Polynomial Approximation and Interpolation Chapter 4

Transcription

1 4.4 LAGRANGE POLYNOMIALS The direct fit polynomial presented in Section 4.3, while quite straightforward in principle, has several disadvantages. It requires a considerable amount of effort to solve the system of equations for the coefficients. For a high-degree polynomial (n greater than about 4), the system of equations can be ill-conditioned, which causes large errors in the values of the coefficients. A simpler, more direct procedure is desired. One such procedure is the Lagrange polynomial, which can be fit to unequally spaced data or equally spaced data. The Lagrange polynomial is presented in Section A variation of the Lagrange polynomial, called Neville s algorithm, which has some computational advantages over the Lagrange polynomial, is presented in Section Nizar Salim 1 lecture 2

2 Lagrange Polynomials Consider two points, [a,f(a)] and [b,f(b)]. The linear Lagrange polynomial P 1 (x) which passes through these two points is given by The Lagrange polynomial can be used for both unequally spaced data and equally spaced data. No system of equations must be solved to evaluate the polynomial. However, a considerable amount of computational effort is involved, especially for higher-degree polynomials. The form of the Lagrange polynomial is quite different in appearance from the form of the direct fit polynomial, Eq. (4.34). However, by the Nizar Salim 2 lecture 2

3 uniqueness theorem, the two forms both represent the unique polynomial that passes exactly through a set of points. Nizar Salim 3 lecture 2

4 The results are summarized below, where the results of linear, quadratic, and cubic interpolation, and the errors, Error(3.44) = P(3.44) = , are tabulated. The advantages of higher-degree interpolation are obvious. P(3.44) = linear interpolation Error = = quadratic interpolation = = cubic interpolation = These results are identical to the results obtained in Example 4.2 by direct fit polynomials, as they should be, since the same data points are used in both examples. The main advantage of the Lagrange polynomial is that the data may be unequally spaced. There are several disadvantages. All of the work must be redone for each degree polynomial. All the work must be redone for each value of x. The first disadvantage is eliminated by Neville's algorithm, which is presented in the next subsection. Both disadvantages are eliminated by using divided differences, which are presented in Section Neville's Algorithm Neville's algorithm is equivalent to a Lagrange polynomial. It is based on a series of linear interpolations. The data do not have to be in monotonic order, or in any structured order. However, the most accurate results are obtained if the data are arranged in order of closeness to the point to be interpolated. Consider the following set of data: Nizar Salim 4 lecture 2

5 where the subscript i denotes the base point of the value (e.g., i,i+ 1, etc.) and the superscript (n) denotes the degree of the interpolation (e.g., zeroth, first, second, etc.). A table of linearly interpolated values is constructed for the original data, which are denoted as f i (0). For the first interpolation of the data, Nizar Salim 5 lecture 2

6 (1) as illustrated in Figure 4.6a. This creates a column of n - 1 values of f i. A (2) second column of n - 2 values of f i is obtained by linearly interpolating (1) the column of f i values. Thus, which is illustrated in Figure 4.6b. This process is repeated to create a third column of f i (3) values, as illustrated in Figure 4.6c, and so on. The form of the resulting table is illustrated in Table 4.1. Nizar Salim 6 lecture 2

7 It can be shown by direct substitution that each specific value in Table 4.1 is identical to a Lagrange polynomial based on the data points used to (2) calculate the specific value. For example,f 1 is identical to a second-degree Lagrange polynomial based on points 1, 2, and 3. The advantage of Neville's algorithm over direct Lagrange polynomial interpolation is now apparent. The third-degree Lagrange polynomial based on points 1 to 4 is obtained simply by applying the linear interpolation formula, Eq. (4.52), to f (2) (2) 1 and f 2 to obtain f (3) 1. None of the prior work must be redone, as it would have to be redone to evaluate a third-degree Lagrange polynomial. If the original data are arranged in order of closeness to the interpolation point, each value in the (n) table, f i, represents a centered interpolation. Example 4.4. Neville's algorithm. Consider the four data points given in Example 4.3. Let's interpolate for f(3.44) using linear, quadratic, and cubic interpolation using Neville's algorithm. Rearranging the data in order of closeness to x = 3.44 yields the following set of data: Nizar Salim 7 lecture 2

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9 Thus, the result of quadratic interpolation is f(3.44) =f 1 (2) = To evaluate,f 1 (3),f 3 (1) and f 2 (2) must first be evaluated. Then f 1 (3) can be evaluated. These results, and the results calculated above, are presented in Table 4.2. These results are the same as the results obtained by Lagrange polynomials in Example 4.3. The advantage of Neville's algorithm over a Lagrange interpolating polynomial, if the data are arranged in order of closeness to the interpolated point, is that none of the work performed to obtain a specific degree result must be redone to evaluate the next higher degree result. Neville's algorithm has a couple of minor disadvantages. All of the work must be redone for each new value of x. The amount of work is essentially the same as for a Lagrange polynomial. The divided difference polynomial presented in Section 4.5 minimizes these disadvantages. 4.5 DIVIDED DIFFERENCE TABLES AND DIVIDED DIFFERENCE POLYNOMIALS A divided difference is defined as the ratio of the difference in the function values at two points divided by the difference in the values of the corresponding independent variable. Thus, the first divided difference at point i is defined as Nizar Salim 9 lecture 2

10 Similar expressions can be obtained for divided differences of any order. Approximating polynomials for no equally spaced data can be constructed using divided differences Divided Difference Tables Consider a table of data: By definition, f[x i ] =f i. The notation presented above is a bit clumsy. A more compact notation is defined in the same manner as the notation used in Neville's method, which is presented in Section Thus, Nizar Salim 10 lecture 2

11 Table 4.3 illustrates the formation of a divided difference table. The first column contains the values of x i and the second column contains the values of f(x i ) =f i, which are denoted by f i (0). The remaining columns contain the values of the divided differences, f i (n) where the subscript i denotes the base point of the value and the superscript (n) denotes the degree of the divided difference. The data points do not have to be in any specific order to apply the divided difference concept. However, just as for the direct fit polynomial, the Lagrange polynomial, and Neville's method, more accurate results are obtained if the data are arranged in order of closeness to the interpolated point. Example 4.5. Divided difference Table. Let's construct a six-place divided difference table for the data presented in Section 4.1. The results are presented in Table 4.4. Nizar Salim 11 lecture 2

12 Divided Difference Polynomials Let's define a power series for P n (x) such that the coefficients are identical to the divided differences, f i (n) Thus, P n (x) is clearly a polynomial of degree n. To demonstrate that P n (x) passes exactly through the data points, let's substitute the data points into Eq. (4.65). Thus, Nizar Salim 12 lecture 2

13 Since P n (x) is a polynomial of degree n and passes exactly through the n + 1 data points, it is obviously one form of the unique polynomial passing through the data points. Example 4.6. Divided difference polynomials. Consider the divided difference table presented in Example 4.5. Let's interpolate for f(3.44) using the divided difference polynomial, Eq. (4.65), using x 0 = 3.35 as the base point. The exact solution is f(3.44) = 1/3.44 = From Eq. (4.65): Nizar Salim 13 lecture 2

14 The advantage of higher-degree interpolation is obvious. The above results are not the most accurate possible since the data points in Table 4.4 are in monotonic order, which make the linear interpolation result actually linear extrapolation. Rearranging the data in order of closeness to x = 3.44 yields the results presented in Table 4.5. From Eq. (4.65): Nizar Salim 14 lecture 2

15 The linear interpolation value is much more accurate due to the centering of the data. The quadratic and cubic interpolation values are the same as before, except for round-off errors, because the same points are used in those two interpolations. These results are the same as the results obtained in the previous examples. Nizar Salim 15 lecture 2

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