This document is kept for historic reasons, but may no longer reflect the current state of the latest release of clifford. For the most up to date source of issues, look at the GitHub issue tracker.

  • Currently, algebras over 6 dimensions are very slow. this is because this module was written for pedagogical purposes. However, because the syntax for this module is so attractive, we plan to fix the performance problems, in the future…

  • Due to Python’s order of operations, the bit operators ^ << | are evaluated after the normal arithmetic operators + - * /, which do not follow the precedence expected in GA

    # written        meaning            possibly intended
    1^e1 + 2^e2   == 1^(e1+2)^e2     != (1^e0) + (2^e1)
    e2 + e1|e2    == (e2 + e1)|e2    != e1 + (e1|e2)

    This can also cause confusion within the bitwise operators:

    # written        meaning            possibly intended
    e1 << e2 ^ e1 == (e1 << e2) ^ e1 != e1 << (e2 ^ e1)
    e1 ^ e2 | e1  == (e1 << e2) ^ e1 != e1 << (e2 ^ e1)
  • Since | is the inner product and the inner product with a scalar vanishes by definition, an expression like:

    (1|e0) + (2|e1)

    is null. Use the outer product or full geometric product, to multiply scalars with MultiVectors. This can cause problems if one has code that mixes Python numbers and MultiVectors. If the code multiplies two values that can each be either type without checking, one can run into problems as 1 | 2 has a very different result from the same multiplication with scalar MultiVectors.

  • Taking the inverse of a MultiVector will use a method proposed by Christian Perwass that involves the solution of a matrix equation. A description of that method follows:

    Representing multivectors as \(2^\text{dims}\)-vectors (in the matrix sense), we can carry out the geometric product with a multiplication table. In pseudo-tensorish language (using summation notation)

    \[m_i g_{ijk} n_k = v_j\]

    Suppose \(m_i\) are known (M is the vector we are taking the inverse of), the \(g_{ijk}\) have been computed for this algebra, and \(v_j = 1\) if the \(j\)’th element is the scalar element and 0 otherwise, we can compute the dot product \(m_i g_{ijk}\). This yields a rank-2 matrix. We can then use well-established computational linear algebra techniques to solve this matrix equation for \(n_k\). The laInv method does precisely that.

    The usual, analytic, method for computing inverses (\(M^{-1} = \tilde M/(M \tilde M)\) iff \(M\tilde M = {|M|}^2\)) fails for those multivectors where M*~M is not a scalar. It is onl)y used if the inv method is manually set to point to normalInv.

    My testing suggests that laInv works. In the cases where normalInv works, laInv returns the same result (within _eps). In all cases, M * M.laInv() == 1.0 (within _eps). Use whichever you feel comfortable with.

    Of course, a new issue arises with this method. The inverses found are sometimes dependant on the order of multiplication. That is:

    M.laInv() * M == 1.0
    M * M.laInv() != 1.0

    XXX Thus, there are two other methods defined, leftInv and rightInv which point to leftLaInv and rightLaInv. The method inv points to rightInv. Should the user choose, leftInv and rightInv will both point to normalInv, which yields a left- and right-inverse that are the same should either exist (the proof is fairly simple).

  • The basis vectors of any algebra will be orthonormal unless you supply your own multiplication tables (which you are free to do after the Layout constructor is called). A derived class could be made to calculate these tables for you (and include methods for generating reciprocal bases and the like).

  • No care is taken to preserve the dtype of the arrays. The purpose of this module is pedagogical. If your application requires so many multivectors that storage becomes important, the class structure here is unsuitable for you anyways. Instead, use the algorithms from this module and implement application-specific data structures.

  • Conversely, explicit typecasting is rare. MultiVectors will have integer coefficients if you instantiate them that way. Dividing them by Python integers will have the same consequences as normal integer division. Public outcry will convince me to add the explicit casts if this becomes a problem.

Happy hacking!

Robert Kern