aimz2 - What is the rate o± convergence c Predict how may...

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AIMS Exercise Set # 2 Peter J. Olver 1. Explain why the equation e - x = x has a solution on the interval [0 , 1]. Use bisection to fnd the root to 4 decimal places. Can you prove that there are no other roots? 2. Find 6 3 to 5 decimal places by setting up an appropriate equation and solving using bisection. 3. Find all real roots o± the polynomial x 5 - 3 x 2 + 1 to 4 decimal places using bisection. 4. Let g ( u ) have a fxed point u ? in the interval [0 , 1], with g 0 ( u ? ) 6 = 1. Defne G ( u ) = u g 0 ( u ) - g ( u ) g 0 ( u ) - 1 . ( a ) Prove that, ±or an initial guess u (0) near u ? , the fxed point iteration scheme u ( n +1) = G ( u ( n ) ) converges to the fxed point u . ( b ) What is the order o± convergence o± this method? ( c ) Test this method on the non-convergent cubic scheme in Example 2.16. 5. Let g ( u ) = 1 + u - 1 8 u 3 . ( a ) Find all fxed points o± g ( u ). ( b ) Does fxed point iteration converge? I± so, to which fxed point(s)?
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Unformatted text preview: What is the rate o± convergence? ( c ) Predict how may iterates will be needed to get the fxed point accurate to 4 decimal places starting with the initial guess u (0) = 1. ( d ) Check your prediction by per±orming the iteration. 6. Solve Exercise 1–3 by Newton’s Method. 7. ( a ) Let u ? be a simple root o± f ( u ) = 0. Discuss the rate o± convergence o± the iterative method (sometimes known as Olver’s Method , in honor o± the author’s ±ather) based on g ( u ) = u + f ( u ) 2 f 00 ( u )-2 f ( u ) f ( u ) 2 2 f ( u ) 3 to u ? . ( b ) Try this method on the equation in Exercise 3, and compare the speed o± convergence with that o± Newton’s Method. 1 c ° 2006 Peter J. Olver...
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