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CSC_349_HANDOUT#13 - COMPUTER SCIENCE 349A Handout Number...

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1 COMPUTER SCIENCE 349A Handout Number 13 HORNER’S ALGORITHM (NESTED MULTIPLICATION, SYNTHETIC DIVISION) Given a polynomial = = n i i i x a x f 0 ) ( and a value 0 x , this algorithm is used to efficiently evaluate ) ( and ) ( 0 0 x f x f . To illustrate the basic idea, consider the case 4 = n : (1) 4 4 3 3 2 2 1 0 ) ( x a x a x a x a a x f + + + + = can be rewritten in the form (2) ))) ( ( ( ) ( 4 3 2 1 0 a x a x a x a x a x f × + × + × + × + = . Evaluation of (1) at x 0 requires 7 multiplications and 4 additions, whereas (2) requires only 4 multiplications and 4 additions. The general case (for a polynomial of order n ): form (1) requires 1 2 n multiplications and n additions, form (2) requires n multiplications and n additions. An algorithm to evaluate ) ( 0 x f , assuming that = = n i i i x a x f 0 ) ( is written in the “nested” form , as in (2): 0 , 1 , , 2 , 1 for compute Then . Let 0 1 0 1 0 0 0 1 2 2 0 1 1 K M = + = + = + = + = = + n n k x b a b x b a b x b a b x b a b a b k k k n n n n n n n n Then ) ( 0 0 x f b = .
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