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7 The trigonometric function sin(x) interpolated by linear up to sixth-order interpolation over the interval λ0 = 2π. 4 (shown by gray dashed-lines). for all possible p values. , for p = 2), the asymptotic behavior is matched for all child ≤ λ0 /3. 8 for the sin(x) and cos(x) functions, respectively. , the linear) interpolation, = λ0 /7 with interpolation of the second and third orders, = λ0 /6 with fourth order of interpolation, and = λ0 /5 with interpolation of the fifth and sixth orders. 60) is about 10% even when using a sixth-order interpolation.

For i ≥ 1 the zeros of Ri ( p, ξ) are at ξ = 0, 1p , 2p , . . i−1 p . To construct the interpolatory vector functions that will be discussed later in Chapter 4, we also use the ( p + 1) modified or shifted Silvester polynomials [5] Rˆ i ( p, ξ) = Ri−1 p, ξ − 1p ⎧ i−1 ⎪ ⎨ 1 ( pξ − k) = (i − 1)! 2 • Interpolatory Polynomials of Silvester 39 that have the following properties (1 ≤ i ≤ p + 1 is understood): 1. They are polynomials of degree (i − 1) in ξ. 2. 2). 3. Rˆ i ( p, ξ) is unity at ξ = i . p 4.

1 First-order approx. Second-order approx. −1 Third-order approx. Fourth-order approx. 0. 15 Absolute errors obtained by approximating the functions f1 (x) = −ln(x), √ f2 (x) = (1/ x − 1)/a, and f3 (x) = (1/x − 1)/b (from top to bottom) with the lowest approximation orders n = 1, 2, 3, and 4. 01. 1 to 4 in the attempt to diminish the error, the error on the fourth interpolation subinterval increases. 15, where we report the errors obtained by using approximations of order n = 1, 2, 3, and 4. From this figure, one can appreciate that, typically, the error of the nth-order approximation is maximum on the nth interpolation subinterval, provided the interpolation points are properly spaced.

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