Trait NormedVectorSpace

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pub trait NormedVectorSpace: VectorSpace {
    // Required method
    fn norm(self) -> Self::Scalar;

    // Provided methods
    fn norm_squared(self) -> Self::Scalar { ... }
    fn distance(self, rhs: Self) -> Self::Scalar { ... }
    fn distance_squared(self, rhs: Self) -> Self::Scalar { ... }
}

Expand description

A type that supports the operations of a normed vector space; i.e. a norm operation in addition to those of VectorSpace. Specifically, the implementor must guarantee that the following relationships hold, within the limitations of floating point arithmetic:

(Nonnegativity) For all v: Self, v.norm() >= 0.0.
(Positive definiteness) For all v: Self, v.norm() == 0.0 implies v == Self::ZERO.
(Absolute homogeneity) For all c: Self::Scalar, v: Self, (v * c).norm() == v.norm() * c.abs().
(Triangle inequality) For all v, w: Self, (v + w).norm() <= v.norm() + w.norm().

Note that, because implementing types use floating point arithmetic, they are not required to actually implement PartialEq or Eq.

Required Methods§

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fn norm(self) -> Self::Scalar

The size of this element. The return value should always be nonnegative.

Provided Methods§

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fn norm_squared(self) -> Self::Scalar

The squared norm of this element. Computing this is often faster than computing NormedVectorSpace::norm.

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fn distance(self, rhs: Self) -> Self::Scalar

The distance between this element and another, as determined by the norm.

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fn distance_squared(self, rhs: Self) -> Self::Scalar

The squared distance between this element and another, as determined by the norm. Note that this is often faster to compute in practice than NormedVectorSpace::distance.