Lazy real and complex numbers¶

These classes are very lazy, in the sense that it doesn’t really do anything but simply sits between exact rings of characteristic 0 and the real numbers. The values are actually computed when they are cast into a field of fixed precision.

The main purpose of these classes is to provide a place for exact rings (e.g. number fields) to embed for the coercion model (as only one embedding can be specified in the forward direction).

sage.rings.real_lazy.ComplexLazyField()[source]¶

Return the lazy complex field.

EXAMPLES:

There is only one lazy complex field:

Sage

sage: ComplexLazyField() is ComplexLazyField()
True

Python

>>> from sage.all import *
>>> ComplexLazyField() is ComplexLazyField()
True

class sage.rings.real_lazy.ComplexLazyField_class[source]¶

Bases: LazyField

This class represents the set of complex numbers to unspecified precision. For the most part it simply wraps exact elements and defers evaluation until a specified precision is requested.

For more information, see the documentation of the RLF.

EXAMPLES:

Sage

sage: a = CLF(-1).sqrt()
sage: a
1*I
sage: CDF(a)
1.0*I
sage: ComplexField(200)(a)
1.0000000000000000000000000000000000000000000000000000000000*I

Python

>>> from sage.all import *
>>> a = CLF(-Integer(1)).sqrt()
>>> a
1*I
>>> CDF(a)
1.0*I
>>> ComplexField(Integer(200))(a)
1.0000000000000000000000000000000000000000000000000000000000*I

construction()[source]¶

Return the functorial construction of self, namely, algebraic closure of the real lazy field.

EXAMPLES:

Sage

sage: c, S = CLF.construction(); S
Real Lazy Field
sage: CLF == c(S)
True

Python

>>> from sage.all import *
>>> c, S = CLF.construction(); S
Real Lazy Field
>>> CLF == c(S)
True

gen(i=0)[source]¶

Return the \(i\)-th generator of self.

EXAMPLES:

Sage

sage: CLF.gen()
1*I
sage: ComplexField(100)(CLF.gen())                                          # needs sage.rings.number_field
1.0000000000000000000000000000*I

Python

>>> from sage.all import *
>>> CLF.gen()
1*I
>>> ComplexField(Integer(100))(CLF.gen())                                          # needs sage.rings.number_field
1.0000000000000000000000000000*I

interval_field(prec=None)[source]¶

Return the interval field that represents the same mathematical field as self.

EXAMPLES:

Sage

sage: CLF.interval_field()
Complex Interval Field with 53 bits of precision
sage: CLF.interval_field(333)                                               # needs sage.rings.complex_interval_field
Complex Interval Field with 333 bits of precision
sage: CLF.interval_field() is CIF
True

Python

>>> from sage.all import *
>>> CLF.interval_field()
Complex Interval Field with 53 bits of precision
>>> CLF.interval_field(Integer(333))                                               # needs sage.rings.complex_interval_field
Complex Interval Field with 333 bits of precision
>>> CLF.interval_field() is CIF
True

class sage.rings.real_lazy.LazyAlgebraic[source]¶

Bases: LazyFieldElement

This represents an algebraic number, specified by a polynomial over \(\QQ\) and a real or complex approximation.

EXAMPLES:

Sage

sage: x = polygen(QQ)
sage: from sage.rings.real_lazy import LazyAlgebraic
sage: a = LazyAlgebraic(RLF, x^2-2, 1.5)
sage: a
1.414213562373095?

Python

>>> from sage.all import *
>>> x = polygen(QQ)
>>> from sage.rings.real_lazy import LazyAlgebraic
>>> a = LazyAlgebraic(RLF, x**Integer(2)-Integer(2), RealNumber('1.5'))
>>> a
1.414213562373095?

eval(R)[source]¶

Convert self into an element of R.

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyAlgebraic
sage: a = LazyAlgebraic(CLF, QQ['x'].cyclotomic_polynomial(7), 0.6+0.8*CC.0)
sage: a
0.6234898018587335? + 0.7818314824680299?*I
sage: ComplexField(150)(a)  # indirect doctest                              # needs sage.rings.number_field
0.62348980185873353052500488400423981063227473 + 0.78183148246802980870844452667405775023233452*I

sage: a = LazyAlgebraic(CLF, QQ['x'].0^2-7, -2.0)
sage: RR(a)                                                                 # needs sage.rings.number_field
-2.64575131106459
sage: RR(a)^2                                                               # needs sage.rings.number_field
7.00000000000000

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyAlgebraic
>>> a = LazyAlgebraic(CLF, QQ['x'].cyclotomic_polynomial(Integer(7)), RealNumber('0.6')+RealNumber('0.8')*CC.gen(0))
>>> a
0.6234898018587335? + 0.7818314824680299?*I
>>> ComplexField(Integer(150))(a)  # indirect doctest                              # needs sage.rings.number_field
0.62348980185873353052500488400423981063227473 + 0.78183148246802980870844452667405775023233452*I

>>> a = LazyAlgebraic(CLF, QQ['x'].gen(0)**Integer(2)-Integer(7), -RealNumber('2.0'))
>>> RR(a)                                                                 # needs sage.rings.number_field
-2.64575131106459
>>> RR(a)**Integer(2)                                                               # needs sage.rings.number_field
7.00000000000000

class sage.rings.real_lazy.LazyBinop[source]¶

Bases: LazyFieldElement

A lazy element representing a binary (usually arithmetic) operation between two other lazy elements.

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyBinop
sage: a = LazyBinop(RLF, 2, 1/3, operator.add)
sage: a
2.333333333333334?
sage: Reals(200)(a)
2.3333333333333333333333333333333333333333333333333333333333

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyBinop
>>> a = LazyBinop(RLF, Integer(2), Integer(1)/Integer(3), operator.add)
>>> a
2.333333333333334?
>>> Reals(Integer(200))(a)
2.3333333333333333333333333333333333333333333333333333333333

depth()[source]¶

Return the depth of self as an arithmetic expression.

This is the maximum number of dependent intermediate expressions when evaluating self, and is used to determine the precision needed to get the final result to the desired number of bits.

It is equal to the maximum of the right and left depths, plus one.

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyBinop
sage: a = LazyBinop(RLF, 6, 8, operator.mul)
sage: a.depth()
1
sage: b = LazyBinop(RLF, 2, a, operator.sub)
sage: b.depth()
2

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyBinop
>>> a = LazyBinop(RLF, Integer(6), Integer(8), operator.mul)
>>> a.depth()
1
>>> b = LazyBinop(RLF, Integer(2), a, operator.sub)
>>> b.depth()
2

eval(R)[source]¶

Convert the operands to elements of R, then perform the operation on them.

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyBinop
sage: a = LazyBinop(RLF, 6, 8, operator.add)
sage: a.eval(RR)
14.0000000000000

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyBinop
>>> a = LazyBinop(RLF, Integer(6), Integer(8), operator.add)
>>> a.eval(RR)
14.0000000000000

A bit absurd:

Sage

sage: a.eval(str)
'68'

Python

>>> from sage.all import *
>>> a.eval(str)
'68'

class sage.rings.real_lazy.LazyConstant[source]¶

Bases: LazyFieldElement

This class represents a real or complex constant (such as pi or I).

eval(R)[source]¶

Convert self into an element of R.

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyConstant
sage: a = LazyConstant(RLF, 'e')
sage: RDF(a) # indirect doctest
2.718281828459045
sage: a = LazyConstant(CLF, 'I')
sage: CC(a)
1.00000000000000*I

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyConstant
>>> a = LazyConstant(RLF, 'e')
>>> RDF(a) # indirect doctest
2.718281828459045
>>> a = LazyConstant(CLF, 'I')
>>> CC(a)
1.00000000000000*I

class sage.rings.real_lazy.LazyField[source]¶

Bases: Field

The base class for lazy real fields.

Warning

LazyField uses __getattr__(), to implement:

Sage

sage: CLF.pi
3.141592653589794?

Python

>>> from sage.all import *
>>> CLF.pi
3.141592653589794?

I (NT, 20/04/2012) did not manage to have __getattr__ call Parent.__getattr__() in case of failure; hence we can’t use this __getattr__ trick for extension types to recover the methods from categories. Therefore, at this point, no concrete subclass of this class should be an extension type (which is probably just fine):

Sage

sage: RLF.__class__
<class 'sage.rings.real_lazy.RealLazyField_class_with_category'>
sage: CLF.__class__
<class 'sage.rings.real_lazy.ComplexLazyField_class_with_category'>

Python

>>> from sage.all import *
>>> RLF.__class__
<class 'sage.rings.real_lazy.RealLazyField_class_with_category'>
>>> CLF.__class__
<class 'sage.rings.real_lazy.ComplexLazyField_class_with_category'>

Element[source]¶: alias of LazyWrapper

algebraic_closure()[source]¶

Return the algebraic closure of self, i.e., the complex lazy field.

EXAMPLES:

Sage

sage: RLF.algebraic_closure()
Complex Lazy Field

sage: CLF.algebraic_closure()
Complex Lazy Field

Python

>>> from sage.all import *
>>> RLF.algebraic_closure()
Complex Lazy Field

>>> CLF.algebraic_closure()
Complex Lazy Field

interval_field(prec=None)[source]¶: Abstract method to create the corresponding interval field.

class sage.rings.real_lazy.LazyFieldElement[source]¶

Bases: FieldElement

approx()[source]¶

Return self as an element of an interval field.

EXAMPLES:

Sage

sage: CLF(1/6).approx()
0.1666666666666667?
sage: CLF(1/6).approx().parent()
Complex Interval Field with 53 bits of precision

Python

>>> from sage.all import *
>>> CLF(Integer(1)/Integer(6)).approx()
0.1666666666666667?
>>> CLF(Integer(1)/Integer(6)).approx().parent()
Complex Interval Field with 53 bits of precision

When the absolute value is involved, the result might be real:

Sage

sage: # needs sage.symbolic
sage: z = exp(CLF(1 + I/2)); z
2.38551673095914? + 1.303213729686996?*I
sage: r = z.abs(); r
2.71828182845905?
sage: parent(z.approx())
Complex Interval Field with 53 bits of precision
sage: parent(r.approx())
Real Interval Field with 53 bits of precision

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> z = exp(CLF(Integer(1) + I/Integer(2))); z
2.38551673095914? + 1.303213729686996?*I
>>> r = z.abs(); r
2.71828182845905?
>>> parent(z.approx())
Complex Interval Field with 53 bits of precision
>>> parent(r.approx())
Real Interval Field with 53 bits of precision

continued_fraction()[source]¶

Return the continued fraction of self.

EXAMPLES:

Sage

sage: # needs sage.symbolic
sage: a = RLF(sqrt(2)) + RLF(sqrt(3))
sage: cf = a.continued_fraction()
sage: cf
[3; 6, 1, 5, 7, 1, 1, 4, 1, 38, 43, 1, 3, 2, 1, 1, 1, 1, 2, 4, ...]
sage: cf.convergent(100)
444927297812646558239761867973501208151173610180916865469/141414466649174973335183571854340329919207428365474086063

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> a = RLF(sqrt(Integer(2))) + RLF(sqrt(Integer(3)))
>>> cf = a.continued_fraction()
>>> cf
[3; 6, 1, 5, 7, 1, 1, 4, 1, 38, 43, 1, 3, 2, 1, 1, 1, 1, 2, 4, ...]
>>> cf.convergent(Integer(100))
444927297812646558239761867973501208151173610180916865469/141414466649174973335183571854340329919207428365474086063

depth()[source]¶

Abstract method for returning the depth of self as an arithmetic expression.

This is the maximum number of dependent intermediate expressions when evaluating self, and is used to determine the precision needed to get the final result to the desired number of bits.

It is equal to the maximum of the right and left depths, plus one.

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyBinop
sage: a = LazyBinop(RLF, 6, 8, operator.mul)
sage: a.depth()
1

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyBinop
>>> a = LazyBinop(RLF, Integer(6), Integer(8), operator.mul)
>>> a.depth()
1

eval(R)[source]¶

Abstract method for converting self into an element of R.

EXAMPLES:

Sage

sage: a = RLF(12)
sage: a.eval(ZZ)
12

Python

>>> from sage.all import *
>>> a = RLF(Integer(12))
>>> a.eval(ZZ)
12

class sage.rings.real_lazy.LazyNamedUnop[source]¶

Bases: LazyUnop

This class is used to represent the many named methods attached to real numbers, and is instantiated by the __getattr__ method of LazyElements.

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyNamedUnop
sage: a = LazyNamedUnop(RLF, 1, 'arcsin')
sage: RR(a)
1.57079632679490
sage: a = LazyNamedUnop(RLF, 9, 'log', extra_args=(3,))
sage: RR(a)
2.00000000000000

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyNamedUnop
>>> a = LazyNamedUnop(RLF, Integer(1), 'arcsin')
>>> RR(a)
1.57079632679490
>>> a = LazyNamedUnop(RLF, Integer(9), 'log', extra_args=(Integer(3),))
>>> RR(a)
2.00000000000000

approx()[source]¶: Do something reasonable with functions that are not defined on the interval fields.

eval(R)[source]¶: Convert self into an element of R.

class sage.rings.real_lazy.LazyUnop[source]¶

Bases: LazyFieldElement

Represent a unevaluated single function of one variable.

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyUnop
sage: a = LazyUnop(RLF, 3, sqrt); a
1.732050807568878?
sage: a._arg
3
sage: a._op
<function sqrt at ...>
sage: Reals(100)(a)
1.7320508075688772935274463415
sage: Reals(100)(a)^2
3.0000000000000000000000000000

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyUnop
>>> a = LazyUnop(RLF, Integer(3), sqrt); a
1.732050807568878?
>>> a._arg
3
>>> a._op
<function sqrt at ...>
>>> Reals(Integer(100))(a)
1.7320508075688772935274463415
>>> Reals(Integer(100))(a)**Integer(2)
3.0000000000000000000000000000

depth()[source]¶

Return the depth of self as an arithmetic expression.

This is the maximum number of dependent intermediate expressions when evaluating self, and is used to determine the precision needed to get the final result to the desired number of bits.

It is equal to one more than the depth of its operand.

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyUnop
sage: a = LazyUnop(RLF, 3, sqrt)
sage: a.depth()
1
sage: b = LazyUnop(RLF, a, sin)
sage: b.depth()
2

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyUnop
>>> a = LazyUnop(RLF, Integer(3), sqrt)
>>> a.depth()
1
>>> b = LazyUnop(RLF, a, sin)
>>> b.depth()
2

eval(R)[source]¶

Convert self into an element of R.

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyUnop
sage: a = LazyUnop(RLF, 3, sqrt)
sage: a.eval(ZZ)                                                            # needs sage.symbolic
sqrt(3)

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyUnop
>>> a = LazyUnop(RLF, Integer(3), sqrt)
>>> a.eval(ZZ)                                                            # needs sage.symbolic
sqrt(3)

class sage.rings.real_lazy.LazyWrapper[source]¶

Bases: LazyFieldElement

A lazy element that simply wraps an element of another ring.

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyWrapper
sage: a = LazyWrapper(RLF, 3)
sage: a._value
3

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyWrapper
>>> a = LazyWrapper(RLF, Integer(3))
>>> a._value
3

continued_fraction()[source]¶

Return the continued fraction of self.

EXAMPLES:

Sage

sage: a = RLF(sqrt(2))                                                      # needs sage.symbolic
sage: a.continued_fraction()                                                # needs sage.symbolic
[1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...]

Python

>>> from sage.all import *
>>> a = RLF(sqrt(Integer(2)))                                                      # needs sage.symbolic
>>> a.continued_fraction()                                                # needs sage.symbolic
[1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...]

depth()[source]¶

Return the depth of self as an expression, which is always 0.

EXAMPLES:

Sage

sage: RLF(4).depth()
0

Python

>>> from sage.all import *
>>> RLF(Integer(4)).depth()
0

eval(R)[source]¶

Convert self into an element of R.

EXAMPLES:

Sage

sage: a = RLF(12)
sage: a.eval(ZZ)
12
sage: a.eval(ZZ).parent()
Integer Ring

Python

>>> from sage.all import *
>>> a = RLF(Integer(12))
>>> a.eval(ZZ)
12
>>> a.eval(ZZ).parent()
Integer Ring

class sage.rings.real_lazy.LazyWrapperMorphism[source]¶

Bases: Morphism

This morphism coerces elements from anywhere into lazy rings by creating a wrapper element (as fast as possible).

EXAMPLES:

Sage

sage: from sage.rings.real_lazy import LazyWrapperMorphism
sage: f = LazyWrapperMorphism(QQ, RLF)
sage: a = f(3); a
3
sage: type(a)
<class 'sage.rings.real_lazy.LazyWrapper'>
sage: a._value
3
sage: a._value.parent()
Rational Field

Python

>>> from sage.all import *
>>> from sage.rings.real_lazy import LazyWrapperMorphism
>>> f = LazyWrapperMorphism(QQ, RLF)
>>> a = f(Integer(3)); a
3
>>> type(a)
<class 'sage.rings.real_lazy.LazyWrapper'>
>>> a._value
3
>>> a._value.parent()
Rational Field

sage.rings.real_lazy.RealLazyField()[source]¶

Return the lazy real field.

EXAMPLES:

There is only one lazy real field:

Sage

sage: RealLazyField() is RealLazyField()
True

Python

>>> from sage.all import *
>>> RealLazyField() is RealLazyField()
True

class sage.rings.real_lazy.RealLazyField_class[source]¶

Bases: LazyField

This class represents the set of real numbers to unspecified precision. For the most part it simply wraps exact elements and defers evaluation until a specified precision is requested.

Its primary use is to connect the exact rings (such as number fields) to fixed precision real numbers. For example, to specify an embedding of a number field \(K\) into \(\RR\) one can map into this field and the coercion will then be able to carry the mapping to real fields of any precision.

EXAMPLES:

Sage

sage: a = RLF(1/3)
sage: a
0.3333333333333334?
sage: a + 1/5
0.5333333333333334?
sage: a = RLF(1/3)
sage: a
0.3333333333333334?
sage: a + 5
5.333333333333334?
sage: RealField(100)(a+5)
5.3333333333333333333333333333

Python

>>> from sage.all import *
>>> a = RLF(Integer(1)/Integer(3))
>>> a
0.3333333333333334?
>>> a + Integer(1)/Integer(5)
0.5333333333333334?
>>> a = RLF(Integer(1)/Integer(3))
>>> a
0.3333333333333334?
>>> a + Integer(5)
5.333333333333334?
>>> RealField(Integer(100))(a+Integer(5))
5.3333333333333333333333333333

Sage

sage: CC.0 + RLF(1/3)
0.333333333333333 + 1.00000000000000*I
sage: ComplexField(200).0 + RLF(1/3)
0.33333333333333333333333333333333333333333333333333333333333 + 1.0000000000000000000000000000000000000000000000000000000000*I

Python

>>> from sage.all import *
>>> CC.gen(0) + RLF(Integer(1)/Integer(3))
0.333333333333333 + 1.00000000000000*I
>>> ComplexField(Integer(200)).gen(0) + RLF(Integer(1)/Integer(3))
0.33333333333333333333333333333333333333333333333333333333333 + 1.0000000000000000000000000000000000000000000000000000000000*I

construction()[source]¶

Return the functorial construction of self, namely, the completion of the rationals at infinity to infinite precision.

EXAMPLES:

Sage

sage: c, S = RLF.construction(); S
Rational Field
sage: RLF == c(S)
True

Python

>>> from sage.all import *
>>> c, S = RLF.construction(); S
Rational Field
>>> RLF == c(S)
True

gen(i=0)[source]¶

Return the \(i\)-th generator of self.

EXAMPLES:

Sage

sage: RLF.gen()
1

Python

>>> from sage.all import *
>>> RLF.gen()
1

interval_field(prec=None)[source]¶

Return the interval field that represents the same mathematical field as self.

EXAMPLES:

Sage

sage: RLF.interval_field()
Real Interval Field with 53 bits of precision
sage: RLF.interval_field(200)
Real Interval Field with 200 bits of precision

Python

>>> from sage.all import *
>>> RLF.interval_field()
Real Interval Field with 53 bits of precision
>>> RLF.interval_field(Integer(200))
Real Interval Field with 200 bits of precision

sage.rings.real_lazy.make_element(parent, *args)[source]¶

Create an element of parent.

EXAMPLES:

Sage

sage: a = RLF(pi) + RLF(sqrt(1/2))  # indirect doctest                          # needs sage.symbolic
sage: bool(loads(dumps(a)) == a)                                                # needs sage.symbolic
True

Python

>>> from sage.all import *
>>> a = RLF(pi) + RLF(sqrt(Integer(1)/Integer(2)))  # indirect doctest                          # needs sage.symbolic
>>> bool(loads(dumps(a)) == a)                                                # needs sage.symbolic
True