Field of Arbitrary Precision Complex Numbers

AUTHORS:

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

sage.rings.complex_field.ComplexField(prec=53, names=None)

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

EXAMPLES:

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

class sage.rings.complex_field.ComplexField_class(prec=53)

Bases: sage.rings.ring.Field

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.

EXAMPLES:

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(RR.pi())
3.14159265358979
sage: C(RR.log2(), RR.pi())
0.693147180559945 + 3.14159265358979*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())
...
TypeError: unable to coerce to a ComplexNumber: <class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_rational_dense'>

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

algebraic_closure()

Return the algebraic closure of self (which is itself).

EXAMPLES:

sage: CC
Complex Field with 53 bits of precision
sage: CC.algebraic_closure()
Complex Field with 53 bits of precision
sage: CC = ComplexField(1000)
sage: CC.algebraic_closure() is CC
True

characteristic()

construction()

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

EXAMPLES:

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

gen(n=0)

is_exact()

is_field(proof=True)

Return True, since the complex numbers are a field.

EXAMPLES:

sage: CC.is_field()
True

is_finite()

Return False, since the complex numbers are infinite.

EXAMPLES:

sage: CC.is_finite()
False

ngens()

pi()

prec()

precision()

random_element(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]).

EXAMPLES:

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]

scientific_notation(status=None)

to_prec(prec)

Returns the complex field to the specified precision.

EXAMPLES:

sage: CC.to_prec(10)
Complex Field with 10 bits of precision
sage: CC.to_prec(100)
Complex Field with 100 bits of precision

zeta(n=2)

Return a primitive -th root of unity.

INPUT:

• n - an integer (default: 2)

OUTPUT: a complex n-th root of unity.

sage.rings.complex_field.is_ComplexField(x)

sage.rings.complex_field.late_import()