Papers › Inverse of multivector: Beyond p+q=5 threshold
Inverse of multivector: Beyond p+q=5 threshold
A. Acus, A. Dargys
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The algorithm of finding inverse multivector (MV) numerically and symbolically is of paramount importance in the applied Clifford geometric algebra (GA) Cl_(p,q). The first general MV inversion algorithm was based on matrix representation of MV. The complexity of calculations and size of the answer in a symbolic form grow exponentially with the GA dimension n=p+q. The breakthrough occurred when D. Lundholm and then P. Dadbeh found compact inverse formulas up to dimension n≤5. The formulas were constructed in a form of Clifford product of initial MV and its carefully chosen grade-negation counterparts. In this report we show that the grade-negation self-product method can be extended beyond n=5 threshold if, in addition, properly constructed linear combinations of such MV products are used. In particular, we present compact explicit MV inverse formulas for algebras of vector space dimension n=6 and show that they embrace all lower dimensional cases as well. For readers convenience, we have also given various MV formulas in a form of grade negations when n≤5.
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