Struct MultivectorHilbertSpace 

pub struct MultivectorHilbertSpace<const P: usize, const Q: usize, const R: usize> { /* private fields */ }

A Hilbert space of multivector elements.

This represents the finite-dimensional Hilbert space Cl(P,Q,R) with the standard inner product inherited from the coefficient representation.

Type Parameters

• P - Number of positive signature basis vectors
• Q - Number of negative signature basis vectors
• R - Number of zero signature basis vectors

Mathematical Background

The Clifford algebra Cl(P,Q,R) is a 2^(P+Q+R)-dimensional real vector space. We equip it with the standard L² inner product on the coefficients:

⟨x, y⟩ = Σᵢ xᵢ yᵢ

This makes Cl(P,Q,R) into a finite-dimensional Hilbert space.

Implementations

impl<const P: usize, const Q: usize, const R: usize> MultivectorHilbertSpace<P, Q, R>

pub const DIM: usize

The dimension of the Clifford algebra.

pub fn new() -> Self

Create a new multivector Hilbert space.

pub fn signature(&self) -> (usize, usize, usize)

Get the signature (P, Q, R) of the algebra.

pub fn algebra_dimension(&self) -> usize

Get the dimension of the algebra.

pub fn from_coefficients( &self, coefficients: &[f64], ) -> Result<Multivector<P, Q, R>>

Create a multivector from coefficients.

pub fn to_coefficients(&self, mv: &Multivector<P, Q, R>) -> Vec<f64>

Get the coefficients of a multivector.

pub fn basis_vector(&self, index: usize) -> Result<Multivector<P, Q, R>>

Create a basis vector (unit multivector in direction i).

pub fn basis(&self) -> Vec<Multivector<P, Q, R>>

Get all basis vectors.

Trait Implementations

impl<const P: usize, const Q: usize, const R: usize> BanachSpace<Multivector<P, Q, R>> for MultivectorHilbertSpace<P, Q, R>

fn is_cauchy_sequence( &self, sequence: &[Multivector<P, Q, R>], tolerance: f64, ) -> bool

Check if a sequence appears to be Cauchy.

fn sequence_limit( &self, sequence: &[Multivector<P, Q, R>], tolerance: f64, ) -> Result<Multivector<P, Q, R>>

Compute the limit of a Cauchy sequence if it exists.

impl<const P: usize, const Q: usize, const R: usize> Clone for MultivectorHilbertSpace<P, Q, R>

fn clone(&self) -> MultivectorHilbertSpace<P, Q, R>

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl<const P: usize, const Q: usize, const R: usize> Debug for MultivectorHilbertSpace<P, Q, R>

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

Formats the value using the given formatter.

impl<const P: usize, const Q: usize, const R: usize> Default for MultivectorHilbertSpace<P, Q, R>

fn default() -> Self

Returns the “default value” for a type.

impl<const P: usize, const Q: usize, const R: usize> HilbertSpace<Multivector<P, Q, R>> for MultivectorHilbertSpace<P, Q, R>

fn riesz_representative<F>(&self, functional: F) -> Result<Multivector<P, Q, R>>

where F: Fn(&Multivector<P, Q, R>) -> f64,

Apply the Riesz representation theorem.

fn orthogonal_complement_projection(&self, x: &V, subspace_basis: &[V]) -> V

where S: Into<f64> + From<f64>, V: Clone,

Compute the orthogonal complement projection.

fn best_approximation(&self, x: &V, subspace_basis: &[V]) -> V

where S: Into<f64> + From<f64>, V: Clone,

Compute the best approximation to x from a subspace.

fn is_orthonormal(&self, vectors: &[V], tolerance: f64) -> bool

where S: Into<f64>,

Check if a set of vectors forms an orthonormal system.

impl<const P: usize, const Q: usize, const R: usize> InnerProductSpace<Multivector<P, Q, R>> for MultivectorHilbertSpace<P, Q, R>

fn inner_product( &self, x: &Multivector<P, Q, R>, y: &Multivector<P, Q, R>, ) -> f64

Compute the inner product of two vectors.

fn project( &self, x: &Multivector<P, Q, R>, y: &Multivector<P, Q, R>, ) -> Multivector<P, Q, R>

Project x onto y.

fn gram_schmidt( &self, vectors: &[Multivector<P, Q, R>], ) -> Vec<Multivector<P, Q, R>>

Gram-Schmidt orthogonalization of a set of vectors.

fn are_orthogonal(&self, x: &V, y: &V, tolerance: f64) -> bool

where S: Into<f64>,

Check if two vectors are orthogonal.

impl<const P: usize, const Q: usize, const R: usize> NormedSpace<Multivector<P, Q, R>> for MultivectorHilbertSpace<P, Q, R>

fn norm(&self, x: &Multivector<P, Q, R>) -> f64

Compute the norm of a vector.

fn normalize(&self, x: &Multivector<P, Q, R>) -> Option<Multivector<P, Q, R>>

Normalize a vector to unit length.

fn distance(&self, x: &V, y: &V) -> f64

Compute the distance between two vectors.

impl<const P: usize, const Q: usize, const R: usize> VectorSpace<Multivector<P, Q, R>> for MultivectorHilbertSpace<P, Q, R>

fn add( &self, x: &Multivector<P, Q, R>, y: &Multivector<P, Q, R>, ) -> Multivector<P, Q, R>

Add two vectors.

fn sub( &self, x: &Multivector<P, Q, R>, y: &Multivector<P, Q, R>, ) -> Multivector<P, Q, R>

Subtract two vectors.

fn scale(&self, scalar: f64, x: &Multivector<P, Q, R>) -> Multivector<P, Q, R>

Multiply a vector by a scalar.

fn zero(&self) -> Multivector<P, Q, R>

Return the zero vector.

fn dimension(&self) -> Option<usize>

Return the dimension of the space (may be infinite).