The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
On the complexity of bilinear forms with commutativity
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Synthesis and Optimization of 2D Filter Designs for Heterogeneous FPGAs
ACM Transactions on Reconfigurable Technology and Systems (TRETS)
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A large class of multiplication problems in arithmetic complexity can be viewed as the simultaneous evaluation of a set of bilinear forms. This class includes the multiplication of matrices, polynomials, quaternions, Cayley and complex numbers. Considering bilinear algorithms, the optimal number of non-scalar multiplications can be described as the rank of a three-tensor or as the smallest number of rank one matrices necessary to include a given set of matrices in their span. In this paper, we attack a rather large subclass of three-tensors, namely that of (p, q, 2) tensors, for arbitrary p and q, and solve it completely. The complexity of a general pair of bilinear forms is determined explicitly in terms of parameters related to Kronecker's theory of pencils and to the theory of invariant polynomials. This reveals unexpected results and shows explicitly the dependence on the algebraic structure of the constants; we display, for example, a pair of n x n bilinear forms whose complexity is n + 1 over infinite fields and which, however, requires exactly n+[(n-1) ¦k] multiplications over a finite field with cardinality k. Another consequence of our results is that any set of m, p x q, bilinear forms require at most m/2(min(p+q/2, q+p/2)) multiplications over any field.