polynomial approximation and interpolation chapter 4 · polynomial approximation and interpolation...

Polynomial Approximation and Interpolation Chapter 4

Nizar Salim 1 lecture 2

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 4.4.1. A variation of the Lagrange polynomial, called Neville s algorithm, which has some computational advantages over the Lagrange polynomial, is presented in Section 4.4.2.



4.4.1. Lagrange Polynomials

Consider two points, [a,f(a)] and [b,f(b)]. The linear Lagrange polynomial P1(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



uniqueness theorem, the two forms both represent the unique polynomial that passes exactly through a set of points.



The results are summarized below, where the results of linear, quadratic, and cubic interpolation, and the errors, Error(3.44) = P(3.44) = 0.290698, are tabulated. The advantages of higher-degree interpolation are obvious.

P(3.44) = 0.290756 linear interpolation Error = 0.000058 = 0.290697 quadratic interpolation = -0.000001 = 0.290698 cubic interpolation = 0.000000

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 4.5.

4.4.2. 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:



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 fi

(0). For the first interpolation of the data,



as illustrated in Figure 4.6a. This creates a column of n - 1 values of fi(1) . A

second column of n - 2 values of fi(2) is obtained by linearly interpolating

the column of fi(1) values. Thus,

which is illustrated in Figure 4.6b. This process is repeated to create a third column of fi

(3) values, as illustrated in Figure 4.6c, and so on. The form of the resulting table is illustrated in Table 4.1.



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 calculate the specific value. For example,f1

(2) 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 f1

(2) and f2(2) to obtain

f1(3). 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 table, fi

(n) , 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:



Thus, the result of quadratic interpolation is f(3.44) =f1(2) = 0.290696. To

evaluate ,f1(3),f3

(1) and f2(2) must first be evaluated. Then f1

(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



Similar expressions can be obtained for divided differences of any order. Approximating polynomials for no equally spaced data can be constructed using divided differences.

4.5.1. Divided Difference Tables

Consider a table of data:

By definition, f[xi] =fi . 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 4.4.2. Thus,



Table 4.3 illustrates the formation of a divided difference table. The first column contains the values of xi and the second column contains the values of f(xi) =fi , which are denoted by fi

(0). The remaining columns contain the values of the divided differences, fi

(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.



4.5.2. Divided Difference Polynomials

Let's define a power series for Pn(x) such that the coefficients are identical to the divided differences, fi

(n) Thus,

Pn(x) is clearly a polynomial of degree n. To demonstrate that Pn(x) passes exactly through the data points, let's substitute the data points into Eq. (4.65). Thus,



Since Pn(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 x0 = 3.35 as the base point. The exact solution is f(3.44) = 1/3.44 = 0.290698. From Eq. (4.65):



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):



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.

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