Struct Complex64Base

pub struct Complex64Base;

An approximate implementation of the complex numbers C, using 64 bit floating point numbers.

Warning

Since floating point numbers do not exactly represent the complex numbers, and this crate follows a mathematically precise approach, we cannot provide any function related to equality. In particular, Complex64Base.eq_el(a, b) is not supported, and will panic. Hence, this ring has only limited use within this crate, and is currently only used for floating-point FFTs.

Implementations

impl Complex64Base

pub fn abs(&self, Complex64El: Complex64El) -> f64

pub fn conjugate(&self, Complex64El: Complex64El) -> Complex64El

pub fn exp(&self, Complex64El: Complex64El) -> Complex64El

pub fn closest_gaussian_int(&self, Complex64El: Complex64El) -> (i64, i64)

pub fn ln_main_branch(&self, Complex64El: Complex64El) -> Complex64El

pub fn is_absolute_approx_eq( &self, lhs: Complex64El, rhs: Complex64El, absolute_threshold: f64, ) -> bool

pub fn is_relative_approx_eq( &self, lhs: Complex64El, rhs: Complex64El, relative_limit: f64, ) -> bool

pub fn is_approx_eq( &self, lhs: Complex64El, rhs: Complex64El, precision: u64, ) -> bool

pub fn from_f64(&self, x: f64) -> Complex64El

pub fn root_of_unity(&self, i: i64, n: i64) -> Complex64El

pub fn re(&self, Complex64El: Complex64El) -> f64

pub fn im(&self, Complex64El: Complex64El) -> f64

Trait Implementations

impl CanHomFrom<Complex64Base> for Complex64Base

type Homomorphism = ()

fn has_canonical_hom(&self, from: &Self) -> Option<()>

fn map_in( &self, _from: &Self, el: <Self as RingBase>::Element, _: &Self::Homomorphism, ) -> <Self as RingBase>::Element

fn map_in_ref( &self, from: &S, el: &S::Element, hom: &Self::Homomorphism, ) -> Self::Element

fn mul_assign_map_in( &self, from: &S, lhs: &mut Self::Element, rhs: S::Element, hom: &Self::Homomorphism, )

fn mul_assign_map_in_ref( &self, from: &S, lhs: &mut Self::Element, rhs: &S::Element, hom: &Self::Homomorphism, )

impl<I: ?Sized + IntegerRing> CanHomFrom<I> for Complex64Base

type Homomorphism = ()

fn has_canonical_hom(&self, _from: &I) -> Option<Self::Homomorphism>

fn map_in( &self, from: &I, el: <I as RingBase>::Element, hom: &Self::Homomorphism, ) -> Self::Element

fn map_in_ref( &self, from: &I, el: &<I as RingBase>::Element, _hom: &Self::Homomorphism, ) -> Self::Element

fn mul_assign_map_in( &self, from: &S, lhs: &mut Self::Element, rhs: S::Element, hom: &Self::Homomorphism, )

fn mul_assign_map_in_ref( &self, from: &S, lhs: &mut Self::Element, rhs: &S::Element, hom: &Self::Homomorphism, )

impl<I> CanHomFrom<RationalFieldBase<I>> for Complex64Base

where I: IntegerRingStore, I::Type: IntegerRing,

type Homomorphism = <Complex64Base as CanHomFrom<<I as RingStore>::Type>>::Homomorphism

fn has_canonical_hom( &self, from: &RationalFieldBase<I>, ) -> Option<Self::Homomorphism>

fn map_in( &self, from: &RationalFieldBase<I>, el: <RationalFieldBase<I> as RingBase>::Element, hom: &Self::Homomorphism, ) -> Self::Element

fn map_in_ref( &self, from: &RationalFieldBase<I>, el: &<RationalFieldBase<I> as RingBase>::Element, hom: &Self::Homomorphism, ) -> Self::Element

fn mul_assign_map_in( &self, from: &S, lhs: &mut Self::Element, rhs: S::Element, hom: &Self::Homomorphism, )

fn mul_assign_map_in_ref( &self, from: &S, lhs: &mut Self::Element, rhs: &S::Element, hom: &Self::Homomorphism, )

impl CanIsoFromTo<Complex64Base> for Complex64Base

type Isomorphism = ()

fn has_canonical_iso(&self, from: &Self) -> Option<()>

fn map_out( &self, _from: &Self, el: <Self as RingBase>::Element, _: &Self::Homomorphism, ) -> <Self as RingBase>::Element

impl Clone for Complex64Base

fn clone(&self) -> Complex64Base

Returns a copy of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl DivisibilityRing for Complex64Base

fn checked_left_div( &self, lhs: &Self::Element, rhs: &Self::Element, ) -> Option<Self::Element>

Checks whether there is an element x such that rhs * x = lhs, and returns it if it exists.

type PreparedDivisorData = ()

Additional data associated to a fixed ring element that can be used to speed up division by this ring element.

fn divides_left(&self, lhs: &Self::Element, rhs: &Self::Element) -> bool

Returns whether there is an element x such that rhs * x = lhs. If you need such an element, consider using DivisibilityRing::checked_left_div().

fn divides(&self, lhs: &Self::Element, rhs: &Self::Element) -> bool

Same as DivisibilityRing::divides_left(), but requires a commutative ring.

fn checked_div( &self, lhs: &Self::Element, rhs: &Self::Element, ) -> Option<Self::Element>

Same as DivisibilityRing::checked_left_div(), but requires a commutative ring.

fn is_unit(&self, x: &Self::Element) -> bool

Returns whether the given element is a unit, i.e. has an inverse.

fn balance_factor<'a, I>(&self, _elements: I) -> Option<Self::Element>

where I: Iterator<Item = &'a Self::Element>, Self: 'a,

Function that computes a “balancing” factor of a sequence of ring elements. The only use of the balancing factor is to increase performance, in particular, dividing all elements in the sequence by this factor should make them “smaller” resp. cheaper to process.

fn prepare_divisor(&self, x: Self::Element) -> PreparedDivisor<Self>

“Prepares” an element of this ring for division.

fn checked_left_div_prepared( &self, lhs: &Self::Element, rhs: &PreparedDivisor<Self>, ) -> Option<Self::Element>

Same as DivisibilityRing::checked_left_div() but for a prepared divisor.

fn divides_left_prepared( &self, lhs: &Self::Element, rhs: &PreparedDivisor<Self>, ) -> bool

Same as DivisibilityRing::divides_left() but for a prepared divisor.

fn is_unit_prepared(&self, x: &PreparedDivisor<Self>) -> bool

Same as DivisibilityRing::is_unit() but for a prepared divisor.

fn invert(&self, el: &Self::Element) -> Option<Self::Element>

If the given element is a unit, returns its inverse, otherwise None.

impl EuclideanRing for Complex64Base

fn euclidean_div_rem( &self, _lhs: Self::Element, _rhs: &Self::Element, ) -> (Self::Element, Self::Element)

Computes euclidean division with remainder.

fn euclidean_deg(&self, _: &Self::Element) -> Option<usize>

Defines how “small” an element is. For details, see EuclideanRing.

fn euclidean_div( &self, lhs: Self::Element, rhs: &Self::Element, ) -> Self::Element

Computes euclidean division without remainder.

fn euclidean_rem( &self, lhs: Self::Element, rhs: &Self::Element, ) -> Self::Element

Computes only the remainder of euclidean division.

impl Field for Complex64Base

fn div(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

impl PartialEq for Complex64Base

fn eq(&self, other: &Complex64Base) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PrincipalIdealRing for Complex64Base

fn checked_div_min( &self, lhs: &Self::Element, rhs: &Self::Element, ) -> Option<Self::Element>

Similar to DivisibilityRing::checked_left_div() this computes a “quotient” q of lhs and rhs, if it exists. However, we impose the additional constraint that this quotient be minimal, i.e. there is no q' with q' | q properly and q' * rhs = lhs.

fn extended_ideal_gen( &self, _lhs: &Self::Element, _rhs: &Self::Element, ) -> (Self::Element, Self::Element, Self::Element)

Computes a Bezout identity for the generator g of the ideal (lhs, rhs) as g = s * lhs + t * rhs.

fn annihilator(&self, val: &Self::Element) -> Self::Element

Returns the (w.r.t. divisibility) smallest element x such that x * val = 0.

fn create_elimination_matrix( &self, a: &Self::Element, b: &Self::Element, ) -> ([Self::Element; 4], Self::Element)

Creates a matrix A of unit determinant such that A * (a, b)^T = (d, 0). Returns (A, d).

fn ideal_gen(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

Computes a generator g of the ideal (lhs, rhs) = (g), also known as greatest common divisor.

fn lcm(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

Computes a generator of the ideal (lhs) ∩ (rhs), also known as least common multiple.

impl RingBase for Complex64Base

type Element = Complex64El

Type of elements of the ring

fn clone_el(&self, val: &Self::Element) -> Self::Element

fn add_assign( &self, Complex64El: &mut Self::Element, Complex64El: Self::Element, )

fn negate_inplace(&self, Complex64El: &mut Self::Element)

fn mul_assign( &self, Complex64El: &mut Self::Element, Complex64El: Self::Element, )

fn from_int(&self, value: i32) -> Self::Element

fn eq_el(&self, _: &Self::Element, _: &Self::Element) -> bool

fn pow_gen<R: IntegerRingStore>( &self, x: Self::Element, power: &El<R>, integers: R, ) -> Self::Element

where R::Type: IntegerRing,

Raises x to the power of an arbitrary, nonnegative integer given by a custom integer ring implementation.

fn is_commutative(&self) -> bool

Returns whether the ring is commutative, i.e. a * b = b * a for all elements a, b. Note that addition is assumed to be always commutative.

fn is_noetherian(&self) -> bool

Returns whether the ring is noetherian, i.e. every ideal is finitely generated.

fn is_approximate(&self) -> bool

Returns whether this ring computes with approximations to elements. This would usually be the case for rings that are based on f32 or f64, to represent real or complex numbers.

fn dbg_within<'a>( &self, Complex64El: &Self::Element, out: &mut Formatter<'a>, env: EnvBindingStrength, ) -> Result

Writes a human-readable representation of value to out, taking into account the possible context to place parenthesis as needed.

fn dbg<'a>(&self, value: &Self::Element, out: &mut Formatter<'a>) -> Result

Writes a human-readable representation of value to out.

fn characteristic<I: IntegerRingStore + Copy>(&self, ZZ: I) -> Option<El<I>>

where I::Type: IntegerRing,

Returns the characteristic of this ring as an element of the given implementation of ZZ.

fn add_assign_ref(&self, lhs: &mut Self::Element, rhs: &Self::Element)

fn sub_assign_ref(&self, lhs: &mut Self::Element, rhs: &Self::Element)

fn mul_assign_ref(&self, lhs: &mut Self::Element, rhs: &Self::Element)

fn zero(&self) -> Self::Element

fn one(&self) -> Self::Element

fn neg_one(&self) -> Self::Element

fn is_zero(&self, value: &Self::Element) -> bool

fn is_one(&self, value: &Self::Element) -> bool

fn is_neg_one(&self, value: &Self::Element) -> bool

fn square(&self, value: &mut Self::Element)

fn negate(&self, value: Self::Element) -> Self::Element

fn sub_assign(&self, lhs: &mut Self::Element, rhs: Self::Element)

fn mul_assign_int(&self, lhs: &mut Self::Element, rhs: i32)

fn mul_int(&self, lhs: Self::Element, rhs: i32) -> Self::Element

fn mul_int_ref(&self, lhs: &Self::Element, rhs: i32) -> Self::Element

fn sub_self_assign(&self, lhs: &mut Self::Element, rhs: Self::Element)

Computes lhs := rhs - lhs.

fn sub_self_assign_ref(&self, lhs: &mut Self::Element, rhs: &Self::Element)

Computes lhs := rhs - lhs.

fn add_ref(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

fn add_ref_fst(&self, lhs: &Self::Element, rhs: Self::Element) -> Self::Element

fn add_ref_snd(&self, lhs: Self::Element, rhs: &Self::Element) -> Self::Element

fn add(&self, lhs: Self::Element, rhs: Self::Element) -> Self::Element

fn sub_ref(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

fn sub_ref_fst(&self, lhs: &Self::Element, rhs: Self::Element) -> Self::Element

fn sub_ref_snd(&self, lhs: Self::Element, rhs: &Self::Element) -> Self::Element

fn sub(&self, lhs: Self::Element, rhs: Self::Element) -> Self::Element

fn mul_ref(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element

fn mul_ref_fst(&self, lhs: &Self::Element, rhs: Self::Element) -> Self::Element

fn mul_ref_snd(&self, lhs: Self::Element, rhs: &Self::Element) -> Self::Element

fn mul(&self, lhs: Self::Element, rhs: Self::Element) -> Self::Element

fn sum<I>(&self, els: I) -> Self::Element

where I: IntoIterator<Item = Self::Element>,

Sums the elements given by the iterator.

fn prod<I>(&self, els: I) -> Self::Element

where I: IntoIterator<Item = Self::Element>,

Computes the product of the elements given by the iterator.

impl RingExtension for Complex64Base

type BaseRing = RingValue<Real64Base>

Type of the base ring;

fn base_ring<'a>(&'a self) -> &'a Self::BaseRing

Returns a reference to the base ring.

fn from(&self, x: El<Self::BaseRing>) -> Self::Element

Maps an element of the base ring into this ring.

fn mul_assign_base(&self, lhs: &mut Self::Element, rhs: &El<Self::BaseRing>)

Computes lhs := lhs * rhs, where rhs is mapped into this ring via RingExtension::from_ref(). Note that this may be faster than self.mul_assign(lhs, self.from_ref(rhs)).

fn from_ref(&self, x: &El<Self::BaseRing>) -> Self::Element

Maps an element of the base ring (given as reference) into this ring.

impl Copy for Complex64Base

impl Domain for Complex64Base

impl StructuralPartialEq for Complex64Base