General Properties of Rational Interpolation

2.2 Interpolation by Rational Functions

2.2.1 General Properties of Rational Interpolation

(2.1.5.12) **f−ϕ**

∞:= max

x∈[a,b]f(x)−ϕ(x)≤ 1

2^2ν(2ν)!**f^(2ν)**

∞∆^2ν where

∆= max

0≤i≤m−1|x_i+1−x_i| is the “fineness” of the partition∆.

The approximation error goes to zero with the 2ν th power of the fineness ∆_j if we consider a sequence of partitions ∆_j of the interval [a, b] with ∆_j →0. Contrast this with the case of ordinary polynomial interpolation, where the approximation error does not necessarily go to zero as∆_j →0 [see Section 2.1.4].

Ciarlet, Schultz, and Varga (1967) were able to show also that the firstν derivatives ofϕare a good approximation to the corresponding derivatives off:

(2.1.5.13)

f^(k)(x)−P_i^(k)(x)≤ |(x−x_i)(x−x_i+1)|^ν−k

k!(2ν−2k)! (x_i+1−x_i)^kmax

ξ∈Ii

f^(2ν)(ξ)

for allx∈I_i,i= 0, 1,. . .,m−1,k= 0, 1,. . .,ν, and therefore (2.1.5.14) **f^(k)−ϕ^(k)**

∞ ≤ ∆^2ν−k

2^2ν−2kk! (2ν−2k)!**f^(2ν)**

∞

fork= 0, 1,. . .,ν.

(2.2.1.1) Φ^µ,ν(x_i) =f_i, i= 0,1, . . . , µ+ν .

We denote by A^µ,ν the problem of calculating the rational function Φ^µ,ν from (2.2.1.1).

It is clearly necessary that the coefficients a_r, b_s, of Φ^µ,ν solve the homogeneous system of linear equations

(2.2.1.2) P^µ,ν(x_i)−f_iQ^µ,ν(x_i) = 0, i= 0,1, . . . , µ+ν, or written out in full,

a₀+a₁x_i+· · ·+a_µx^µ_i −f_i(b₀+b₁x_i+· · ·+b_νx^ν_i) = 0, i= 0,1, . . . , µ+ν.

We denote this linear system byS^µ,ν.

At first glace, substituting S^µ,ν for A^µ,ν does not seem to present a problem. The next example will show, however, that this is not the case, and that rational interpolation is inherently more complicated than polynomial interpolation.

Example.For support points

xi: 0 1 2 fi: 1 2 2 andµ=ν= 1:

a₀ −1·b₀ = 0, a₀+a₁ −2(b₀+b₁) = 0, a₀+ 2a₁−2(b₀+ 2b₁) = 0.

Up to a common nonzero factor, solving the above systemS^1,1 yields the coeffi- cients

a₀= 0, b₀= 0, a₁ = 2, b₁= 1, and therefore the rational expression

Φ^1,1(x)≡2x x,

which forx= 0 leads to the indeterminate expression 0/0. After cancelling the factorx, we arrive at the rational expression

Φ˜^1,1(x)≡2.

Both expressions Φ^1,1 and ˜Φ^1,1 represent the same rational function, namely the constant function of value 2. This function misses the first support point (x₀, f₀) = (0,1). Therefore it does not solveA^1,1. Since solvingS^1,1 is necessary for any solution ofA^1,1, we conclude that no such solution exists.

The above example shows that the rational interpolation problemA^µ,ν need not be solvable. Indeed, ifS^µ,νhas a solution which leads to a rational function that does not solveA^µ,ν— as was the case in the example — then

the rational interpolation problem is not solvable. In order to examine this situation more closely, we have to distinguish between different represen- tations of the same rational functionΦ^µ,ν, which arise from each other by canceling or by introducing a common polynomial factor in numerator and denominator. We say that two rational expressions (that is pairs of two polynomials, the numerator and denominator polynomials)

Φ₁(x)..≡ P₁(x)

Q₁(x), Φ₂(x)..≡ P₂(x)

Q₂(x), Q₁(x)≡0, Q₂(x)≡0, areequivalent, and write

Φ₁∼Φ₂, if

P₁(x)Q₂(x)≡P₂(x)Q₁(x).

This is precisely when the two rational expressions represent the same rational function.

A rational expression is called relatively prime if its numerator and denominator are relatively prime, i.e., not both divisible by the same poly- nomial of positive degree. If a rational expression is not relatively prime, then canceling all common polynomial factors leads to an equivalent ratio- nal which is.

Finally we say that a rational expressionΦ^µ,νis a solution ofS^µ,νif its coefficients solve S^µ,ν. As noted before,Φ^µ,ν solvesS^µ,ν if it solves A^µ,ν. Rational interpolation is complicated by the fact that the converse need not hold.

(2.2.1.3) Theorem. The homogeneous linear system of equations S^µ,ν always has nontrivial solutions. For each such solution

Φ^µ,ν≡ P^µ,ν(x) Q^µ,ν(x),

Q^µ,ν(x)≡0 holds, i.e., all nontrivial solutions define rational functions.

Proof.The homogeneous linear systemS^µ,ν hasµ+ν+ 1 equations for µ+ν+ 2 unknowns. As a homogeneous linear system with more unknowns than equations,S^µ,ν has nontrivial solutions

(a₀, a₁, . . . , a_µ, b₀, . . . , b_ν)= (0, . . . ,0,0, . . . ,0). For any such solution,Q^µ,ν(x)≡0, since

Q^µ,ν(x)≡b₀+b₁x+· · ·+b_νx^ν ≡0

would imply that the polynomialP^µ,ν(x)≡a₀+a₁+· · ·+a_µx^µ has the zeros

P^µ,ν(x_i) = 0, i= 0,1, . . . , µ+ν .

It would follow thatP^µ,ν(x)≡0, since the polynomialP^µ,ν has at most degreeµ, and vanishes atµ+ν+ 1≥µ+ 1 different locations, contradicting (a₀, a₁, . . . , a_µ, b₀, . . . , b_ν)= (0, . . . ,0). The following theorem shows that the rational interpolation problem has a unique solution if it has a solution at all.

(2.2.1.4) Theorem. If Φ₁ and Φ₂ are both (nontrivial) solutions of the homogeneous linear system S^µ,ν, then they are equivalent (Φ₁ ∼Φ₂), that is, they determine the same rational function.

Proof.If bothΦ₁(x)≡P₁(x)/Q₁(x) andΦ₂(x)≡P₂(x)/Q₂(x) solveS^µ,ν, then the polynomial

P(x) :≡P₁(x)Q₂(x)−P₂(x)Q₁(x) hasµ+ν+ 1 different zeros

P(x_i) =P₁(x_i)Q₂(x_i)−P₂(x_i)Q₁(x_i)

=f_iQ₁(x_i)Q₂(x_i)−f_iQ₂(x_i)Q₁(x_i)

= 0, i= 0, . . . , µ+ν.

Since the degree of polynomial P does not exceed µ+ν, it must vanish identically, and it follows thatΦ₁(x)∼Φ₂(x).

Note that the converse of the above theorem does not hold: a rational ex- pressionΦ₁may well solveS^µ,νwhereas some equivalent rational expression Φ₂ does not. The previously considered example furnishes a case in point.

Combining Theorems (2.2.1.3) and (2.2.1.4), we find that there exists for each rational interpolation problem A^µ,ν a unique rational function, which is represented by any rational expressionΦ^µ,ν that solves the corre- sponding linear systemS^µ,ν. Either this rational function satisfies (2.2.1.1), thereby solvingA^µ,ν, orA^µ,νis not solvable at all. In the latter case, there must be some support point (x_i, f_i) which is “missed” by the rational func- tion. Such a support point is called inaccessible. Thus A^µ,ν is solvable if there are no inaccessible points.

SupposeΦ^µ,ν(x)≡P^µ,ν(x)/Q^µ,ν(x) is a solution to S^µ,ν. For anyi∈ {0,1, . . . , µ+ν}we distinguish the two cases:

1) Q^µ,ν(x_i)= 0, 2) Q^µ,ν(x_i) = 0.

In the first case, clearly Φ^µ,ν(x_i) = f_i. In the second case, however, the support point (x_i, f_i) may be inaccessible. Here

P^µ,ν(x_i) = 0

must hold by (2.2.1.2). Therefore, bothP^µ,ν and Q^µ,ν contain the factor x−x_i and are consequently not relatively prime. Thus:

(2.2.1.5).IfS^µ,ν has a solutionΦ^µ,ν which is relatively prime, then there are no inaccessible points: A^µ,ν is solvable.

GivenΦ^µ,ν, let ˜Φ^µ,νbe an equivalent rational expression which is relatively prime. We then have the general result:

(2.2.1.6) Theorem. SupposeΦ^µ,ν solves S^µ,ν. Then A^µ,ν is solvable — andΦ^µ,ν represents the solution — if and only ifΦ˜^µ,ν solves S^µ,ν. Proof.If ˜Φ^µ,ν solvesS^µ,ν, thenA^µ,ν is solvable by (2.2.1.5). If ˜Φ^µ,ν does not solveS^µ,ν, its corresponding rational function does not solveA^µ,ν. Even if the linear system S^µ,ν has full rank µ+ν+ 1, the rational inter- polation problem A^µ,ν may not solvable. However, since the solutions of S^µ,νare, in this case, uniquely determined up to a common constant factor ρ= 0, we have:

(2.2.1.7) Corollary to (2.2.1.6).IfS^µ,ν has full rank, thenA^µ,ν is solv- able if and only if the solutionΦ^µ,ν of S^µ,ν is relatively prime.

We say that the support points (x_i, f_i),i= 0, 1,. . .,σare inspecial po- sition if they are interpolated by a rational expression of degree type (κ, λ) with κ+λ < σ. In other words, the interpolation problem is solvable for a smaller combined degree of numerator and denominator than suggested by the number of support points. We observe that

(2.2.1.8) Theorem.The accessible support points of a nonsolvable inter- polation problemA^µ,ν are in special position.

Proof.Leti₁, . . . , i_α be the subscripts of the inaccessible points, and let Φ^µ,ν be a solution ofS^µ,ν. The numerator and the denominator of Φ^µ,ν were seen above to have the common factors x−x_i1, . . ., x−x_iα, whose cancellation leads to an equivalent rational expressionΦ^κ,λwithκ=µ−α, λ=ν−α. Φ^κ,λ solves the interpolation problemA^κ,λ which just consists of theµ+ν+ 1−αaccessible points. As

κ+λ+ 1 =µ+ν+ 1−2α < µ+ν+ 1−α,

the accessible points ofA^µ,ν are clearly in special position.

The observation (2.2.1.8) makes it clear that nonsolvability of the ra- tional interpolation problem is a degeneracy phenomenon: solvability can be restored by arbitrarily small perturbations of the support points. In what follows, we will therefore restrict our attention tofully nondegenerate problems, that is, problems for which no subset of the support points is in special position. Not only isA^µ,ν solvable in this case, but so are all prob- lemsA^κ,λofκ+λ+ 1 of the original support points whereκ+λ≤µ+ν.

For further details see Milne (1950) and Maehly and Witzgall (1960).

Most of the following discussion will be of recursive procedures for solv- ing rational interpolation problemsA^µ,ν. With each step of such recursions there will be associated a rational expression Π^µ,ν, of degree type (µ, ν) with µ≤ mand ν ≤n, and either the numerator or the denominator of Φ^µ,ν will be increased by 1. Because of the availability of this choice, the recursion methods for rational interpolation are more varied than those for polynomial interpolation. It will be helpful to plot the sequence of degree types (µ, ν) which are encountered in a particular recursion as paths in a diagram:

µ= 0 1 2 3 . . .

ν= 0 ^{• • •}

1 ^{• •}

2 ^{• •}

3 ^•

...

We will distinguish two kinds of algorithms. The first kind is analogous to Newton’s method of interpolation: A tableau of quantities analogous to divided differences is generated from which coefficients are gathered for an interpolating rational expression. The second kind corresponds to the Neville-Aitken approach of generating a tableau of values of intermediate rational functionsΦ^µ,ν. These values relate to each other directly.

2.2.2 Inverse and Reciprocal Differences. Thiele’s Continued