx was copied to complex_interval.pyx and heavily modified:

• Carl Witty (2007-10-24): rewrite to become a complex interval package

• Travis Scrimshaw (2012-10-18): Added documentation to get full coverage.

Warning

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

Sage

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

Python

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

Todo

Implement ComplexIntervalFieldElement multiplicative order similar to ComplexNumber multiplicative order with _set_multiplicative_order(n) and ComplexNumber.multiplicative_order() methods.

class sage.rings.complex_interval.ComplexIntervalFieldElement

Bases: FieldElement

A complex interval.

EXAMPLES:

Sage

sage: I = CIF.gen()
sage: b = 3/2 + 5/2*I
sage: TestSuite(b).run()

Python

>>> from sage.all import *
>>> I = CIF.gen()
>>> b = Integer(3)/Integer(2) + Integer(5)/Integer(2)*I
>>> TestSuite(b).run()

arg()

Same as argument().

EXAMPLES:

Sage

sage: i = CIF.0
sage: (i^2).arg()
3.141592653589794?

Python

>>> from sage.all import *
>>> i = CIF.gen(0)
>>> (i**Integer(2)).arg()
3.141592653589794?

argument()

The argument (angle) of the complex number, normalized so that \(-\pi < \theta.lower() \leq \pi\).

We raise a ValueError if the interval strictly contains 0, or if the interval contains only 0.

Warning

We do not always use the standard branch cut for argument! If the interval crosses the negative real axis, then the argument will be an interval whose lower bound is less than \(\pi\) and whose upper bound is more than \(\pi\); in effect, we move the branch cut away from the interval.

EXAMPLES:

Sage

sage: i = CIF.0
sage: (i^2).argument()
3.141592653589794?
sage: (1+i).argument()
0.785398163397449?
sage: i.argument()
1.570796326794897?
sage: (-i).argument()
-1.570796326794897?
sage: (-1/1000 - i).argument()
-1.571796326461564?
sage: CIF(2).argument()
0
sage: CIF(-2).argument()
3.141592653589794?

Python

>>> from sage.all import *
>>> i = CIF.gen(0)
>>> (i**Integer(2)).argument()
3.141592653589794?
>>> (Integer(1)+i).argument()
0.785398163397449?
>>> i.argument()
1.570796326794897?
>>> (-i).argument()
-1.570796326794897?
>>> (-Integer(1)/Integer(1000) - i).argument()
-1.571796326461564?
>>> CIF(Integer(2)).argument()
0
>>> CIF(-Integer(2)).argument()
3.141592653589794?

Here we see that if the interval crosses the negative real axis, then the argument can exceed \(\pi\), and we we violate the standard interval guarantees in the process:

Sage

sage: CIF(-2, RIF(-0.1, 0.1)).argument().str(style='brackets')
'[3.0916342578678501 .. 3.1915510493117365]'
sage: CIF(-2, -0.1).argument()
-3.091634257867851?

Python

>>> from sage.all import *
>>> CIF(-Integer(2), RIF(-RealNumber('0.1'), RealNumber('0.1'))).argument().str(style='brackets')
'[3.0916342578678501 .. 3.1915510493117365]'
>>> CIF(-Integer(2), -RealNumber('0.1')).argument()
-3.091634257867851?

bisection()

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

EXAMPLES:

Sage

sage: z = CIF(RIF(2, 3), RIF(-5, -4))
sage: z.bisection()
(3.? - 5.?*I, 3.? - 5.?*I, 3.? - 5.?*I, 3.? - 5.?*I)
sage: for z in z.bisection():
....:     print(z.real().endpoints())
....:     print(z.imag().endpoints())
(2.00000000000000, 2.50000000000000)
(-5.00000000000000, -4.50000000000000)
(2.50000000000000, 3.00000000000000)
(-5.00000000000000, -4.50000000000000)
(2.00000000000000, 2.50000000000000)
(-4.50000000000000, -4.00000000000000)
(2.50000000000000, 3.00000000000000)
(-4.50000000000000, -4.00000000000000)

sage: # needs sage.symbolic
sage: z = CIF(RIF(sqrt(2), sqrt(3)), RIF(e, pi))
sage: a, b, c, d = z.bisection()
sage: a.intersection(b).intersection(c).intersection(d) == CIF(z.center())
True
sage: zz = a.union(b).union(c).union(c)
sage: zz.real().endpoints() == z.real().endpoints()
True
sage: zz.imag().endpoints() == z.imag().endpoints()
True

Python

>>> from sage.all import *
>>> z = CIF(RIF(Integer(2), Integer(3)), RIF(-Integer(5), -Integer(4)))
>>> z.bisection()
(3.? - 5.?*I, 3.? - 5.?*I, 3.? - 5.?*I, 3.? - 5.?*I)
>>> for z in z.bisection():
...     print(z.real().endpoints())
...     print(z.imag().endpoints())
(2.00000000000000, 2.50000000000000)
(-5.00000000000000, -4.50000000000000)
(2.50000000000000, 3.00000000000000)
(-5.00000000000000, -4.50000000000000)
(2.00000000000000, 2.50000000000000)
(-4.50000000000000, -4.00000000000000)
(2.50000000000000, 3.00000000000000)
(-4.50000000000000, -4.00000000000000)

>>> # needs sage.symbolic
>>> z = CIF(RIF(sqrt(Integer(2)), sqrt(Integer(3))), RIF(e, pi))
>>> a, b, c, d = z.bisection()
>>> a.intersection(b).intersection(c).intersection(d) == CIF(z.center())
True
>>> zz = a.union(b).union(c).union(c)
>>> zz.real().endpoints() == z.real().endpoints()
True
>>> zz.imag().endpoints() == z.imag().endpoints()
True

center()

Return the closest floating-point approximation to the center of the interval.

EXAMPLES:

Sage

sage: CIF(RIF(1, 2), RIF(3, 4)).center()
1.50000000000000 + 3.50000000000000*I

Python

>>> from sage.all import *
>>> CIF(RIF(Integer(1), Integer(2)), RIF(Integer(3), Integer(4))).center()
1.50000000000000 + 3.50000000000000*I

conjugate()

Return the complex conjugate of this complex number.

EXAMPLES:

Sage

sage: i = CIF.0
sage: (1+i).conjugate()
1 - 1*I

Python

>>> from sage.all import *
>>> i = CIF.gen(0)
>>> (Integer(1)+i).conjugate()
1 - 1*I

contains_zero()

Return True if self is an interval containing zero.

EXAMPLES:

Sage

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

Python

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

cos()

Compute the cosine of this complex interval.

EXAMPLES:

Sage

sage: CIF(1,1).cos()
0.833730025131149? - 0.988897705762865?*I
sage: CIF(3).cos()
-0.9899924966004455?
sage: CIF(0,2).cos()
3.762195691083632?

Python

>>> from sage.all import *
>>> CIF(Integer(1),Integer(1)).cos()
0.833730025131149? - 0.988897705762865?*I
>>> CIF(Integer(3)).cos()
-0.9899924966004455?
>>> CIF(Integer(0),Integer(2)).cos()
3.762195691083632?

Check that Issue #17285 is fixed:

Sage

sage: CIF(cos(2/3))                                                         # needs sage.symbolic
0.7858872607769480?

Python

>>> from sage.all import *
>>> CIF(cos(Integer(2)/Integer(3)))                                                         # needs sage.symbolic
0.7858872607769480?

ALGORITHM:

The implementation uses the following trigonometric identity

\[\cos(x + iy) = \cos(x) \cosh(y) - i \sin(x) \sinh(y)\]

cosh()

Return the hyperbolic cosine of this complex interval.

EXAMPLES:

Sage

sage: CIF(1,1).cosh()
0.833730025131149? + 0.988897705762865?*I
sage: CIF(2).cosh()
3.762195691083632?
sage: CIF(0,2).cosh()
-0.4161468365471424?

Python

>>> from sage.all import *
>>> CIF(Integer(1),Integer(1)).cosh()
0.833730025131149? + 0.988897705762865?*I
>>> CIF(Integer(2)).cosh()
3.762195691083632?
>>> CIF(Integer(0),Integer(2)).cosh()
-0.4161468365471424?

ALGORITHM:

The implementation uses the following trigonometric identity

\[\cosh(x+iy) = \cos(y) \cosh(x) + i \sin(y) \sinh(x)\]

crosses_log_branch_cut()

Return True if this interval crosses the standard branch cut for log() (and hence for exponentiation) and for argument. (Recall that this branch cut is infinitesimally below the negative portion of the real axis.)

EXAMPLES:

Sage

sage: z = CIF(1.5, 2.5) - CIF(0, 2.50000000000000001); z
1.5000000000000000? + -1.?e-15*I
sage: z.crosses_log_branch_cut()
False
sage: CIF(-2, RIF(-0.1, 0.1)).crosses_log_branch_cut()
True

Python

>>> from sage.all import *
>>> z = CIF(RealNumber('1.5'), RealNumber('2.5')) - CIF(Integer(0), RealNumber('2.50000000000000001')); z
1.5000000000000000? + -1.?e-15*I
>>> z.crosses_log_branch_cut()
False
>>> CIF(-Integer(2), RIF(-RealNumber('0.1'), RealNumber('0.1'))).crosses_log_branch_cut()
True

diameter()

Return a somewhat-arbitrarily defined “diameter” for this interval.

The diameter of an interval is the maximum of the diameter of the real and imaginary components, where diameter on a real interval is defined as absolute diameter if the interval contains zero, and relative diameter otherwise.

EXAMPLES:

Sage

sage: CIF(RIF(-1, 1), RIF(13, 17)).diameter()
2.00000000000000
sage: CIF(RIF(-0.1, 0.1), RIF(13, 17)).diameter()
0.266666666666667
sage: CIF(RIF(-1, 1), 15).diameter()
2.00000000000000

Python

>>> from sage.all import *
>>> CIF(RIF(-Integer(1), Integer(1)), RIF(Integer(13), Integer(17))).diameter()
2.00000000000000
>>> CIF(RIF(-RealNumber('0.1'), RealNumber('0.1')), RIF(Integer(13), Integer(17))).diameter()
0.266666666666667
>>> CIF(RIF(-Integer(1), Integer(1)), Integer(15)).diameter()
2.00000000000000

edges()

Return the 4 edges of the rectangle in the complex plane defined by this interval as intervals.

OUTPUT: a 4-tuple of complex intervals (left edge, right edge, lower edge, upper edge)

See also

endpoints() which returns the 4 corners of the rectangle.

EXAMPLES:

Sage

sage: CIF(RIF(1,2), RIF(3,4)).edges()
(1 + 4.?*I, 2 + 4.?*I, 2.? + 3*I, 2.? + 4*I)
sage: ComplexIntervalField(20)(-2).log().edges()
(0.69314671? + 3.14160?*I,
 0.69314766? + 3.14160?*I,
 0.693147? + 3.1415902?*I,
 0.693147? + 3.1415940?*I)

Python

>>> from sage.all import *
>>> CIF(RIF(Integer(1),Integer(2)), RIF(Integer(3),Integer(4))).edges()
(1 + 4.?*I, 2 + 4.?*I, 2.? + 3*I, 2.? + 4*I)
>>> ComplexIntervalField(Integer(20))(-Integer(2)).log().edges()
(0.69314671? + 3.14160?*I,
 0.69314766? + 3.14160?*I,
 0.693147? + 3.1415902?*I,
 0.693147? + 3.1415940?*I)

endpoints()

Return the 4 corners of the rectangle in the complex plane defined by this interval.

OUTPUT: a 4-tuple of complex numbers (lower left, upper right, upper left, lower right)

See also

edges() which returns the 4 edges of the rectangle.

EXAMPLES:

Sage

sage: CIF(RIF(1,2), RIF(3,4)).endpoints()
(1.00000000000000 + 3.00000000000000*I,
 2.00000000000000 + 4.00000000000000*I,
 1.00000000000000 + 4.00000000000000*I,
 2.00000000000000 + 3.00000000000000*I)
sage: ComplexIntervalField(20)(-2).log().endpoints()
(0.69315 + 3.1416*I,
 0.69315 + 3.1416*I,
 0.69315 + 3.1416*I,
 0.69315 + 3.1416*I)

Python

>>> from sage.all import *
>>> CIF(RIF(Integer(1),Integer(2)), RIF(Integer(3),Integer(4))).endpoints()
(1.00000000000000 + 3.00000000000000*I,
 2.00000000000000 + 4.00000000000000*I,
 1.00000000000000 + 4.00000000000000*I,
 2.00000000000000 + 3.00000000000000*I)
>>> ComplexIntervalField(Integer(20))(-Integer(2)).log().endpoints()
(0.69315 + 3.1416*I,
 0.69315 + 3.1416*I,
 0.69315 + 3.1416*I,
 0.69315 + 3.1416*I)

exp()

Compute \(e^z\) or \(\exp(z)\) where \(z\) is the complex number self.

EXAMPLES:

Sage

sage: i = ComplexIntervalField(300).0
sage: z = 1 + i
sage: z.exp()
1.46869393991588515713896759732660426132695673662900872279767567631093696585951213872272450? + 2.28735528717884239120817190670050180895558625666835568093865811410364716018934540926734485?*I

Python

>>> from sage.all import *
>>> i = ComplexIntervalField(Integer(300)).gen(0)
>>> z = Integer(1) + i
>>> z.exp()
1.46869393991588515713896759732660426132695673662900872279767567631093696585951213872272450? + 2.28735528717884239120817190670050180895558625666835568093865811410364716018934540926734485?*I

imag()

Return imaginary part of self.

EXAMPLES:

Sage

sage: i = ComplexIntervalField(100).0
sage: z = 2 + 3*i
sage: x = z.imag(); x
3
sage: x.parent()
Real Interval Field with 100 bits of precision

Python

>>> from sage.all import *
>>> i = ComplexIntervalField(Integer(100)).gen(0)
>>> z = Integer(2) + Integer(3)*i
>>> x = z.imag(); x
3
>>> x.parent()
Real Interval Field with 100 bits of precision

intersection(other)

Return the intersection of the two complex intervals self and other.

EXAMPLES:

Sage

sage: CIF(RIF(1, 3), RIF(1, 3)).intersection(CIF(RIF(2, 4), RIF(2, 4))).str(style='brackets')
'[2.0000000000000000 .. 3.0000000000000000] + [2.0000000000000000 .. 3.0000000000000000]*I'
sage: CIF(RIF(1, 2), RIF(1, 3)).intersection(CIF(RIF(3, 4), RIF(2, 4)))
Traceback (most recent call last):
...
ValueError: intersection of non-overlapping intervals

Python

>>> from sage.all import *
>>> CIF(RIF(Integer(1), Integer(3)), RIF(Integer(1), Integer(3))).intersection(CIF(RIF(Integer(2), Integer(4)), RIF(Integer(2), Integer(4)))).str(style='brackets')
'[2.0000000000000000 .. 3.0000000000000000] + [2.0000000000000000 .. 3.0000000000000000]*I'
>>> CIF(RIF(Integer(1), Integer(2)), RIF(Integer(1), Integer(3))).intersection(CIF(RIF(Integer(3), Integer(4)), RIF(Integer(2), Integer(4))))
Traceback (most recent call last):
...
ValueError: intersection of non-overlapping intervals

is_NaN()

Return True if this is not-a-number.

EXAMPLES:

Sage

sage: CIF(2, 1).is_NaN()
False
sage: CIF(NaN).is_NaN()                                                     # needs sage.symbolic
True
sage: (1 / CIF(0, 0)).is_NaN()
True

Python

>>> from sage.all import *
>>> CIF(Integer(2), Integer(1)).is_NaN()
False
>>> CIF(NaN).is_NaN()                                                     # needs sage.symbolic
True
>>> (Integer(1) / CIF(Integer(0), Integer(0))).is_NaN()
True

is_exact()

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

EXAMPLES:

Sage

sage: CIF(3).is_exact()
True
sage: CIF(0, 2).is_exact()
True
sage: CIF(-4, 0).sqrt().is_exact()
True
sage: CIF(-5, 0).sqrt().is_exact()
False
sage: CIF(0, 2*pi).is_exact()                                               # needs sage.symbolic
False
sage: CIF(e).is_exact()                                                     # needs sage.symbolic
False
sage: CIF(1e100).is_exact()
True
sage: (CIF(1e100) + 1).is_exact()
False

Python

>>> from sage.all import *
>>> CIF(Integer(3)).is_exact()
True
>>> CIF(Integer(0), Integer(2)).is_exact()
True
>>> CIF(-Integer(4), Integer(0)).sqrt().is_exact()
True
>>> CIF(-Integer(5), Integer(0)).sqrt().is_exact()
False
>>> CIF(Integer(0), Integer(2)*pi).is_exact()                                               # needs sage.symbolic
False
>>> CIF(e).is_exact()                                                     # needs sage.symbolic
False
>>> CIF(RealNumber('1e100')).is_exact()
True
>>> (CIF(RealNumber('1e100')) + Integer(1)).is_exact()
False

is_square()

Return True as \(\CC\) is algebraically closed.

EXAMPLES:

Sage

sage: CIF(2, 1).is_square()
True

Python

>>> from sage.all import *
>>> CIF(Integer(2), Integer(1)).is_square()
True

lexico_cmp(left, right)

Intervals are compared lexicographically on the 4-tuple: (x.real().lower(), x.real().upper(), x.imag().lower(), x.imag().upper())

EXAMPLES:

Sage

sage: a = CIF(RIF(0,1), RIF(0,1))
sage: b = CIF(RIF(0,1), RIF(0,2))
sage: c = CIF(RIF(0,2), RIF(0,2))
sage: a.lexico_cmp(b)
-1
sage: b.lexico_cmp(c)
-1
sage: a.lexico_cmp(c)
-1
sage: a.lexico_cmp(a)
0
sage: b.lexico_cmp(a)
1

Python

>>> from sage.all import *
>>> a = CIF(RIF(Integer(0),Integer(1)), RIF(Integer(0),Integer(1)))
>>> b = CIF(RIF(Integer(0),Integer(1)), RIF(Integer(0),Integer(2)))
>>> c = CIF(RIF(Integer(0),Integer(2)), RIF(Integer(0),Integer(2)))
>>> a.lexico_cmp(b)
-1
>>> b.lexico_cmp(c)
-1
>>> a.lexico_cmp(c)
-1
>>> a.lexico_cmp(a)
0
>>> b.lexico_cmp(a)
1

log(base=None)

Complex logarithm of \(z\).

Warning

This does always not use the standard branch cut for complex log! See the docstring for argument() to see what we do instead.

EXAMPLES:

Sage

sage: a = CIF(RIF(3, 4), RIF(13, 14))
sage: a.log().str(style='brackets')
'[2.5908917751460420 .. 2.6782931373360067] + [1.2722973952087170 .. 1.3597029935721503]*I'
sage: a.log().exp().str(style='brackets')
'[2.7954667135098274 .. 4.2819545928390213] + [12.751682453911920 .. 14.237018048974635]*I'
sage: a in a.log().exp()
True

Python

>>> from sage.all import *
>>> a = CIF(RIF(Integer(3), Integer(4)), RIF(Integer(13), Integer(14)))
>>> a.log().str(style='brackets')
'[2.5908917751460420 .. 2.6782931373360067] + [1.2722973952087170 .. 1.3597029935721503]*I'
>>> a.log().exp().str(style='brackets')
'[2.7954667135098274 .. 4.2819545928390213] + [12.751682453911920 .. 14.237018048974635]*I'
>>> a in a.log().exp()
True

If the interval crosses the negative real axis, then we don’t use the standard branch cut (and we violate the interval guarantees):

Sage

sage: CIF(-3, RIF(-1/4, 1/4)).log().str(style='brackets')
'[1.0986122886681095 .. 1.1020725100903968] + [3.0584514217013518 .. 3.2247338854782349]*I'
sage: CIF(-3, -1/4).log()
1.102072510090397? - 3.058451421701352?*I

Python

>>> from sage.all import *
>>> CIF(-Integer(3), RIF(-Integer(1)/Integer(4), Integer(1)/Integer(4))).log().str(style='brackets')
'[1.0986122886681095 .. 1.1020725100903968] + [3.0584514217013518 .. 3.2247338854782349]*I'
>>> CIF(-Integer(3), -Integer(1)/Integer(4)).log()
1.102072510090397? - 3.058451421701352?*I

Usually if an interval contains zero, we raise an exception:

Sage

sage: CIF(RIF(-1,1),RIF(-1,1)).log()
Traceback (most recent call last):
...
ValueError: Can...t take the argument of interval strictly containing zero

Python

>>> from sage.all import *
>>> CIF(RIF(-Integer(1),Integer(1)),RIF(-Integer(1),Integer(1))).log()
Traceback (most recent call last):
...
ValueError: Can...t take the argument of interval strictly containing zero

But we allow the exact input zero:

Sage

sage: CIF(0).log()
[-infinity .. -infinity]

Python

>>> from sage.all import *
>>> CIF(Integer(0)).log()
[-infinity .. -infinity]

If a base is passed from another function, we can accommodate this:

Sage

sage: CIF(-1,1).log(2)
0.500000000000000? + 3.39927010637040?*I

Python

>>> from sage.all import *
>>> CIF(-Integer(1),Integer(1)).log(Integer(2))
0.500000000000000? + 3.39927010637040?*I

magnitude()

The largest absolute value of the elements of the interval, rounded away from zero.

OUTPUT: a real number with rounding mode RNDU

EXAMPLES:

Sage

sage: CIF(RIF(-1,1), RIF(-1,1)).magnitude()
1.41421356237310
sage: CIF(RIF(1,2), RIF(3,4)).magnitude()
4.47213595499958
sage: parent(CIF(1).magnitude())
Real Field with 53 bits of precision and rounding RNDU

Python

>>> from sage.all import *
>>> CIF(RIF(-Integer(1),Integer(1)), RIF(-Integer(1),Integer(1))).magnitude()
1.41421356237310
>>> CIF(RIF(Integer(1),Integer(2)), RIF(Integer(3),Integer(4))).magnitude()
4.47213595499958
>>> parent(CIF(Integer(1)).magnitude())
Real Field with 53 bits of precision and rounding RNDU

mignitude()

The smallest absolute value of the elements of the interval, rounded towards zero.

OUTPUT: a real number with rounding mode RNDD

EXAMPLES:

Sage

sage: CIF(RIF(-1,1), RIF(-1,1)).mignitude()
0.000000000000000
sage: CIF(RIF(1,2), RIF(3,4)).mignitude()
3.16227766016837
sage: parent(CIF(1).mignitude())
Real Field with 53 bits of precision and rounding RNDD

Python

>>> from sage.all import *
>>> CIF(RIF(-Integer(1),Integer(1)), RIF(-Integer(1),Integer(1))).mignitude()
0.000000000000000
>>> CIF(RIF(Integer(1),Integer(2)), RIF(Integer(3),Integer(4))).mignitude()
3.16227766016837
>>> parent(CIF(Integer(1)).mignitude())
Real Field with 53 bits of precision and rounding RNDD

multiplicative_order()

Return the multiplicative order of this complex number, if known, or raise a NotImplementedError.

EXAMPLES:

Sage

sage: C = CIF
sage: i = C.0
sage: i.multiplicative_order()
4
sage: C(1).multiplicative_order()
1
sage: C(-1).multiplicative_order()
2
sage: (i^2).multiplicative_order()
2
sage: (-i).multiplicative_order()
4
sage: C(2).multiplicative_order()
+Infinity
sage: w = (1 + C(-3).sqrt())/2 ; w
0.50000000000000000? + 0.866025403784439?*I
sage: w.multiplicative_order()
Traceback (most recent call last):
...
NotImplementedError: order of element not known

Python

>>> from sage.all import *
>>> C = CIF
>>> i = C.gen(0)
>>> i.multiplicative_order()
4
>>> C(Integer(1)).multiplicative_order()
1
>>> C(-Integer(1)).multiplicative_order()
2
>>> (i**Integer(2)).multiplicative_order()
2
>>> (-i).multiplicative_order()
4
>>> C(Integer(2)).multiplicative_order()
+Infinity
>>> w = (Integer(1) + C(-Integer(3)).sqrt())/Integer(2) ; w
0.50000000000000000? + 0.866025403784439?*I
>>> w.multiplicative_order()
Traceback (most recent call last):
...
NotImplementedError: order of element not known

norm()

Return the norm of this complex number.

If \(c = a + bi\) is a complex number, then the norm of \(c\) is defined as the product of \(c\) and its complex conjugate:

\[\text{norm}(c) = \text{norm}(a + bi) = c \cdot \overline{c} = a^2 + b^2.\]

The norm of a complex number is different from its absolute value. The absolute value of a complex number is defined to be the square root of its norm. A typical use of the complex norm is in the integral domain \(\ZZ[i]\) of Gaussian integers, where the norm of each Gaussian integer \(c = a + bi\) is defined as its complex norm.

See also

• sage.rings.complex_double.ComplexDoubleElement.norm()

EXAMPLES:

Sage

sage: CIF(2, 1).norm()
5
sage: CIF(1, -2).norm()
5

Python

>>> from sage.all import *
>>> CIF(Integer(2), Integer(1)).norm()
5
>>> CIF(Integer(1), -Integer(2)).norm()
5

overlaps(other)

Return True if self and other are intervals with at least one value in common.

EXAMPLES:

Sage

sage: CIF(0).overlaps(CIF(RIF(0, 1), RIF(-1, 0)))
True
sage: CIF(1).overlaps(CIF(1, 1))
False

Python

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

plot(pointsize=10, **kwds)

Plot a complex interval as a rectangle.

EXAMPLES:

Sage

sage: sum(plot(CIF(RIF(1/k, 1/k), RIF(-k, k))) for k in [1..10])            # needs sage.plot
Graphics object consisting of 20 graphics primitives

Python

>>> from sage.all import *
>>> sum(plot(CIF(RIF(Integer(1)/k, Integer(1)/k), RIF(-k, k))) for k in (ellipsis_range(Integer(1),Ellipsis,Integer(10))))            # needs sage.plot
Graphics object consisting of 20 graphics primitives

Exact and nearly exact points are still visible:

Sage

sage: # needs sage.plot sage.symbolic
sage: plot(CIF(pi, 1), color='red') + plot(CIF(1, e), color='purple') + plot(CIF(-1, -1))
Graphics object consisting of 6 graphics primitives

Python

>>> from sage.all import *
>>> # needs sage.plot sage.symbolic
>>> plot(CIF(pi, Integer(1)), color='red') + plot(CIF(Integer(1), e), color='purple') + plot(CIF(-Integer(1), -Integer(1)))
Graphics object consisting of 6 graphics primitives

A demonstration that \(z \mapsto z^2\) acts chaotically on \(|z|=1\):

Sage

sage: # needs sage.plot sage.symbolic
sage: z = CIF(0, 2*pi/1000).exp()
sage: g = Graphics()
sage: for i in range(40):
....:     z = z^2
....:     g += z.plot(color=(1./(40-i), 0, 1))
...
sage: g
Graphics object consisting of 80 graphics primitives

Python

>>> from sage.all import *
>>> # needs sage.plot sage.symbolic
>>> z = CIF(Integer(0), Integer(2)*pi/Integer(1000)).exp()
>>> g = Graphics()
>>> for i in range(Integer(40)):
...     z = z**Integer(2)
...     g += z.plot(color=(RealNumber('1.')/(Integer(40)-i), Integer(0), Integer(1)))
...
>>> g
Graphics object consisting of 80 graphics primitives

prec()

Return precision of this complex number.

EXAMPLES:

Sage

sage: i = ComplexIntervalField(2000).0
sage: i.prec()
2000

Python

>>> from sage.all import *
>>> i = ComplexIntervalField(Integer(2000)).gen(0)
>>> i.prec()
2000

real()

Return real part of self.

EXAMPLES:

Sage

sage: i = ComplexIntervalField(100).0
sage: z = 2 + 3*i
sage: x = z.real(); x
2
sage: x.parent()
Real Interval Field with 100 bits of precision

Python

>>> from sage.all import *
>>> i = ComplexIntervalField(Integer(100)).gen(0)
>>> z = Integer(2) + Integer(3)*i
>>> x = z.real(); x
2
>>> x.parent()
Real Interval Field with 100 bits of precision

sin()

Compute the sine of this complex interval.

EXAMPLES:

Sage

sage: CIF(1,1).sin()
1.298457581415978? + 0.634963914784736?*I
sage: CIF(2).sin()
0.909297426825682?
sage: CIF(0,2).sin()
3.626860407847019?*I

Python

>>> from sage.all import *
>>> CIF(Integer(1),Integer(1)).sin()
1.298457581415978? + 0.634963914784736?*I
>>> CIF(Integer(2)).sin()
0.909297426825682?
>>> CIF(Integer(0),Integer(2)).sin()
3.626860407847019?*I

Check that Issue #17825 is fixed:

Sage

sage: CIF(sin(2/3))                                                         # needs sage.symbolic
0.618369803069737?

Python

>>> from sage.all import *
>>> CIF(sin(Integer(2)/Integer(3)))                                                         # needs sage.symbolic
0.618369803069737?

ALGORITHM:

The implementation uses the following trigonometric identity

\[\sin(x + iy) = \sin(x) \cosh(y) + i \cos (x) \sinh(y)\]

sinh()

Return the hyperbolic sine of this complex interval.

EXAMPLES:

Sage

sage: CIF(1,1).sinh()
0.634963914784736? + 1.298457581415978?*I
sage: CIF(2).sinh()
3.626860407847019?
sage: CIF(0,2).sinh()
0.909297426825682?*I

Python

>>> from sage.all import *
>>> CIF(Integer(1),Integer(1)).sinh()
0.634963914784736? + 1.298457581415978?*I
>>> CIF(Integer(2)).sinh()
3.626860407847019?
>>> CIF(Integer(0),Integer(2)).sinh()
0.909297426825682?*I

ALGORITHM:

The implementation uses the following trigonometric identity

\[\sinh(x+iy) = \cos(y) \sinh(x) + i \sin(y) \cosh(x)\]

sqrt(all=False, **kwds)

The square root function.

Warning

We approximate the standard branch cut along the negative real axis, with sqrt(-r^2) = i*r for positive real r; but if the interval crosses the negative real axis, we pick the root with positive imaginary component for the entire interval.

INPUT:

• all – boolean (default: False); if True, return a list of all square roots

EXAMPLES:

Sage

sage: CIF(-1).sqrt()^2
-1
sage: sqrt(CIF(2))
1.414213562373095?
sage: sqrt(CIF(-1))
1*I
sage: sqrt(CIF(2-I))^2
2.00000000000000? - 1.00000000000000?*I
sage: CC(-2-I).sqrt()^2
-2.00000000000000 - 1.00000000000000*I

Python

>>> from sage.all import *
>>> CIF(-Integer(1)).sqrt()**Integer(2)
-1
>>> sqrt(CIF(Integer(2)))
1.414213562373095?
>>> sqrt(CIF(-Integer(1)))
1*I
>>> sqrt(CIF(Integer(2)-I))**Integer(2)
2.00000000000000? - 1.00000000000000?*I
>>> CC(-Integer(2)-I).sqrt()**Integer(2)
-2.00000000000000 - 1.00000000000000*I

Here, we select a non-principal root for part of the interval, and violate the standard interval guarantees:

Sage

sage: CIF(-5, RIF(-1, 1)).sqrt().str(style='brackets')
'[-0.22250788030178321 .. 0.22250788030178296] + [2.2251857651053086 .. 2.2581008643532262]*I'
sage: CIF(-5, -1).sqrt()
0.222507880301783? - 2.247111425095870?*I

Python

>>> from sage.all import *
>>> CIF(-Integer(5), RIF(-Integer(1), Integer(1))).sqrt().str(style='brackets')
'[-0.22250788030178321 .. 0.22250788030178296] + [2.2251857651053086 .. 2.2581008643532262]*I'
>>> CIF(-Integer(5), -Integer(1)).sqrt()
0.222507880301783? - 2.247111425095870?*I

str(base=10, style=None)

Return a string representation of self.

EXAMPLES:

Sage

sage: CIF(1.5).str()
'1.5000000000000000?'
sage: CIF(1.5, 2.5).str()
'1.5000000000000000? + 2.5000000000000000?*I'
sage: CIF(1.5, -2.5).str()
'1.5000000000000000? - 2.5000000000000000?*I'
sage: CIF(0, -2.5).str()
'-2.5000000000000000?*I'
sage: CIF(1.5).str(base=3)
'1.1111111111111111111111111111111112?'
sage: CIF(1, pi).str(style='brackets')                                      # needs sage.symbolic
'[1.0000000000000000 .. 1.0000000000000000] + [3.1415926535897931 .. 3.1415926535897936]*I'

Python

>>> from sage.all import *
>>> CIF(RealNumber('1.5')).str()
'1.5000000000000000?'
>>> CIF(RealNumber('1.5'), RealNumber('2.5')).str()
'1.5000000000000000? + 2.5000000000000000?*I'
>>> CIF(RealNumber('1.5'), -RealNumber('2.5')).str()
'1.5000000000000000? - 2.5000000000000000?*I'
>>> CIF(Integer(0), -RealNumber('2.5')).str()
'-2.5000000000000000?*I'
>>> CIF(RealNumber('1.5')).str(base=Integer(3))
'1.1111111111111111111111111111111112?'
>>> CIF(Integer(1), pi).str(style='brackets')                                      # needs sage.symbolic
'[1.0000000000000000 .. 1.0000000000000000] + [3.1415926535897931 .. 3.1415926535897936]*I'

See also

• RealIntervalFieldElement.str()

tan()

Return the tangent of this complex interval.

EXAMPLES:

Sage

sage: CIF(1,1).tan()
0.27175258531952? + 1.08392332733870?*I
sage: CIF(2).tan()
-2.185039863261519?
sage: CIF(0,2).tan()
0.964027580075817?*I

Python

>>> from sage.all import *
>>> CIF(Integer(1),Integer(1)).tan()
0.27175258531952? + 1.08392332733870?*I
>>> CIF(Integer(2)).tan()
-2.185039863261519?
>>> CIF(Integer(0),Integer(2)).tan()
0.964027580075817?*I

tanh()

Return the hyperbolic tangent of this complex interval.

EXAMPLES:

Sage

sage: CIF(1,1).tanh()
1.08392332733870? + 0.27175258531952?*I
sage: CIF(2).tanh()
0.964027580075817?
sage: CIF(0,2).tanh()
-2.185039863261519?*I

Python

>>> from sage.all import *
>>> CIF(Integer(1),Integer(1)).tanh()
1.08392332733870? + 0.27175258531952?*I
>>> CIF(Integer(2)).tanh()
0.964027580075817?
>>> CIF(Integer(0),Integer(2)).tanh()
-2.185039863261519?*I

union(other)

Return the smallest complex interval including the two complex intervals self and other.

EXAMPLES:

Sage

sage: CIF(0).union(CIF(5, 5)).str(style='brackets')
'[0.0000000000000000 .. 5.0000000000000000] + [0.0000000000000000 .. 5.0000000000000000]*I'

Python

>>> from sage.all import *
>>> CIF(Integer(0)).union(CIF(Integer(5), Integer(5))).str(style='brackets')
'[0.0000000000000000 .. 5.0000000000000000] + [0.0000000000000000 .. 5.0000000000000000]*I'

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(CIF(2, 3))
0.7980219851462757? - 0.1137443080529385?*I
sage: _.parent()
Complex Interval Field with 53 bits of precision
sage: CIF(2, 3).zeta(1/2)
-1.955171567161496? + 3.123301509220897?*I

Python

>>> from sage.all import *
>>> zeta(CIF(Integer(2), Integer(3)))
0.7980219851462757? - 0.1137443080529385?*I
>>> _.parent()
Complex Interval Field with 53 bits of precision
>>> CIF(Integer(2), Integer(3)).zeta(Integer(1)/Integer(2))
-1.955171567161496? + 3.123301509220897?*I

sage.rings.complex_interval.create_ComplexIntervalFieldElement(s_real, s_imag=None, pad=0, min_prec=53)

Return the complex number defined by the strings s_real and s_imag as an element of ComplexIntervalField(prec=n), where \(n\) potentially has slightly more (controlled by pad) bits than given by \(s\).

INPUT:

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

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

• pad – integer at least 0

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

EXAMPLES:

Sage

sage: ComplexIntervalFieldElement('2.3')
2.300000000000000?
sage: ComplexIntervalFieldElement('2.3','1.1')
2.300000000000000? + 1.100000000000000?*I
sage: ComplexIntervalFieldElement(10)
10
sage: ComplexIntervalFieldElement(10,10)
10 + 10*I
sage: ComplexIntervalFieldElement(1.000000000000000000000000000,2)
1 + 2*I
sage: ComplexIntervalFieldElement(1,2.000000000000000000000)
1 + 2*I
sage: ComplexIntervalFieldElement(1.234567890123456789012345, 5.4321098654321987654321)
1.234567890123456789012350? + 5.432109865432198765432000?*I

Python

>>> from sage.all import *
>>> ComplexIntervalFieldElement('2.3')
2.300000000000000?
>>> ComplexIntervalFieldElement('2.3','1.1')
2.300000000000000? + 1.100000000000000?*I
>>> ComplexIntervalFieldElement(Integer(10))
10
>>> ComplexIntervalFieldElement(Integer(10),Integer(10))
10 + 10*I
>>> ComplexIntervalFieldElement(RealNumber('1.000000000000000000000000000'),Integer(2))
1 + 2*I
>>> ComplexIntervalFieldElement(Integer(1),RealNumber('2.000000000000000000000'))
1 + 2*I
>>> ComplexIntervalFieldElement(RealNumber('1.234567890123456789012345'), RealNumber('5.4321098654321987654321'))
1.234567890123456789012350? + 5.432109865432198765432000?*I

sage.rings.complex_interval.is_ComplexIntervalFieldElement(x)

Check if x is a ComplexIntervalFieldElement.

EXAMPLES:

Sage

sage: from sage.rings.complex_interval import is_ComplexIntervalFieldElement as is_CIFE
sage: is_CIFE(CIF(2))
doctest:warning...
DeprecationWarning: The function is_ComplexIntervalFieldElement is deprecated;
use 'isinstance(..., ComplexIntervalFieldElement)' instead.
See https://github.com/sagemath/sage/issues/38128 for details.
True
sage: is_CIFE(CC(2))
False

Python

>>> from sage.all import *
>>> from sage.rings.complex_interval import is_ComplexIntervalFieldElement as is_CIFE
>>> is_CIFE(CIF(Integer(2)))
doctest:warning...
DeprecationWarning: The function is_ComplexIntervalFieldElement is deprecated;
use 'isinstance(..., ComplexIntervalFieldElement)' instead.
See https://github.com/sagemath/sage/issues/38128 for details.
True
>>> is_CIFE(CC(Integer(2)))
False

sage.rings.complex_interval.make_ComplexIntervalFieldElement0(fld, re, im)

Construct a ComplexIntervalFieldElement for pickling.