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CSC_349_HANDOUT#12 - COMPUTER SCIENCE 349A Handout Number...

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1 COMPUTER SCIENCE 349A Handout Number 12 MULTIPLE ROOTS AND THE MULTIPLICITY OF A ZERO (Section 6.4 in 5 th ed. or Section 6.5 in 6 th ed.) If Newton's method converges to a zero t x of ) ( x f , a necessary condition for quadratic convergence is that 0 ) ( t x f . We now relate this condition on the derivative of ) ( x f to the multiplicity of the zero t x . Definition (not in the textbook) If t x is a zero of any analytic function ) ( x f , then there exists a positive integer m and a function ) ( x q such that 0 ) ( lim where ), ( ) ( ) ( = x q x q x x x f t x x m t . (In particular, if ) ( t x q is defined, note that 0 ) ( t x q .) The value m is called the multiplicity of the zero t x . If 1 = m , then t x is called a simple zero of ) ( x f . Example 1 Consider ) 5 . 2 ( ) 4 ( 160 56 18 5 . 9 ) ( 3 2 3 4 + = + + = x x x x x x x f The zero at ) 0 ) 4 ( and 5 . 2 ) ( here ( 3 has 4 = = = q x x q m x t . The zero at ) 0 ) 5 . 2 ( and ) 4 ( ) ( here ( 1 has 5 . 2 3 + = = = q x x q m x t . Example 2 Consider 1 ) ( = x e x f x . Since 0 ) 0 ( = f , 0 = t x is a zero of ) ( x f . This zero has multiplicity 2 = m since ) ( ) 0 ( ) ( 2 x q x x f = with 2 1 ) ( x x e x q x =
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