Hermite Interpolation

4.3.1 Hermite Interpolation

Newton’s interpolation formula can also be used in generalized interpolation problems, where one or more derivatives are matched at the interpolation points. This problem is known as Hermite interpolation. 132

Theorem 4.3.1.

be m distinct real or complex points. Let f (z) be a given real- or complex- valued function that is defined and has derivatives up to order r i −1 (r i ≥ 1) at z i . The

Let {z i m } i =1

Hermite interpolation problem

, to find a polynomial p(z) of degree ≤ n − 1, where =

i =1 r i such that p(z) and its first r i − 1 derivatives agree with those of f (z) at z i , i.e.,

i ) =f (z i ), j =0:r i − 1, i = 1 : m, (4.3.1) is uniquely solvable. We use here the notation f ( 0) (z) for f (z).

p (j ) (z

(j )

Proof. Note that (4.3.1) are precisely n conditions on p(z). The conditions can be expressed by a system of n linear equations for the coefficients p, with respect to some basis. This has

a unique solution for any right-hand side, unless the corresponding homogeneous problem has a nontrivial solution. Suppose that a polynomial p ∈ P n comes from such a solution of

the homogeneous problem, that is,

p (j ) (z i ) = 0, i = 1 : m, j =0:r i − 1.

Then z i must be a zero of multiplicity r i of p(x), hence p(z) must have at least r i =n zeros (counting the multiplicities). But this is impossible, because the degree of p is less than n. This contradiction proves the theorem.

Hermite interpolation can be viewed as the result of passages to the limit in interpo- lation at n points, where for i = 1 : m, r i interpolation points coalesce into the point z i . We

132 Charles Hermite (1822–1901), a French mathematician, made important contributions to number theory, orthogonal polynomials, and elliptic functions.

382 Chapter 4. Interpolation and Approximation say that the point z i has multiplicity r i . For example, the Taylor polynomial in P n ,

n −1 f (j ) (z

p(z) j = (z −z 1 ) ,

j =0

interpolates f (z) at the point z 1 with multiplicity n (or z 1 is repeated n times).

Example 4.3.1.

Consider the problem of finding a polynomial p(x) ∈ P 4 that interpolates the function

f and its first derivative f ′ at the two points z 1 and z 2 , and also its second derivative at z 1 . In the notations of Sec. 4.1.1 the linear system for the coefficient vector c becomes V T c =f,

where f = (f (z T 1 ), f ′ (z 1 ), f ′′ (z 1 ), f (z 2 ), f ′ (z 2 )) , and  1 0 0 1 0 

is a confluent Vandermonde matrix. Note that the second, third, and fifth column of V is obtained by “differentiating” the previous column. From Theorem 4.3.1 we conclude that such confluent Vandermonde matrices are nonsingular. We remark that the fast algorithms in Sec. 4.2.5 can be generalized to confluent Vandermone systems; see [29].

For the determinant of the general confluent Vandermonde matrix one can show that

m r & j & −1 &

i det(V ) = j n ! (z i −z ) .

When there are gaps in the sequence of derivatives that are known at a point the interpolation problem is called Birkhoff interpolation or lacunary interpolation. Such a problem may not have a solution, as is illustrated by the following example.

Example 4.3.2.

Find a polynomial p ∈ P T 3 that interpolates the data f = (f (−1), f ′ ( 0), f (1)) . The new feature is that f (0) is missing. If we use the power basis, then we obtain the linear

system

The determinant is evidently zero, so there is no solution for most data. An explanation is

that hf ′ = µδf for all f ∈ P 3 .

Newton’s interpolation formula can be generalized to the confluent case quite easily. We will require some continuity and differentiability properties for divided differences.

4.3. Generalizations and Applications 383 These can be obtained through an alternative expression for divided differences that we

now give.

Definition 4.3.2.

A set S of points in C is called convex if for any z, u ∈ S, the straight line {tz+(1−t)u | t

∈ (0, 1)} is also contained in S. The convex hull of a set S in R d is the smallest convex subset of C which contains S.

Let D be the convex hull of the set of points z, u 1 ,...,u n in C. Assume that f is defined in D and has that its nth derivative exists and is continuous on D. Set u 0 (z) = f (z) and consider the functions

u 1 + (u 2 −u 1 )t 1 + · · · + (z − u k )t k dt k · · · dt 1 , (4.3.5)

k = 1 : n. The argument of the integrand lies in the convex hull D, since

ζ k =u 1 + (z − u 1 )t 1 + · · · + (z − u k )t k

= (1 − t 1 )u 1 + (t 1 −t 2 )u 2 + · · · + (t k −1 −t k )u k +t k z =λ 1 u 1 +λ 2 u 2 +···+λ k u k +λ k +1 z,

where 1 ≤ k ≤ n. From 1 ≥ t 1 ≥t 2 ≥···≥t n ≥ 0, it follows that

k +1 λ i = 1, λ i ≥ 0, i = 1 : k + 1.

i =1

If in (4.3.5) we carry out the integration with respect to t k and express the right-hand side using u k −1 we find that the functions u k (z) can be defined through the recursion

, k = 1 : n, (4.3.6)

z −u k

with w 0 (z) = f (z). But this is the same recursion (4.3.5) that was used before to define the divided differences and thus

- [z, u 1 ,...,u k ]f =

holds for arbitrary distinct points z, u 1 ,...,u n .

Notice that the integrand on the right-hand side of (4.3.7) is a continuous function of the variables z, u 1 ,...,u n and hence the right-hand side is a continuous function of these variables. Thus, when the nth derivative of f exists and is continuous on D, (4.3.7) defines the continuous extension of [z, u 1 ,...,u k ]f to the confluent case. From the continuity of the divided differences it follows that the remainder term for interpolation given in Theorem 4.2.3 remains valid for Hermitian interpolation. Provided the points x, z 1 ,...,z m are real we have

f (n)

(ξ m

f (x) − p(x) =

M n (x), M n (x) =

(x −z i ) r i , (4.3.8)

i =1

384 Chapter 4. Interpolation and Approximation with ξ x ∈ int (x, z 1 ,...,z m ) . From (4.3.7) follows

( t ( ( ( 1 1 n −1 ([z, x 1 ,...,x

which can be used to give an upper bound for the remainder in polynomial interpolation for arbitrary interpolation points.

From (4.3.6) it follows that the divided difference [u 1 ,...,u k +s ,x ]f is equal to the divided difference of w k (x) at [u k +1 ,...,u k +s ,x ]f , so that by (4.3.7) we can write

If all points u k +1 ,...,u k +s all tend to the limit z and w k (z) has a continuous sth derivative at z, then it holds that

0 0 s ! k , the divided differences belong to C k +1−max r It can be shown that if f ∈ C i , and that the

interpolation polynomial has this kind of differentiability with respect to the u i (nota bene, if the “groups” do not coalesce further).

Example 4.3.3.

As established above, the usual recurrence formula for divided differences can still

be used for the construction of the divided-difference table in case of multiple points. The limit process is just applied to the divided differences, for example,

For the interpolation problem considered in Example 4.3.1 we construct the generalized

divided-difference table, where x 1 0 .

x 0 f 0 f ′′

f 0 ′ [x 0 ,x 0 ,x 0 ,x 1 ]f

[x 0 ,x 0 ,x 0 ,x 1 ,x 1 ]f [x 0 ,x 1 ]f

x 0 f 0 [x 0 ,x 0 ,x 1 ]f

[x 0 ,x 0 ,x 1 ,x 1 ]f

x 1 f 1 [x 0 ,x 1 ,x 1 ]f

4.3. Generalizations and Applications 385 The interpolating polynomial is

2 p(x) 1

and the remainder is

f (x)

,x ,x ,x ,x ,x

) 3 (x ) − p(x) = [x 2 0 0 0 1 1 ]f (x − x 0 −x 1

( 5) (ξ x )(x

) =f 3 −x 0 (x −x 1 ) 2 / 5.

An important case of the Hermite interpolation problem is when the given data are

f i = f (x i ) ,f i ′ =f ′ (x i ) , i = 0, 1. We can then write the interpolation polynomial as p(x) =f 0 + (x − x 0 ) [x 0 ,x 1 ]f + (x − x 0 )(x −x 1 ) [x 0 ,x 0 ,x 1 ]f

+ (x − x 0 ) 2 (x −x 1 ) [x 0 ,x 0 ,x 1 ,x 1 ]f.

Set x 1 =x 0 + h and x = x 0 + θh, and denote the remainder f (x) − p(x) by R T . Then one can show (Problem 4.3.1) that

p(x) =f 0 + θ4f 0 + θ(1 − θ)(hf 0 ′ − 4f 0 )

+ −θ ( 1 − θ) (hf 0 ′ − 4f 0 ) + (hf 1 ′ − 4f 0 )

+ = (1 − θ)f 0 + θf 1 + θ(1 − θ) ( 1 − θ)(hf 0 ′ − 4f 0 ) − θ(hf 1 ′ − 4f 0 ) .

For x ∈ [x 0 ,x 1 ] we get the error bound

h |f (x) − p(x)| ≤ 4 max

Setting t = 1/2, we get the useful approximation formula