Rational Interpolation

4.3.3 Rational Interpolation

The rational interpolation problem is to determine a rational function

with numerator of degree m and denominator of degree n so that at distinct points x 0 ,x 1 ,...,x n agrees with a function f

(4.3.25) Rational approximation is often superior to polynomial approximation in the neighborhood

r m,n (x i ) =f i , i = 0 : N, N = m + n.

of a point at which the function has a singularity. Since the coefficients can be determined only up to a common nonzero factor, the number of unknown constants in (4.3.24) equals the number of interpolation points N + 1 = m + n + 1.

A necessary condition for (4.3.25) to hold clearly is that the linearized condition

(4.3.26) is satisfied, i.e., for i = 0 : m + n, p x

P m (x i ) −f i Q n (x i ) = 0, i = 0 : N,

i +p 1 i +···+p m i −f i 0 i +q 1 x i +···+q n x i ) = 0. (4.3.27) This is a homogeneous linear system of (m+n+1) equations for the (m+n+2) coefficients

(q x

in P m and Q n . If we introduce the Vandermonde matrices

this system can be written in matrix form as

(A FB)

q = 0,

F = diag (f 0 ,f 1 ,...,f N ), (4.3.28) where p = (p 0 ,p 1 ,...,p m ) T , q = (q 0 ,q 1 ,...,q n ) T . This is a homogeneous linear system

of N + 1 equations in N + 2 unknowns. Such a system always has a nontrivial solution.

390 Chapter 4. Interpolation and Approximation We note that the rational interpolant is fully determined by the denominator polyno-

mial. By (4.3.26) P m (x) is the unique polynomial determined by the interpolation conditions

P (x i ) =f i Q(x i ), i = 1 : m.

While for polynomial interpolation there is always a unique solution to the interpola- tion problem, this cannot be guaranteed for rational interpolation, as shown by the example below. A further drawback is that the denominator polynomial Q(x) may turn out to have zeros in the interval of interpolation.

Example 4.3.6.

Assume that we want to interpolate the four points

by a rational function

p 0 +p 1 x +p 2 x 2

r(x) =

q 0 +q 1 x

Then we must solve the homogeneous linear system  100 

Setting p 2 = 1 we find the solution p 0 = 8, p 1 = −6, q 0 = 4, q 1 = −2. The corresponding rational function

8 − 6x + x

( 4 − x)(2 − x)

has the common factor (2 − x) and is reducible to f 2,1 = (4 − x)/2. The original form is indeterminate 0/0 at x = 2, while the reduced form does not take on the prescribed value at x = 2.

As shown in the above example, for given data (x 0 ,f 0 ), . . . , (x n ,f n ) there can be certain points x j where the given function value f j cannot be attained. Such a point x j is called unattainable. This can occur only if x j is a zero of the denominator Q n (x) . From (4.3.26) it also follows that P m (x j ) = 0. Hence the polynomials P (x) and Q(x) have a

common factor (x − x j ) d , where d is chosen maximal. The polynomials pair

Q(x) P ∗ (x) = (4.3.29)

then satisfies (4.3.26) for all points x k j . Since d was chosen maximal it holds that Q ∗ (x j )

∗ (x j )/Q ∗ (x j ) must be finite. But since when Q n (x j ) = 0, (4.3.26) is satisfied for any choice of f j , one cannot expect the rational function to interpolate

a particular value at x j .

4.3. Generalizations and Applications 391 If there are unattainable points, then the polynomials defined in (4.3.29) only solve

the linearized equations in the attainable points. It can also happen that the polynomials given by the linearized equations have a common factor (x − x ∗ ) d , d ≥ 1, with x ∗

i = 1 : n. In this case all polynomials of the form P ∗

P (x)

Q(x)

, ν = 0 : d, satisfy the linearized system and the matrix ( A FB ) in (4.3.28) has at most rank

N + 1 − d. Conversely we have the following result.

Theorem 4.3.4.

If the rank of the matrix ( A FB ) equals N + 1 − d, then there exists a unique solution p, q corresponding to polynomials P ∗ and Q ∗ with degrees at most m − d and n − d. Further, all solutions have the form

s(x)P ∗ (x)

r(x) =

s(x)Q ∗

(x)

where s(x) is a polynomial of degree at most d. A point x j is unattainable if and only if Q ∗ (x j ) = 0.

Proof. See Schaback and Werner [313, Theorem 12.1.8]. Another complication of rational interpolation is that zeros may occur in Q n (x) , which

are not common to P m (x) . These zeros correspond to poles in r(x), which if they lie inside the interval [min x k , max x k ] may cause trouble. In general it is not possible to determine a

priori if the given data (x k ,f k ) will give rise to such poles. Neither do the coefficients in the representation of r(x) give any hint of such occurrences.

An algorithm similar to Newton’s algorithm can be used for finding rational inter- polants in continued fraction form. Set v 0 (x) = f (x), and use a sequence of substitutions:

The first two substitutions give x −x 0 x −x 0

In general this gives a continued fraction x −x 0 x −x 1 x −x 2 x −x 3

where a k =v k (x k ) , and we have used the compact notation introduced in Sec. 3.5.1. This becomes an identity if the expansion is terminated by replacing a n in the last denominator

392 Chapter 4. Interpolation and Approximation by a n + (x − x n )/v n +1 (x) . If we set x = x k , k ≤ n, then the fraction terminates before

the residual (x − x n )/v n +1 (x) is introduced. This means that setting 1/v k +1 = 0 will give

a rational function which agrees with f (x) at the points x i , i = 0 : k ≤ n, assuming that the constants a 0 ,...,a k exist. These continued fractions give a sequence of rational approximations f k,k ,f k +1,k , k = 0, 1, 2, . . . . Introducing the notation

(4.3.32) we have a k = [x 0 ,x 1 ,...,x k −1 ,x k ]φ. Then by (4.3.30) we have x −x 0 x −x 0

v k (x) = [x 0 ,x 1 ,...,x k −1 ,x ]φ

[x]φ = f (x),

[x 0 ,x ]φ =

[x]φ − [x 0 ]φ

and in general

[x −1 0 ,x 1 ,...,x k −1 ,x

−x k

. (4.3.33) [x 0 ,...,x k −2 ,x ]φ − [x 0 ,...,x k −2 ,x k −1 ]φ

Therefore, we also have

a −x k k −1 = . (4.3.34) [x 0 ,...,x k −2 ,x k ]φ − [x 0 ,...,x k −2 ,x k −1 ]φ

We call the quantity defined by (4.3.34) the kth inverse divided difference of f (x). Note that certain inverse differences can become infinite if the denominator vanishes. They are, in general, symmetrical only in their last two arguments.

The inverse divided differences of a function f (x) can conveniently be computed recursively and arranged in a table similar to the divided-difference table.

Here the upper diagonal elements are the desired coefficients in the expansion (4.3.31).

4.3. Generalizations and Applications 393

Example 4.3.7.

Assume that we want to interpolate the four points given in Example 4.3.6 and the additional points (4, 6/17). Forming the inverse differences we get the following table.

This gives a sequence of rational approximations. If we terminate the expansion

after a 3 we recover the solution of the previous example. Note that the degeneracy of the approximation is shown by the entry a 3 = 0. Adding the last fraction gives the (degenerate) approximation

f 2,2 2+x = 2 . 1+x

It is verified directly that this rational function interpolates all the given points. Because the inverse differences are not symmetric in all their arguments the reciprocal

differences are often preferred. These are recursively defined by

i ,x i +1 ,...,x

+k ]ρ =

+k

[x i ,...,x i +k−1 ]ρ − [x i +1 ,...,x i +k ]ρ

(4.3.36) See Hildebrand [201, p. 406]. While this formula is less simple than (4.3.34), the reciprocal

+ [x i +1 ,...,x i +k−1 ]ρ.

differences are symmetric functions of all their arguments. The symmetry permits the calculation of the kth reciprocal difference from any two (k − 1)th reciprocal differences

having k −1 arguments in common, together with the (k −2)th reciprocal difference formed with this argument. Using (4.3.36) a reciprocal difference table may be constructed.

The coefficients in the continued fraction (4.3.31) can then be determined by an interpolation formula due to Thiele [350]:

a 3 = [x 0 ,x 1 ,x 2 ,x 3 ]ρ − [x 0 ,x 1 ]ρ, . . . . The formulas using inverse or reciprocal differences are useful if one wants to deter-

mine the coefficients of the rational approximation, and use it to compute approximations

394 Chapter 4. Interpolation and Approximation for several arguments. If one only wants the value of the rational interpolating function for

a single argument, then it is more convenient to use an alternative algorithm of Neville type. This is the case in the ρ-algorithm, which is a convergence acceleration procedure using rational interpolation to extrapolate to infinity with the same degree in the numerator and denominator.

If we consider the sequence of rational approximations of degrees (m, n)

the following recursive algorithm results (Stoer and Bulirsch [338, Sec. 2.2]):

For i = 0, 1, 2, . . ., set T i, −1 = 0, T i, 0 =f i , and

1 ≤ k ≤ i. (4.3.37) −x −k

T i,k

−1 −T i −1,k−1

ik =T i,k −1 + x

0 T i,k

−1 −T 1− −1,k−1

x −x i

T i,k −1 −T i −1,k−2

As in Neville interpolation the calculations can be arranged in a table of the following form. (m, n)

Here any entry is determined by a rhombus rule from three entries in the preceding two columns. Note that it is easy to add a new interpolation point in this scheme.

As shown by Berrut and Mittelmann [23], every rational interpolant can be written in barycentric form

r(x) =

Let q k =Q n (x k ) , k = 1 : N, be the values of the denominator at the nodes. Then the barycentric representation of the denominator Q n (x) is

Hence r(x) can be written as in (4.3.38), where u k =w k q k is the weight corresponding to the node x k . Since w k

k = 0 at a node if and only if the corresponding weight equals zero.

4.3. Generalizations and Applications 395 The barycentric form has the advantage that the barycentric weights give information

about possible unattainable points. However, the determination of the parameter vector u = (u 0 ,u 1 ,...,u N ) T is more complicated. An elimination method for the computation of u in O(n 3 ) operations is given by Berrut and Mittelmann [23].