By Walter Gautschi

ISBN-10: 0817682589

ISBN-13: 9780817682583

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**Additional resources for Instructor's Solutions Manual to Numerical Analysis**

**Example text**

Determine the best constant c (or constants c, if there is nonuniqueness) and the minimum error. 2. Consider the data f (ti ) = 1, i = 1, 2, . . , N − 1; f (tN ) = y ≫ 1. (a) Determine the discrete L∞ approximant to f by means of a constant c (polynomial of degree zero). (b) Do the same for discrete (equally weighted) least square approximation. (c) Compare and discuss the results, especially as N → ∞. 3. Let x0 , x1 , . . , xn be pairwise distinct points in [a, b], −∞ < a < b < ∞, and f ∈ C 1 [a, b].

Consider the function f (t) = tα on 0 ≤ t ≤ 1, where α > 0. Suppose we want to approximate f best in the Lp norm by a constant c, 0 < c < 1, that is, minimize the Lp error Ep (c) = tα − c 1 p = 0 1/p |tα − c|p dt 64 Exercises to Chapter 2 as a function of c. Find the optimal c = cp for p = ∞, p = 2, and p = 1, and determine Ep (cp ) for each of these p-values. 5. Taylor expansion yields the simple approximation ex ≈ 1 + x, 0 ≤ x ≤ 1. Suppose you want to improve this by seeking an approximation of the form ex ≈ 1 + cx, 0 ≤ x ≤ 1, for some suitable c.

8. , which by Stirling’s formula is asymptotically equivalent to 1011 n + (n + 21 ) ln(n + 1) − (n + 1) + 1 2 ln 2π as n → ∞. 000000000128155 × 1017 . 000000000128155e+17. The last two are accurate to all 16 digits shown, the first only in the first eleven digits. One might argue that many of the initial terms of the series in the first summation are shifted out of range when added to 1011 n = 1017 and only those which are large enough will contribute to the sum. In the second and third summations, all terms in the series are fully accounted for.

### Instructor's Solutions Manual to Numerical Analysis by Walter Gautschi

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