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 = ()

Data required to compute the action of the canonical homomorphism on ring elements.

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

If there is a canonical homomorphism from -> self, returns Some(data), where data is additional data that can be used to compute the action of the homomorphism on ring elements. Otherwise, None is returned.

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

Evaluates the homomorphism.

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

Evaluates the homomorphism, taking the element by reference.

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

Evaluates the homomorphism on rhs, and multiplies the result to lhs.

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

Evaluates the homomorphism on rhs, taking it by reference, and multiplies the result to lhs.

fn fma_map_in( &self, from: &S, lhs: &Self::Element, rhs: &S::Element, summand: Self::Element, hom: &Self::Homomorphism, ) -> Self::Element

Fused-multiply-add. Computes summand + lhs * rhs, where rhs is mapped into the ring via the homomorphism.

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

type Homomorphism = ()

Data required to compute the action of the canonical homomorphism on ring elements.

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

If there is a canonical homomorphism from -> self, returns Some(data), where data is additional data that can be used to compute the action of the homomorphism on ring elements. Otherwise, None is returned.

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

Evaluates the homomorphism.

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

Evaluates the homomorphism, taking the element by reference.

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

Evaluates the homomorphism on rhs, and multiplies the result to lhs.

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

Evaluates the homomorphism on rhs, taking it by reference, and multiplies the result to lhs.

fn fma_map_in( &self, from: &S, lhs: &Self::Element, rhs: &S::Element, summand: Self::Element, hom: &Self::Homomorphism, ) -> Self::Element

Fused-multiply-add. Computes summand + lhs * rhs, where rhs is mapped into the ring via the homomorphism.

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

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

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

Data required to compute the action of the canonical homomorphism on ring elements.

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

If there is a canonical homomorphism from -> self, returns Some(data), where data is additional data that can be used to compute the action of the homomorphism on ring elements. Otherwise, None is returned.

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

Evaluates the homomorphism.

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

Evaluates the homomorphism, taking the element by reference.

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

Evaluates the homomorphism on rhs, and multiplies the result to lhs.

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

Evaluates the homomorphism on rhs, taking it by reference, and multiplies the result to lhs.

fn fma_map_in( &self, from: &S, lhs: &Self::Element, rhs: &S::Element, summand: Self::Element, hom: &Self::Homomorphism, ) -> Self::Element

Fused-multiply-add. Computes summand + lhs * rhs, where rhs is mapped into the ring via the homomorphism.

impl CanIsoFromTo<Complex64Base> for Complex64Base

type Isomorphism = ()

Data required to compute a preimage under the canonical homomorphism.

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

If there is a canonical homomorphism from -> self, and this homomorphism is an isomorphism, returns Some(data), where data is additional data that can be used to compute preimages under the homomorphism. Otherwise, None is returned.

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

Computes the preimage of el under the canonical homomorphism from -> self.

impl Clone for Complex64Base

fn clone(&self) -> Complex64Base

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for Complex64Base

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

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, _: &Self::Element) -> Self::PreparedDivisorData

“Prepares” an element of this ring for division.

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

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

fn divides_left_prepared( &self, lhs: &Self::Element, rhs: &Self::Element, _rhs_prep: &Self::PreparedDivisorData, ) -> 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

Computes the division lhs / rhs, where rhs != 0.

impl<K, Impl, I> Homomorphism<NumberFieldBase<Impl, I>, Complex64Base> for ComplexEmbedding<K, Impl, I>

where K: RingStore<Type = NumberFieldBase<Impl, I>>, Impl: RingStore, Impl::Type: Field + FreeAlgebra, <Impl::Type as RingExtension>::BaseRing: RingStore<Type = RationalFieldBase<I>>, I: RingStore, I::Type: IntegerRing,

type DomainStore = K

The type of the RingStore used by this object to store the domain ring.

type CodomainStore = RingValue<Complex64Base>

The type of the RingStore used by this object to store the codomain ring.

fn codomain<'a>(&'a self) -> &'a Self::CodomainStore

Returns a reference to the codomain ring.

fn domain<'a>(&'a self) -> &'a Self::DomainStore

Returns a reference to the domain ring.

fn map_ref( &self, x: &<NumberFieldBase<Impl, I> as RingBase>::Element, ) -> <Complex64Base as RingBase>::Element

Applies this homomorphism to the given element from the domain ring, resulting in an element in the codomain ring.

fn map( &self, x: <NumberFieldBase<Impl, I> as RingBase>::Element, ) -> <Complex64Base as RingBase>::Element

Applies this homomorphism to the given element from the domain ring, resulting in an element in the codomain ring.

fn mul_assign_map(&self, lhs: &mut Codomain::Element, rhs: Domain::Element)

Multiplies the given element in the codomain ring with an element obtained by applying this homomorphism to a given element from the domain ring.

fn mul_assign_ref_map(&self, lhs: &mut Codomain::Element, rhs: &Domain::Element)

Multiplies the given element in the codomain ring with an element obtained by applying this homomorphism to a given element from the domain ring.

fn mul_map( &self, lhs: Codomain::Element, rhs: Domain::Element, ) -> Codomain::Element

Multiplies the given element in the codomain ring with an element obtained by applying this homomorphism to a given element from the domain ring.

fn fma_map( &self, lhs: &Codomain::Element, rhs: &Domain::Element, summand: Codomain::Element, ) -> Codomain::Element

Fused-multiply-add. This computes lhs * rhs + summand, where rhs is mapped into the ring via this homomorphism.

fn mul_ref_fst_map( &self, lhs: &Codomain::Element, rhs: Domain::Element, ) -> Codomain::Element

Multiplies the given element in the codomain ring with an element obtained by applying this homomorphism to a given element from the domain ring.

fn mul_ref_snd_map( &self, lhs: Codomain::Element, rhs: &Domain::Element, ) -> Codomain::Element

Multiplies the given element in the codomain ring with an element obtained by applying this homomorphism to a given element from the domain ring.

fn mul_ref_map( &self, lhs: &Codomain::Element, rhs: &Domain::Element, ) -> Codomain::Element

Multiplies the given element in the codomain ring with an element obtained by applying this homomorphism to a given element from the domain ring.

fn compose<F, PrevDomain: ?Sized + RingBase>( self, prev: F, ) -> ComposedHom<PrevDomain, Domain, Codomain, F, Self>

where Self: Sized, F: Homomorphism<PrevDomain, Domain>,

Constructs the homomorphism x -> self.map(prev.map(x)).

fn mul_assign_ref_map_through_hom<First: ?Sized + RingBase, H: Homomorphism<First, Domain>>( &self, lhs: &mut Codomain::Element, rhs: &First::Element, hom: H, )

Multiplies the given element in the codomain ring with an element obtained by applying this and another homomorphism to a given element from another ring.

fn mul_assign_map_through_hom<First: ?Sized + RingBase, H: Homomorphism<First, Domain>>( &self, lhs: &mut Codomain::Element, rhs: First::Element, hom: H, )

Multiplies the given element in the codomain ring with an element obtained by applying this and another homomorphism to a given element from another ring.

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 ideal_gen_with_controller<Controller>( &self, lhs: &Self::Element, rhs: &Self::Element, _: Controller, ) -> Self::Element

where Controller: ComputationController,

As PrincipalIdealRing::ideal_gen(), this computes a generator of the ideal (lhs, rhs). However, it additionally accepts a ComputationController to customize the performed computation.

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 fma( &self, lhs: &Self::Element, rhs: &Self::Element, summand: Self::Element, ) -> Self::Element

Fused-multiply-add. This computes summand + lhs * rhs.

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 fma_int( &self, lhs: &Self::Element, rhs: i32, summand: Self::Element, ) -> Self::Element

Fused-multiply-add with an integer. This computes summand + lhs * rhs.

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.

fn fma_base( &self, lhs: &Self::Element, rhs: &El<Self::BaseRing>, summand: Self::Element, ) -> Self::Element

fn mul_assign_base_through_hom<S: ?Sized + RingBase, H: Homomorphism<S, <Self::BaseRing as RingStore>::Type>>( &self, lhs: &mut Self::Element, rhs: &S::Element, hom: H, )

Computes lhs := lhs * rhs, where rhs is mapped into this ring via the given homomorphism, followed by the inclusion (as specified by RingExtension::from_ref()).

impl Copy for Complex64Base

impl Domain for Complex64Base

impl StructuralPartialEq for Complex64Base