By Abdelwahab Kharab

**Praise for earlier Editions**Kharab and Guenther supply an enticing, transparent, error-free, and well-written advent to numerical equipment ... hugely recommended.

-J.H. Ellison,

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**Extra resources for An Introduction to Numerical Methods : A MATLAB Approach, Third Edition**

**Example text**

15. How many terms must be taken in the previous exercise to get an error of less than 10−2 ? 10−4 ? ✐ ✐ ✐ ✐ ✐ ✐ “k” — 2011/11/22 — 10:14 — page 20 — ✐ 20 ✐ INTRODUCTION 16. Construct a Taylor polynomial approximation to f (x) = e−x with x0 = 0 that is accurate to within 10−3 on the interval [0, 1]. Find a value M such that |f (x1 ) − f (x2 )| ≤ M |x1 − x2 | for all x1 , x2 on the interval. 17. Find the Taylor polynomial of order 2 around x0 = 1 for f (x) = x3 + x. 18. Find intervals containing solutions to the following equations (a) x − 3−x = 0, (b) 4x2 − ex = 0.

K” — 2011/11/22 — 10:14 — page 31 — ✐ ROUND-OFF ERRORS ✐ 31 8. Suppose p˜ must approximate p with relative error at most 10−3 . Find the largest interval in which p˜ must lie for each value of p (a) 150 (b) 1500. 9. Use 3-digits chopping after each addition for N i=1 1 . i (a) Find N such that N is the smallest integer for which your sum is not the exact solution. (b) What is the absolute error for this value of N ? (c) What is the relative error for this value of N ? 10. The number e can be calculated as ∞ e= 1 .

B1 b2 . . bk bk+1 . )10 × 10e . 3) We say that the number x is chopped to k digits when all digits following the kth digits are discarded; that is, the digits bk+1 bk+2 . . b1 b2 . . bk )10 × 10e . Conversely, x is rounded to k digits when f l(x) is obtained by choosing f l(x) nearest to x; that is, adding one to bk if bk+1 ≥ 5 and chop oﬀ all but the ﬁrst k digits if bk+1 < 5. 2718281828 . . × 101 . 718282. 718281. The error that results from replacing a number x with its ﬂoating-point form f l(x) is called round-oﬀ error.