28.4 Field of Arbitrary Precision Complex Numbers

Module: sage.rings.complex_field

Field of Arbitrary Precision Complex Numbers

Author: William Stein (2006-01-26): complete rewrite

Module-level Functions

ComplexField( [prec=53], [names=None])

Return the complex field with real and imaginary parts having prec *bits* of precision.

sage: ComplexField()
Complex Field with 53 bits of precision
sage: ComplexField(100)
Complex Field with 100 bits of precision
sage: ComplexField(100).base_ring()
Real Field with 100 bits of precision
sage: i = ComplexField(200).gen()
sage: i^2
-1.0000000000000000000000000000000000000000000000000000000000

is_ComplexField( x)

late_import( )

Class: ComplexField_class

class ComplexField_class

An approximation to the field of complex numbers using floating point numbers with any specified precision. Answers derived from calculations in this approximation may differ from what they would be if those calculations were performed in the true field of complex numbers. This is due to the rounding errors inherent to finite precision calculations.

sage: C = ComplexField(); C
Complex Field with 53 bits of precision
sage: Q = RationalField()
sage: C(1/3)
0.333333333333333
sage: C(1/3, 2)
0.333333333333333 + 2.00000000000000*I
sage: C(RQDF.pi())
3.14159265358979
sage: C(RQDF.log2(), RQDF.e())
0.693147180559945 + 2.71828182845905*I

We can also coerce rational numbers and integers into C, but coercing a polynomial will raise an exception.

sage: Q = RationalField()
sage: C(1/3)
0.333333333333333
sage: S = PolynomialRing(Q, 'x')
sage: C(S.gen())
Traceback (most recent call last):
...
TypeError: unable to coerce to a ComplexNumber

This illustrates precision.

sage: CC = ComplexField(10); CC(1/3, 2/3)
0.33 + 0.67*I
sage: CC
Complex Field with 10 bits of precision
sage: CC = ComplexField(100); CC
Complex Field with 100 bits of precision
sage: z = CC(1/3, 2/3); z
0.33333333333333333333333333333 + 0.66666666666666666666666666667*I

We can load and save complex numbers and the complex field.

sage: loads(z.dumps()) == z
True
sage: loads(CC.dumps()) == CC
True
sage: k = ComplexField(100)
sage: loads(dumps(k)) == k
True

This illustrates basic properties of a complex field.

sage: CC = ComplexField(200)
sage: CC.is_field()
True
sage: CC.characteristic()
0
sage: CC.precision()
200
sage: CC.variable_name()
'I'
sage: CC == ComplexField(200)
True
sage: CC == ComplexField(53)
False
sage: CC == 1.1
False

ComplexField_class( self, [prec=53])

Functions: characteristic, $\,$ construction, $\,$ gen, $\,$ is_exact, $\,$ is_field, $\,$ is_finite, $\,$ ngens, $\,$ pi, $\,$ prec, $\,$ precision, $\,$ random_element, $\,$ scientific_notation, $\,$ zeta

construction( self)

Returns the functorial construction of self, namely, algebraic closure of the real field with the same precision.

sage: c, S = CC.construction(); S
Real Field with 53 bits of precision
sage: CC == c(S)
True

is_field( self)

Return True, since the complex numbers are a field.

sage: CC.is_field()
True

is_finite( self)

Return False, since the complex numbers are infinite.

sage: CC.is_finite()
False

random_element( self, [component_max=1])

Returns a uniformly distributed random number inside a square centered on the origin (by default, the square [-1,1]x[-1,1]).

sage: [CC.random_element() for _ in range(5)]
[-0.306077326077253 - 0.0759291930543202*I, -0.838081254900233 -
0.207006276657392*I, -0.757827933063776 - 0.530834220505783*I,
0.918013195263849 - 0.805114150788948*I, 0.116924427170636 +
0.203592757069680*I]
sage: CC6 = ComplexField(6)
sage: [CC6.random_element(2^-20) for _ in range(5)]
[-5.7e-7 + 5.4e-7*I, 8.6e-7 + 9.2e-7*I, -5.7e-7 + 6.9e-7*I, -1.2e-7 -
6.9e-7*I, 2.7e-7 + 8.3e-7*I]
sage: [CC6.random_element(pi^20) for _ in range(5)]
[-5.0e9*I, 2.8e9 - 5.1e9*I, 2.7e8 + 6.3e9*I, 2.7e8 - 6.4e9*I, 6.7e8 +
1.7e9*I]

zeta( self, [n=2])

Return a primitive -th root of unity.

Input:

n: - an integer (default: 2)

Output: a complex n-th root of unity.

Special Functions: __call__, $\,$ __cmp__, $\,$ __init__, $\,$ __reduce__, $\,$ _coerce_impl, $\,$ _latex_, $\,$ _real_field, $\,$ _repr_

__call__( self, x, [im=None])

sage: CC(2)
2.00000000000000
sage: CC(CC.0)
1.00000000000000*I
sage: CC('1+I')
1.00000000000000 + 1.00000000000000*I
sage: CC(2,3)
2.00000000000000 + 3.00000000000000*I
sage: CC(QQ[I].gen())
1.00000000000000*I
sage: CC.gen() + QQ[I].gen()
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for '+': 'Complex Field with 53
bits of precision' and 'Number Field in I with defining polynomial x^2 + 1'

_coerce_impl( self, x)

Return the canonical coerce of x into this complex field, if it is defined, otherwise raise a TypeError.

The rings that canonicaly coerce to the MPFS complex field are: * this MPFR complex field, or any other of higher precision * anything that canonically coerces to the mpfr real field with this prec

sage: ComplexField(200)(1) + RealField(90)(1)
2.0000000000000000000000000

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