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]#

Returns the lazy complex field.

EXAMPLES:

There is only one lazy complex field:

sage: ComplexLazyField() is ComplexLazyField()
True

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

>>> 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]#

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

EXAMPLES:

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

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

>>> 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]#

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

EXAMPLES:

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

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

>>> 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: 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

>>> 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: 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

>>> 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: 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

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

>>> 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: a.eval(str)
'68'

>>> 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: 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

>>> 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: CLF.pi
3.141592653589794?

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

>>> 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]#

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

EXAMPLES:

sage: RLF.algebraic_closure()
Complex Lazy Field

sage: CLF.algebraic_closure()
Complex Lazy Field

>>> 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]#

Returns self as an element of an interval field.

EXAMPLES:

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

>>> 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: # 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

>>> 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: # 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

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

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

>>> 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: 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

>>> 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]#

Does 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

Represents a unevaluated single function of one variable.

EXAMPLES:

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

>>> 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: 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

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

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

>>> 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: 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, ...]

>>> 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]#

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

EXAMPLES:

sage: RLF(4).depth()
0

>>> from sage.all import *
>>> RLF(Integer(4)).depth()
0
eval(R)[source]#

Convert self into an element of R.

EXAMPLES:

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

>>> 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: 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

>>> 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: RealLazyField() is RealLazyField()
True

>>> 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: 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

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

>>> 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]#

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

EXAMPLES:

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

>>> 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: RLF.gen()
1

>>> from sage.all import *
>>> RLF.gen()
1
interval_field(prec=None)[source]#

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

EXAMPLES:

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

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

>>> 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