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Efficient Polynomial Evaluation: Methods and Techniques

This document explores various methods for polynomial evaluation, detailing straightforward evaluation, smarter techniques reducing multiplications and additions, and the use of Horner's rule. It emphasizes the efficiency of each method, from a basic polynomial evaluation using multiple multiplications and additions to advanced methods like exponentiation by squaring and factorization. By comparing the number of operations required, this guide aids developers and mathematicians in selecting the best approach for polynomial computations.

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Efficient Polynomial Evaluation: Methods and Techniques

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  1. Polynomial Evaluation

  2. Straightforward Evaluation • P(x) = 3x5+2x4+7x3+8x2+2x+4 • t1 = (3*x*x*x*x*x) • t2 = t1 + (2*x*x*x*x) • t3 = t2 + (7*x*x*x) • t4 = t3 + (8*x*x) • P = t4 +(2*x) + 4 • 15 Multiplications, 5 Additions

  3. A Little Smarter • P(x) = 3x5+2x4+7x3+8x2+2x+4 • t1 := 4; xp := x; • t2 := (2*xp) + t1; xp := xp * x; • t3 := (8*xp) + t2; xp := xp * x; • t4 := (7*xp) + t3; xp := xp * x; • t5 := (2*xp) + t4; xp := xp * x; • P := (3*xp) + t5; • 9 Multiplications, 5 Additions

  4. Horner’s Rule • P(x) = 3x5+2x4+7x3+8x2+2x+4 • t1 = (3*x) + 2 • t2 = (t1*x) + 7 • t3 = (t2*x) + 8 • t4 = (t3*x) + 2 • P = (t4*x) + 4 • 5 Multiplications, 5 Additions

  5. Computing Powers xp := x; pwork := power; Res := 1; While pwork > 0 Do If pwork mod 2 = 1 then Res := Res * xp; End If xp := xp * xp; pwork := pwork / 2; /* integer division */ End While

  6. For Power = 15 • 6 Multiplications by Squaring Algorithm • An Alternative Procedure: • p = x * x • p = p * p *x • p = p * p * p • 5 Multiplications by Factorization

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