Determining the Rolle function in Lagrange interpolatory approximation

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1 Determining the Rolle function in Lagrange interpolatory approimation arxiv: v1 [math.na] Oct 18 J. S. C. Prentice Department of Pure and Applied Mathematics University of Johannesburg South Africa October 4, 18 Abstract We determine the Rolle function in Lagrange polynomial approimation using a suitable differential equation. We then propose a device for improving the Lagrange approimation by eploiting our knowledge of the Rolle function. 1 Introduction Approimation of nonlinear functions is of fundamental importance in applied mathematics, especially by means of polynomials. Those techniques for which the approimation error is well understood are particularly useful. In this paper, we describe how the error in Lagrange polynomial interpolation can be precisely determined, by solving an appropriate initial-value problem. We then consider using the knowledge so obtained to improve the quality of the original approimation. Relevant Concepts, Terminology and Notation Let f ) be a real-valued univariate function. The Lagrange interpolating polynomial P n ) of degree n, at most, that interpolates the data {f ), 1

2 f 1 ),...,f n )} at the nodes {, 1,..., n }, where < 1 < < n, has the property P n k ) = f k ) for k =,1,...,n. Naturally, we regard P n ) as an approimation to f ). The pointwise error in Lagrange interpolation, on [, n ], is ;P n ) f ) P n ) = fn+1) ξ)) n+1)! n k ), 1) where < ξ) < n, and is derived by invoking Rolle s Theorem [1, ]. Clearly, we assume here that f ) is n+1)-times differentiable and, as will be seen, we must assume that f ) is, in fact, n+)-times differentiable. We refer to ξ generically as the Rolle number, and to ξ) as the Rolle function. k= 3 Determining the Rolle Function Using the notation π) n k) we have, by differentiating with k= respect to, n+1)!f ) P n )) = f n+1) ξ))π) n+1)!f ) P n)) = f n+1) ξ)π )+π) dfn+1) ξ) dξ dξ d = f n+1) ξ)π )+π)f n+) ξ) dξ d. We have used the well-known prime notation for differentiation with respect to. In this epression, the factor f n+1) ξ) denotes the n+1)th derivative of f ξ) with respect to ξ, and similarly for f n+) ξ). We now find dξ d = n+1)!f ) P n )) fn+1) ξ)π ) π)f n+) ξ) and, if we have a particular value ξ z = ξ z ) at our disposal, we then have an initial-value problem that can, in principle, be solved to yield the Rolle function ξ). Note the necessity of our assumption that f ) is n+)- times differentiable.

3 4 Calculations Consider the Lagrange interpolation of f ) = e sin over the nodes { },. We have n = 1 so that the Lagrange polynomial is e sin )) ) e P 1 ) = =. Furthermore, we have ;P 1 ) = e sin e ) = ) e ξ) cosξ)) ) and ) dξ e cos+sin) e e ξ cosξ ) ) d = ). 3) eξ cosξ sinξ)) We solve this differential equation using an initial value chosen close to the node =. Observe that it is not possible to find the Rolle number at any interpolation node, because the factor n k) in 1) ensures that ;P n ) = at the interpolation nodes, irrespective of the value k= of ξ. Hence, we choose here z = 1 5 and determine the corresponding Rolle number by applying Newton s ) method to ). Naturally, we compute z ;P 1 ) = e z sin z z to facilitate this calculation. In fact, we e find two values for ξ z, giving the initial values z,ξ z ) = 1 5,.1931) and z,ξ z ) = 1 5,4.6631). Note to the reader: we quote numerical values correct to no more than four decimal places throughout this paper, simply for ease of presentation, but all calculations are performed in double precision). Using these two initial values we solve 3) to find ξ), shown respectively in Figures 1a) and 1b). We are then able to compute the pointwise error ;P 1 ) using ), and this is shown in Figure 1c) for z,ξ z ) = 1 5,.1931). Of course, we can compare this error curve with the actual error, and the magnitude of the difference between the two - the error in, so to speak - is shown in Figure 1d). Clearly, this error is small, indicating the accuracy of our numerical estimate of ξ). Similar results obtain for z,ξ z ) = 1 5,4.6631), with a maimum magnitude in the error in of This accuracy is a consequence of using a fourth-order Runge-Kutta method [3, 4] to solve 3) with a small stepsize ). 3

4 As a second eample, let us use the same objective function f ) as above, but now with three interpolatory nodes { },,. So, we have n = which gives P ) = a +b+c, where a = ,b = and c =. Also, with π) = ) ) ) = 3 + ) ) + π ) = 3 + 4)+, we have and ;P ) = eξ cosξ sinξ)π) 3 dξ d = 6e cos+sin) a+b)) e ξ cosξ sinξ)π ). 4) π) 4e ξ sinξ) We perform similar calculations as above, with results depicted in Figure. Again, we find two initial values z,ξ z ) = 1 5,1.7845) and z,ξ z ) = 1 5,3.8165), and the error plot in Figure d) corresponds to z,ξ z ) = 1 5,1.7845). For z,ξ z ) = 1 5,3.8165) the maimum magnitude in the error in is Our second eample allows us to make a point about the continuity of ξ). Since P ) is continuous and the assumed n+)-times differentiability of f ) implies the continuity of both f ) and f n+1) ), we will assume, from 1), that ξ) is continuous. Hence, even though it is not really meaningful to ask for the value of ξ) at an interpolatory node, the continuity of ξ) allows us to infer a value at such a node. We refer to such a value as an implied Rolle number. In our second eample we determine the implied values ξ) =.991 from Figure a) and ξ) = from Figure b). Our technique for doing this is as follows: we ensure that = is not amongst the nodes used for the Runge-Kutta calculation, because this would lead to a zero in the denominator on the RHS of 4), but we use the values of ξ) at the Runge-Kutta nodes on either side of = to estimate ξ) using linear interpolation. Of course, we acknowledge that it is not necessary to know the Rolle number at any interpolatory node, because the pointwise error at interpolatory nodes is always zero. As such, ξ) is actually arbitrary at interpolatory nodes, but our technique does allow a 4

5 sensible estimate to be made, if only for completeness sake. Note also that simple linear etrapolation will allow an estimate of ξ) to be made at the endpoints of the interval. 5 A Possible Application Our ability to determine the Rolle function suggests an interesting possibility. If we know ξ) then we know f n+1) ξ)). If we approimate f n+1) ξ)) by means of a polynomial - a least-squares fit, or a truncated Taylor series, for eample - then, using 1), we find f ) P n )+ P ξ) n+1)! n k ), where P ξ ) denotes the polynomial that approimates f n+1) ξ)). Notice that the RHS of this epression is simply a polynomial, and so constitutes a polynomial approimation to f ). Thus, our knowledge of ξ) allows us to improve the approimation P n ) by adding a polynomial term that approimates the error in P n ). By way of eample, we return to the first of our earlier calculations. Here, we have f ) = e sin ) e P 1 ) = ;P 1 ) = ) e ξ) cosξ)) using the nodes { },. Once we have found ξ) - using z,ξ z ) = 1 5,.1931) and shown in Figure 1a)- we determine a polynomial approimation to e ξ) cosξ)) using a least-squares fit. For the purpose of demonstration, we choose to fit a degree si polynomial so that ;P 1 )+P 1 ) becomes an eighth-degree polynomial approimation, which we denote 8 ). In Figure 3a) we show the pointwise error in P 1 ), and observe it has a maimum magnitude of 75. In Figure 3b) we plot the pointwise error k= e sin 8 ). This error is significantly smaller than that in Figure 3a) and has maimum magnitude In other words, 8 ) is an approimation more 5

6 than four orders of magnitude!) more accurate than P 1 ), and this was achieved only through our knowledge of the Rolle function ξ). This eample serves to illustrate the value of knowing ξ), and certainly warrants further investigation, but we will reserve a detailed study thereof for future research. 6 Conclusion We have described how the Rolle function in Lagrange interpolatory polynomial approimation can be determined by solving an initial-value problem. Knowledge of the Rolle function permits the calculation of the approimation error. In particular, the Rolle term in the epression for the approimation error can itself be approimated by means of a polynomial, once the Rolle function is known, and this can lead to a significant improvement in the quality of the Lagrange approimation overall. Of course, the ideas presented in this paper are not restricted to Lagrange approimation, but could also be applied to Hermite interpolation, for eample, in which the error term is also derived through the use of Rolle s Theorem. References [1] Isaacson, E. and Keller, H. B. 1994). Analysis of Numerical Methods, New York: Dover. [] Kincaid, D. and Cheney, W. ). Numerical Analysis: Mathematics of Scientific Computing, 3rd ed., Pacific Grove: Brooks/Cole. [3] Butcher, J. C. 3). Numerical Methods for Ordinary Differential Equations, Chichester: Wiley. [4] Hairer, E., Norsett, S.P., and Wanner, G. ). Solving Ordinary Differential Equations I: Nonstiff Problems, Berlin: Springer. 6

7 3 Figure 1a) 4.7 Figure 1b) Rolle number Rolle number Figure 1c) Figure 1d) 7 4 Delta Error in Delta

8 .4 Figure a) 3.84 Figure b) Rolle number Rolle number Figure c) Figure d) Delta Error in Delta

9 8 Figure 3a) Error Figure 3b) 4 Error