On the substitution of polynomial forms

  • Authors:

  • Ellis Horowitz

  • Affiliations:

  • -

  • Venue:
  • ACM '73 Proceedings of the ACM annual conference
  • Year:
  • 1973

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Abstract

The problem of devising efficient algorithms for computing Q(x1,...,xr-1, P(x1,...,xr-1)) where P and Q are multivariate polynomials is considered. It is shown that for polynomials which are completely dense an algorithm based upon evaluation and interpolation is more efficient than Horner's method. Then various characterizations for sparse polynomials are made and the subsequent methods are re-analyzed. In conclusion a test is devised which takes only linear time to compute and by which a decision can automatically be made concerning whether to use a substitution algorithm which exploits sparsity or one which assumes relatively dense inputs. This choice yields the method which takes the fewest arithmetic operations.