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SET A = a² + 4b IF A = 0 SET A TO 20 SET B TO B = 2x₁/a - xo SET TO (A + Bn)(a/2)" x RETURN x ELSE SET 0₁ TO (a + √A)/2 SET 0₂ TO (a - √A)/2 SET A TO your value of A from 3(a). SET B TO your value of B from 3(a). SET TO A0+ B02 RETURN x 6. In this question, the rounding errors incurred by the exact-formula algorithm are analysed. You may assume that a, b, xo, ₁ are all integers no larger in magnitude than 106 and that the computer is using double precision. (a) Each time a computer calculates a value x which is not in the registry, instead of storing the true value x, it stores the computed value æº. Set d to be the relative rounding error incurred by doing the computation i.e. da = xc - - I Express in terms of x and d, and give bounds on d in terms of machine precision t (so t = 52 for double precision). [2] (b) Suppose that a and b are such that A = 0. By expressing the values computed at each stage in the form given in (a), show that the total rounding error incurred by the exact-formula algorithm is O(2-t), so the algorithm does not significantly amplify the rounding errors. (c) Analyse the rounding errors incurred by the exact-formula algorithm when A ‡ 0. (d) For which values of a, b, x0, x₁, n would you expect this algorithm to be the most susceptible to rounding errors? Is the size of the error a concern in this case? [5] [7] [4] SET A = a² + 4b IF A = 0 SET A TO 20 SET B TO B = 2x₁/a - xo SET TO (A + Bn)(a/2)" x RETURN x ELSE SET 0₁ TO (a + √A)/2 SET 0₂ TO (a - √A)/2 SET A TO your value of A from 3(a). SET B TO your value of B from 3(a). SET TO A0+ B02 RETURN x 6. In this question, the rounding errors incurred by the exact-formula algorithm are analysed. You may assume that a, b, xo, ₁ are all integers no larger in magnitude than 106 and that the computer is using double precision. (a) Each time a computer calculates a value x which is not in the registry, instead of storing the true value x, it stores the computed value æº. Set d to be the relative rounding error incurred by doing the computation i.e. da = xc - - I Express in terms of x and d, and give bounds on d in terms of machine precision t (so t = 52 for double precision). [2] (b) Suppose that a and b are such that A = 0. By expressing the values computed at each stage in the form given in (a), show that the total rounding error incurred by the exact-formula algorithm is O(2-t), so the algorithm does not significantly amplify the rounding errors. (c) Analyse the rounding errors incurred by the exact-formula algorithm when A ‡ 0. (d) For which values of a, b, x0, x₁, n would you expect this algorithm to be the most susceptible to rounding errors? Is the size of the error a concern in this case? [5] [7] [4]
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Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
Posted Date:
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