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Cl3 is a Haskell Library implementing standard functions for the Algebra of Physical Space Cl(3,0)
The goal of the Cl3 library is to provide a specialized, safe, high performance, Algebra of Physical Space implementation. This library is suitable for physics simulations. The library integrates into Haskell's standard prelude and has few dependencies. The library uses a ADT data type to specialize to specific graded elements in the Algebra of Physical Space.
The constructors are specialized to single and double grade combinations and the general case of APS. Using the specialized constructors helps the compiler to compile to code similar to that you would hand write. The constructors follow the following conventions for basis.
scalar = R e0 vector = V3 e1 e2 e3 bivector = BV e23 e31 e12 trivectorPseudoScalar = I e123 paravector = PV e0 e1 e2 e3 quarternion = H e0 e23 e31 e12 complex = C e0 e123 biparavector = BPV e1 e2 e3 e23 e31 e12 oddparavector = ODD e1 e2 e3 e123 triparavector = TPV e23 e31 e12 e123 aps = APS e0 e1 e2 e3 e23 e31 e12 e123
In MATLAB or Octave one can write:
sqrt(-25) and get
In standard Haskell
sqrt (-25) will produce
But using the Cl3 library
sqrt (-25) :: Cl3 will produce
I 5.0, and likewise
(I 5.0)^2 will produce
If the unit imaginary is defined as
i = I 1, expressions very similar to MATLAB can be formed
1.2 + 2.3*i will produce
C 1.2 2.3
Vector addition is also natural, two arbitrary vectors
v1 = V3 a1 a2 a3 and
v2 = V3 b1 b2 b3 can be added
v1 + v2 and scaled
The dot product (inner product) of two arbitrary vectors is
toR $ v1 * v2, that is the scalar part of the geometric product of two vectors.
The cross product is the Hodge Dual of the wedge product (outer product)
-i * toBV (v1*v2)
The multiplication of two unit vectors is related to the rotor rotating from
u_to like so
rot = sqrt $ u_to * u_from
Any arbitrary vector can be rotated by a rotor with the equation of
v' = rot * v * dag rot
Rotors can also be formed with an axis unit vector
u and real scalar angle
theta in units of radians, it produces the versor (unit quaternion)
rot = exp $ (-i/2) * theta * u
For special relativity with the velocity vector
v and speed of light scalar
- Beta is
beta = v / c
- Rapidity is
rapidity = atanh beta
- Gamma is
gamma = cosh rapidity
- Composition of velocities is simply adding the two rapidities and converting back to velocity
- Proper Velocity is
w = c * sinh rapidityor
w = gamma * v
- Four Velocity is a paravector
u = exp rapiditywhere the real scalar part is
gamma * cand the vector part is
w / c
- The Boost is
boost = exp $ rapidity / 2
Where e0 is the scalar basis frequently refered to as "1", in other texts.
e1, e2, and e3 are the vector basis of 3 orthonormal vectors.
e23, e31, and e12 are the bivector basis, these are formed by the outer product of two vector basis. For instance in the case of e23, the outer product, or wedge product, is e2 /\ e3, but because this can be simplified to the geometric product of e2 * e3 because the scalar part is zero for orthoginal vector basis'. The geometric product of the two basis vectors is further shortened for brevity to e23.
e123 is the trivector basis, and is formed by the wedge product of e1 /\ e2 /\ e3, and likewise shortened to e123
Multiplication of the basis elements
The basis vectors multiply with the following multiplication table:
Multiplication of the ADT Constructors
The grade specialized type constructors multiply with the following multiplication table:
A benchmark has been developed based on the Haskell entry for the N-Body Benchmark in the The Computer Language Benchmarks Game with some modifications to run with Criterion. On my machine the current fastest implementation completes 50M steps with a mean time of 13.35 seconds. The benchmark uses a hand rolled implementation of vector math. The Cl3 implementation completes 50M steps with a mean time of 15.73 seconds. This 2.38 second difference amounts to a 23.8 ns difference in the inner loop.
The design space for Clifford Algebra libraries was explored quite a bit before the development of this library. Initially the isomorphism of APS with 2x2 Complex Matrices was used, this had the draw back that multiplying the scalar 2 * 2 would incur all of the computational cost of multiplying two 2x2 complex matrices.
Then the design was changed to lists that contained the basis' values, but lists are computationally slow and do not produce well optimized code.
Then a single constructor data type for APS was developed, but this had all of the drawbacks of 2x2 complex matrices.
The specialized ADT Constructor version of the library was developed and it showed that it had some promise.
More of the design space was explored, a version of the Cl3 library was developed using Multi-parameter Type Classes and Functional Dependencies, this didn't appear to have much gained over the specialized ADT Syntax interface and it didn't use the standard Prelude classes like Num, Float, etc. It was also difficult for me to figure out how to code a
So the specialized ADT Constructor design of the Cl3 library was finished and released.
How does this fit in with the existing Haskell ecosystem?
Cl3 is meant to be a Linear killer based on Geometric Algebra. The linear package consists of many different types that are not easily combinable using the Num Class, and require many specialized functions each to multiply a different combination of types.
The clif is for symbolic computing using symbolic and numeric computations with finite and infinite-dimensional Clifford algebras arising from arbitrary bilinear forms. The libraries representation of a Cliffor also makes use of lists.