CSC_349_HANDOUT#12

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/10/2011 for the course CSC 349 taught by Professor Oadje during the Spring '11 term at University of Victoria.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online