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Polynomial Approximation and Interpolation Chapter 4
Nizar Salim 1 lecture 3
4.6 DIFFERENCE TABLES AND DIFFERENCE POLYNOMIALS
Fitting approximating polynomials to tabular data is considerably simpler when the
values of the independent variable are equally spaced. Implementation of polynomial
fitting for equally spaced data is best accomplished in terms of differences. Consequently,
the concept of differences, difference tables, and difference polynomials are introduced in
this section.
4.6.1. Difference Tables
A difference table is an arrangement of a set of data, [x,f(x)], in a table with the x values
in monotonic ascending order, with additional columns composed of the differences of
the numbers in the preceding column. A triangular array is obtained, as illustrated in
Table 4.6.
The numbers appearing in a difference table are unique. However, three different
interpretations can be assigned to these numbers, each with its unique notation. The
forward difference relative to point i is (fi+1 - fi), the backward difference relative to point
i+ 1 is (fi+1 - fi), and the centered difference relative to point i+l/2 is (fi+1 - fi). The forward
difference operator is defined as
A difference table, such as Table 4.6, can be interpreted as a forward-difference table,
a backward-difference table, or a centered-difference table, as illustrated in Figure 4.7.
Polynomial Approximation and Interpolation Chapter 4
Nizar Salim 2 lecture 3
The numbers in the tables are identical. Only the notation is different. The three
different types of interpretation and notation simplify the use of difference tables in the
construction of approximating polynomials, which is discussed in Sections 4.6.2 to 4.6.4.
Example 4.7. Difference table.
Let s construct a six-place difference table for the function f(x) = 1/x for 3.1 x 3.9
with x = 0.1. The results are presented in Table 4.7, which uses the forward-difference
notation to denote the columns of differences.
Polynomial Approximation and Interpolation Chapter 4
Nizar Salim 3 lecture 3
4.6.2, The Newton Forward-Difference Polynomial
Given n + 1 data points, [x,f(x)], one form of the unique nth-degree polynomial that
passes through the n + 1 points is given by
Equation (4.88) does not look anything like the direct fit polynomial [see Eq. (4.34)], the
Lagrange polynomial [see Eq. (4.46)], or the divided difference polynomial [see Eq.
(4.65)]. However, if Eq. (4.88) is a polynomial of degree n and passes exactly through the
n + 1 data points, it must be one form of the unique polynomial that passes through this
set of data.
The interpolating variable, s = (x - xo)/h, is linear in x. Consequently, the last term
in Eq. (4.88) is order n, and Eq. (4.88) is an nth-degree polynomial. Let s = 0. Then x=xo,
f =fo, and Pn(xo) =fo. Let s = 1.
Then x = x0 + h = x1, f =fl , and Pn(x1) = fo+ fo=fo + (f1-fo) =f1. In a similar manner, it
can be shown that Pn(x) =f(x) for the n + 1 discrete points. Therefore, Pn(x) is the desired
unique nth-degree polynomial. Equation (4.88) is called the Newton forward-difference
polynomial.
Polynomial Approximation and Interpolation Chapter 4
Nizar Salim 4 lecture 3
A major advantage of the Newton forward-difference polynomial, in addition to its
simplicity, is that each higher-degree polynomial is obtained from the previous lowerdegree
polynomial simply by adding the next term. The work already performed for the
lower-degree polynomial does not have to be repeated. This feature is in sharp contrast to
the direct fit polynomial and the Lagrange polynomial, where all of the work must be
repeated each time the degree of the polynomial is changed. This feature makes it simple
to determine when the desired accuracy has been obtained. When the next term in the
polynomial is less than some prespecified value, the desired accuracy has been obtained.
Example 4.8. Newton forward-difference polynomial.
From the six-place difference table for f(x) = l/x, Table 4.7, calculate P(3.44) by the
Newton forward-difference polynomial. The exact solution is f(3.44) = 1/3.44 = 0.290698
... In Table 4.7, h = 0.1. Choose x0 = 3.40. Then,
Evaluating Eq. (4.94) term by term yields the following results and errors:
P(3.44) = 0.290756 linear interpolation ErrorC.44) = 0.000058
= 0.290700 quadratic interpolation = 0.000002
= 0.290698 cubic interpolation = 0.000000
The advantage of higher-degree interpolation is obvious.
In this example, the base point, x0 = 3.4, was selected so that the point of
interpolation, x = 3.44, falls within the range of data used to determine the polynomial,
that is, interpolation occurs. If x0 is chosen so that x does not fall within the range of fit,
extrapolation occurs, and the results are less accurate. For example, let x0 = 3.2, for which
s = 2.4. The following results and errors are obtained:
(3.44) = 0.289772 linear extrapolation Error = -0.000926
= 0.290709 quadratic extrapolation = 0.000011
= 0.290698 cubic interpolation = 0.000000
Polynomial Approximation and Interpolation Chapter 4
Nizar Salim 5 lecture 3
The increase in error is significant for linear and quadratic extrapolation. For x0 = 3.2, the
cubic yields an interpolating polynomial.
4.6.3 The Newton Backward-Difference Polynomial
The Newton forward-difference polynomial, Eq. (4.88), can be applied at the top or in the
middle of a set of tabular data, where the downward-sloping forward differences
illustrated in Figure 4.7a exist. However, at the bottom of a set of tabular data, the
required forward differences do not exist, and the Newton forward-difference polynomial
cannot be used. In that case, an approach that uses the upward-sloping backward
differences illustrated in Figure 4.7b is required. Such a polynomial is developed in this
section.
Given n + 1 data points, [x,f(x)], one form of the unique nth-degree polynomial that
passes through the n + 1 points is given by
Polynomial Approximation and Interpolation Chapter 4
Nizar Salim 6 lecture 3
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المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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