Transcript Polynomial Evaluation - Baylor University || School of

Polynomial Evaluation
Straightforward Evaluation
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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
A Little Smarter
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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
Horner’s Rule
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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
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
For Power = 15
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6 Multiplications by Squaring Algorithm
An Alternative Procedure:
p=x*x
p = p * p *x
p=p*p*p
5 Multiplications by Factorization