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A unified view of the complexity of evaluation and interpolation

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Summary

Four problems are considered: 1) from an n-precision integer compute its residues modulo n single precision primes; 2) from an n-degree polynomial compute its values at n points; 3) from n residues compute the unique n-precision integer congruent to the residues; 4) from n points compute the unique interpolating polynomial through those points. If M (n) is the time for n-precision integer multiplication, then the time for problems 1 and 2 is shown to be M (n) log n and for problems 3 and 4 to be M (n) (log n) 2. Moreover it is shown that each of the four algorithms are really all instances of the same general algorithm. Finally it is shown how preconditioning or a change of domain will reduce the time for problems 3 and 4 to M (n) (log n).

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Authors and Affiliations

  1. Computer Science Program University of Southern California Powell Hall, Room 224, 90007, Los Angeles, California, USA

    Ellis Horowitz

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  1. Ellis Horowitz
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Additional information

This research was supported by the National Science Foundation under grant GJ-33169.

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Horowitz, E. A unified view of the complexity of evaluation and interpolation. Acta Informatica 3, 123–133 (1974). https://doi.org/10.1007/BF00264033

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  • DOI: https://doi.org/10.1007/BF00264033

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