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Quaternion

This module provides a Quaternion type generic over an underlying RealType:

1> import QuaternionModule
2> let q = Quaternion(real: 1, imaginary: 1, 1, 1) // q = 1 + i + j + k

The usual arithmetic operators are provided for Quaternions, many useful properties, plus conformances to the obvious usual protocols: Equatable, Hashable, Codable (if the underlying RealType is), and AlgebraicField (hence also AdditiveArithmetic and SignedNumeric).

Dependencies:

  • RealModule.

The magnitude property

The Numeric protocol requires a .magnitude property, but (deliberately) does not fully specify the semantics. The most obvious choice for Quaternion would be to use the Euclidean norm (aka the "2-norm", given by sqrt(real*real + i*i + k*k + j*j)). However, in practice there are good reasons to use something else instead:

  • The 2-norm requires special care to avoid spurious overflow/underflow, but the naive expressions for the 1-norm ("taxicab norm") or ∞-norm ("sup norm") are always correct.
  • Even when care is used, near the overflow boundary the 2-norm and the 1-norm are not representable. As an example, consider q = Quaternion(big, (big, big, big)), where big is Double.greatestFiniteMagnitude. The 1-norm and 2-norm of q both overflow (the 1-norm would be 4*big, and the 2-norm would be sqrt(4)*big, neither of which are representable as Double), but the ∞-norm is always equal to either real, i, j or k, so it is guaranteed to be representable. Because of this, the ∞-norm is the obvious alternative; it gives the nicest API surface.
  • If we consider the magnitude of more exotic types, like operators, the 1-norm and ∞-norm are significantly easier to compute than the 2-norm (O(n) vs. "no closed form expression, but O(n^3) iterative methods"), so it is nice to establish a precedent of .magnitude binding one of these cheaper-to-compute norms.
  • The ∞-norm is heavily used in other computational libraries; for example, it is used by the izamax and icamax functions in BLAS.

The 2-norm still needs to be available, of course, because sometimes you need it. This functionality is accessed via the .length and .lengthSquared properties.