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128a_su09_hw7_sol

128a_su09_hw7_sol - MATH 128A SUMMER 2009 HOMEWORK 7...

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Unformatted text preview: MATH 128A, SUMMER 2009: HOMEWORK 7 SOLUTIONS (1) Let’s use the notation w AB3 for the Adams-Bashforth(3) predictor’s estimate of y i +1 , w AM2 for the Adams-Moulton(2) corrector’s estimate of y i +1 , and τ AB3 and τ AM2 for their respective truncation errors. From the textbook, we have the truncation error formulas τ AB3 = y i +1- w AB3 h = 3 8 y (4) ( ξ AB3 ) h 3 , ξ AB3 ∈ [ t i- 2 ,t i +1 ] τ AM2 = y i +1- w AM2 h ≈- 1 24 y (4) ( ξ AM2 ) h 3 , ξ AM2 ∈ [ t i- 1 ,t i +1 ] (Technically speaking, the equation in the second line is approximate: it gives the predictor- corrector method’s truncation error as that of the corrector step. The difference ends up being O ( h 4 ), and may be ignored.) Proceed by assuming that y (4) ( ξ ) is approximately constant on the relevant interval. When we subtract the equations, the y i +1 terms cancel and we’re left with w AM2- w AB3 h ≈ 10 24 y (4) (?) h 3 The left-hand side is- 1, so y (4) (?) h 3 ≈ - 24 / 10. Plugging back into the formula for10....
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128a_su09_hw7_sol - MATH 128A SUMMER 2009 HOMEWORK 7...

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