CSC_349_HANDOUT#9 - convergence in the first column above,...

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1 COMPUTER SCIENCE 349A Handout Number 9 An illustration of order of convergence when α = 1 and = 2 . The limit of each sequence is 097 3414 3001 3652 . 1 = t x . Note: in the following computed approximations, the underlined digits are correct. Example 1 Computed approximations Linear convergence ( = 1) _____________________ ______________________ = 0 x 1 .5 = 1 x 1 .2869 53768 5808 . 0 / 0 1 = x x x x t t = 2 x 1 .4025 40804 4767 . 0 / 1 2 = x x x x t t = 3 x 1.3 454 58374 5299 . 0 / 2 3 = x x x x t t = 4 x 1.3 751 70253 5028 . 0 / 3 4 = x x x x t t 5 x = 1.36 00 94193 5167 . 0 / 4 5 = x x x x t t = 6 x 1.36 78 46968 5095 . 0 / 5 6 = x x x x t t = 7 x 1.36 38 87004 5132 . 0 / 6 7 = x x x x t t = 8 x 1.365 9 16734 5113 . 0 / 7 8 = x x x x t t = 9 x 1.36 48 78217 5123 . 0 / 8 9 = x x x x t t = 10 x 1.365 4 10062 5118 . 0 / 9 10 = x x x x t t constant 0.51 t h a t i s , constant lim 1 = + i i i e e λ Note that the above ratios i t i t x x x x + / 1 can be computed only if you know the exact zero t x . In practice these ratios are never computed; they are given here to illustrate the definition of “order of convergence” (that is, how these ratios determine the kind of slow
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Unformatted text preview: convergence in the first column above, where the number of correct significant digits increases by some constant amount with each successive iteration). 2 Example 2 Computed approximations Quadratic convergence ( α = 2 ) _____________________ __________________________ = x 1 .5 = 1 x 1.3 733 3333 3333 333 4461 . / 2 1 = − − x x x x t t = 2 x 1.3652 6201 4874 627 4874 . / 2 1 2 = − − x x x x t t = 3 x 1.3652 3001 3 916 147 4902 . / 2 2 3 = − − x x x x t t ↓ constant 49 . ≈ t h a t i s , constant lim 2 1 = + ∞ → i i i e e λ and this reflects the approximate doubling of the number of correct significant digits with each i t e r a t i o n...
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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.

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CSC_349_HANDOUT#9 - convergence in the first column above,...

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