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Lecture 6 Notes

125 archimedes 250 bc zu chongzhi liu hui 480 ad

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Unformatted text preview: beJer” En and x* Bounds •  What is “beJer” –  Accuracy, Speed, Precision •  Want x* accurate as quickly as possible –  Least number of steps ~= least number of computa3ons (or fastest execu3on 3me), but it is a start –  How do you analyze approaches? •  Run experiments •  Look at how fast error bound shrinks (rate of convergence) –  Check absolute error for special cases –  Analy3c Methods •  Compare rates of convergence Convergence Rates of Convergence Ayer 5 terms, •  The Gregory–Leibniz series is within 0.2 •  The Nilakantha's series is within 0.002 Some really great series exist (>14 correct decimal digits per term) great En, but diﬃcult En Approxima3ng Func3ons Evolu3on of x* for log(1+y) A func3on at a par3cular point is a number (like pi), so numerical approxima3on material we have studied applies, e.g., x: = ln(1+y) = y – y^2/2 + y^3/3 – y^4/4 + y^5/5 + … x* = y (N=1) = y – y^2/2 (N=2) = y – y^2/2 + y^3/3 (N=3) = y – y^2/2 + y^3/3 – y^4/4 (N=4) = y – y^2/2 + y^3/3 – y^4/4 + y^5/5 (N=5) … En = En(y) (<- bound generally depe...
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