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Unformatted text preview: 2 , (2) where , , are unknowns. Find numerical values for , , so that in terms of the new variable X , Eq. (1) becomes X = r + X 2 + O ( 3 ) . (3) (c) Solve Eq. (2) to nd an explicit expression for X in terms of x . (d) Theres a simpler way to dene X in terms of x and r which gives Eq. (1) with no error terms . What is it? (e) Extra credit: If = 1 / 10 , roughly how big is the size of the O ( 3 ) remainder in Eq. (3)? Hint: First, nd the exact formula for the transformed equation X = F ( X, r ) . Since X = x + O ( 2 ) , its reasonable to guess that | X | is not much bigger than | x | itself, say | X | c for some reasonably small value of c . (You will have to guess a value for c .) How big can | r + X 2-F ( X, r ) | get for | X | c and | r | ? A numerical estimate (by plugging in a few values of X and r ) is ne....
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This note was uploaded on 02/17/2010 for the course MATH 142532 taught by Professor Glasner during the Spring '10 term at University of Arizona- Tucson.
- Spring '10