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Jim Lambers Math 105A Summer Session I 2003- Lecture 9 Notes
These notes correspond to Sections 3.2 and 3.3 in the text.
Divided Differences
In the previous lecture, we learned how to compute the value of an interpolating polynomial at a given point, using Neville’s Method. However, the theorem that serves as the basis for Neville’s Method can easily be used to compute the interpolating polynomial itself. The basic idea is to represent interpolating polynomials using the Newton form, discussed in the Lecture 7 notes, with the interpolation points used as the centers. Recall that if the interpolation points x 0 ,... , xn are distinct, then the process of finding a polynomial that passes through the points (xi, yi), i = 0,... , n, is equivalent to solving a system of linear equations Ax = b that has a unique solution. The matrix A is determined by the choice of basis for the space of polynomials of degree n or less. Each entry aij of A is the value of the jth polynomial in the basis at the point xi. In Newton interpolation, the matrix A is upper triangular, and the basis functions are defined to be the set {Nj (x)}nj=0, where
N 0 (x) = 1, Nj (x) =
j∏− 1
k=
(x − xk), j = 1,... , n.
The advantage of Newton interpolation is that the interpolating polynomial is easily updated as interpolation points are added, since the basis functions {Nj (x)}, j = 0,... , n, do not change from the addition of the new points. The coefficients cj of the Newton interpolating polynomial
pn(x) =
∑^ n
j=
cj Nj (x)
are given by cj = f [x 0 ,... , xj ]
where f [x 0 ,... , xj ] denotes the divided difference of x 0 ,... , xj. The divided difference is defined as follows:
f [xi] = yi,
f [xi, xi+1] =
yi+1 − yi xi+1 − xi
f [xi, xi+1,... , xi+k] = f [xi+1,... , xi+k] − f [xi,... , xi+k− 1 ] xi+k − xi
This definition implies that for each nonnegative integer j, the divided difference f [x 0 , x 1 ,... , xj ] only depends on the interpolation points x 0 , x 1 ,... , xj and the value of f (x) at these points. It follows that the addition of new interpolation points does not change the coefficients c 0 ,... , cn. Specifically, we have
pn+1(x) = pn(x) +
yn+1 − pn(xn+1) Nn+1(xn+1) Nn+1(x).
This ease of updating makes Newton interpolation the most commonly used method of obtaining the interpolating polynomial. The following result shows how the Newton interpolating polynomial bears a resemblance to a Taylor polynomial.
Theorem Let f be n times continuously differentiable on [a, b], and let x 0 , x 1 ,... , xn be distinct points in [a, b]. Then there exists a number ξ ∈ [a, b] such that
f [x 0 , x 1 ,... , xn] = f (n)(ξ) n!
Computing the Newton Interpolating Polynomial
We now describe in detail how to compute the coefficients cj = f [x 0 , x 1 ,... , xj ] of the Newton interpolating polynomial pn(x), and how to evaluate pn(x) efficiently using these coefficients. The computation of the coefficients proceeds by filling in the entries of a divided-difference table. This is a triangular table consisting of n + 1 columns, where n is the degree of the interpolating polynomial to be computed. For j = 0, 1 ,... , n, the jth column contains n − j entries, which are the divided differences f [xk, xk+1,... , xk+j ], for k = 0, 1 ,... , n − j. We construct this table by filling in the n + 1 entries in column 0, which are the trivial divided differences f [xj ] = f (xj ), for j = 0, 1 ,... , n. Then, we use the recursive definition of the divided differences to fill in the entries of subsequent columns. Once the construction of the table is complete, we can obtain the coefficients of the Newton interpolating polynomial from the first entry in each column, which is f [x 0 , x 1 ,... , xj ], for j = 0, 1 ,... , n. In a practical implementation of this algorithm, we do not need to store the entire table, because we only need the first entry in each column. Because each column has one fewer entry than the previous column, we can overwrite all of the other entries that we do not need. The following algorithm implements this idea.
Algorithm (Divided-Difference Table) Given n distinct interpolation points x 0 , x 1 ,... , xn, and the values of a function f (x) at these points, the following algorithm computes the coefficients cj = f [x 0 , x 1 ,... , xj ] of the Newton interpolating polynomial.
the denominators of all of the divided differences can be expressed in terms of the spacing h. If we recall the forward difference operator ∆, defined by
∆xk = xk+1 − xk,
where {xk} is any sequence, then the divided differences f [x 0 , x 1 ,... , xk] are given by
f [x 0 , x 1 ,... , xk] =
k!hk^
∆kf (x 0 ).
The interpolating polynomial can then be described by the Newton forward-difference formula
pn(x) = f [x 0 ] +
∑^ n
k=
s k
∆kf (x 0 ),
where the new variable s is related to x by
s = x − x 0 h
and the extended binomial coefficient
s k
is defined by ( s k
s(s − 1)(s − 2) · · · (s − k + 1) k!
where k is a nonnegative integer. If we define the backward difference operator ∇ by
∇xk = xk − xk− 1 ,
for any sequence {xk}, then we obtain the Newton backward-difference formula
pn(x) = f [xn] +
∑^ n
k=
(−1)k
−s k
∇kf (xn),
where s = (x − xn)/h, and the preceding definition of the extended binomial coefficient applies.
Osculatory Interpolation
Suppose that the interpolation points are perturbed so that two neighboring points xi and xi+1, 0 ≤ i < n, approach each other. What happens to the interpolating polynomial? In the limit, as xi+1 → xi, the interpolating polynomial pn(x) not only satisfies pn(xi) = yi, but also the condition
p′ n(xi) = (^) xlim i+1→xi
yi+1 − yi xi+1 − xi
It follows that in order to ensure uniqueness, the data must specify the value of the derivative of the interpolating polynomial at xi. In general, the inclusion of an interpolation point xi k times within the set x 0 ,... , xn must be accompanied by specification of p( nj )(xi), j = 0,... , k − 1, in order to ensure a unique solution. These values are used in place of divided differences of identical interpolation points in Newton interpolation. Interpolation with repeated interpolation points is called osculatory interpolation, since it can be viewed as the limit of distinct interpolation points approaching one another, and the term “osculatory” is based on the Latin word for “kiss”.
Hermite Interpolation
In the case where each of the interpolation points x 0 , x 1 ,... , xn is repeated exactly once, the interpolating polynomial for a differentiable function f (x) is called the Hermite polynomial of f (x), and is denoted by H 2 n+1(x), since this polynomial must have degree 2n + 1 in order to satisfy the 2 n + 2 constraints
H 2 n+1(xi) = f (xi), H 2 ′n+1(xi) = f ′(xi), i = 0, 1 ,... , n.
The Hermite polynomial can be described using Lagrange polynomials and their derivatives, but this representation is not practical because of the difficulty of differentiating and evaluating these polynomials. Instead, one can construct the Hermite polynomial using a divided-difference table, as discussed previously, in which each entry corresponding to two identical interpolation points is filled with the value of f ′(x) at the common point. Then, the Hermite polynomial can be represented using the Newton divided-difference formula.
