CSC_349_HANDOUT#14

CSC_349_HANDOUT#14 - COMPUTER SCIENCE 349A Handout Number...

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1 COMPUTER SCIENCE 349A Handout Number 14 ZEROS OF POLYNOMIALS USING NEWTON'S METHOD WITH HORNER’S ALGORITHM, AND POLYNOMIAL DEFLATION Outline of a procedure to compute a zero of a polynomial ) ( x f using Newton's method and Horner’s algorithm: ) ( ) ( set ) ( and ) ( evaluate to algorithm s Horner' use imax to 1 for ) ( of zero a ion to approximat initial an be Let 1 1 1 1 1 0 = i i i i i i x f x f x x x f x f i x f x if ε < i i x x / 1 1 exit end output failed to converge in imax iterations Polynomial Deflation . Suppose that the values K , , , 2 1 0 x x x computed above converge in N iterations. Then N x is the final computed approximation to some zero, say ) ( of , 1 x f r . Now the final computation in the above procedure with Newton's method (after N iterations) is ) ( ) ( 1 1 1 N N N N x f x f x x . If 0 1 , , , b b b n n K are the values computed by Horner’s algorithm to evaluate ) ( 1 N x f -- that is, in the last step of the above procedure (when N i = ), then from page 2 of Handout Number 13 it follows that (1) 0 1 ) ( ) ( ) ( b x Q x x x f N + = , where (2) 1 2 3 2 1 ) ( + + + + = n n x b x b x b b x Q L . On letting 1 = N x x in (1), we obtain ). ( of zero the since 0 ) ( 1 1 1 0 x f r x x x f b N N N = Therefore, from (1), ) ( ) ( ) ( 1 x Q x x x f N and consequently
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2 1 ) ( ) ( N x x x f x Q . That is, the polynomial
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CSC_349_HANDOUT#14 - COMPUTER SCIENCE 349A Handout Number...

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