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SIAM J. COMPUT. Vol. 9, No. 4, November 1980
1980 Society for Industrial and Applied Mathematics 0097-5397/80/0904-0004 $01.00/0
APPROXIMATE SOLUTIONS FOR THE BILINEAR FORM COMPUTATIONAL PROBLEM* DARIO BINH, GRAZIA LOTTI$
AND
FRANCESCO ROMANI
Abstract. A set of bilinear forms can be evaluated with a multiplicative complexity lower than the rank of the associated tensor by allowing an arbitrarily small error. A topological interpretation of this fact is presented together with the error analysis. A complexity measure is introduced which takes into account the numerical stability of algorithms. Relations are established between the complexities of exact and approximate algorithms.
1. Introduction. The nonscalar complexity of a problem is commonly defined as the minimal number of nonscalar multiplications required to solve it exactly by a straight-line algorithm. Here "exactly" means that no error is introduced by the algorithm. We call these Exactly Computing (EC) algorithms. In the computation with a d-digit floating point arithmetic, an error depending on d is introduced in the result. In computing bilinear forms the complexity measure is related to the rank of the associated tensor. We show here that in some cases it is possible to decrease the number of the required multiplications by allowing an arbitrarily small error. We call these Arbitrary Precision Approximating (APA) algorithms. Thus we get algorithms of complexity lower than the rank of the associated tensor which solve the problem with the desired accuracy (superoptimal APA-algorithms). In 2 an example of a superoptimal algorithm is given for a simple problem. In 3 a theoretical interpretation is given related to several notions in topology. The concepts of border tensor, border tensorial basis, and border rank are introduced corresponding to the analogous well known concepts of tensor, tensorial basis, and tensorial rank. In 4 an analysis of the error of APA-algorithms is made. In 5 relations between exact and approximate algorithms are investigated. 2. Superoptimal algorithms. We introduce the concept of superoptimal APAalgorithm with an example. Let
fz(_X, y) x2y q- XlY2, be bilinear forms where _x (x, x2), yr (y, y2). Some results in arithmetic complexity theory (see for example [9]) allow us to show a lower bound of 3 multiplications when we compute (2.1). An optimal bilinear
