CSC_349_HANDOUT#10

CSC_349_HANDOUT#10 - COMPUTER SCIENCE 349A Handout Number...

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COMPUTER SCIENCE 349A Handout Number 10 QUADRATIC CONVERGENCE OF NEWTON'S METHOD The following result is proved on p. 141 of the 5 th ed. (p. 150 of the 6 th ed.) Theorem If Newton's method is applied to 0 ) ( = x f producing a sequence {} i x that converges to a root t x and if 0 ) ( t x f , then the order of convergence is 2. If 0 ) ( = t x f and Newton's method converges to a root t x , then we will see later that the order of convergence is NOT quadratic. Example 1. An illustration of quadratic convergence of Newton’s method. Here x x x f = cos ) ( . This was computed in MATLAB, so at most 16 correct digits are possible. The underlined digits are all correct. i i x no. of correct digits ______ _________________________ ________________ 0 = 4 π 0.7 853 98 1 1 0.739 5 361 3 2 0.7390 851 7 81 7 3 0.7390 8513 3215 16 11 14 4 0.7390 8513 3215 1607 16 Example 2 . The following illustrates the possible effect of a poor initial approximation with Newton's method, yet the eventual characteristic quadratic convergence. Here
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CSC_349_HANDOUT#10 - COMPUTER SCIENCE 349A Handout Number...

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