Newton’s Interpolation Formula: Divided Differences

2.1 Interpolation by Polynomials

2.1.3 Newton’s Interpolation Formula: Divided Differences

The recursion (2.1.2.5) then translates into (2.1.2.6)

Q_ik:= (D_i,k−1−Q_i−1,k−1) x_i−x x_i−k−x_i D_ik:= (D_i,k−1−Q_i−1,k−1)x_i−k−x x_i−k−x_i











1≤k≤i, i= 0,1, . . . .

Starting with Q_i0 := D_i0 := f_i, one calculates Q_ik, D_ik from the above recursion. Finally

P_nn := f_n+ n k=1

Q_nk.

If the valuesf₀, . . . , f_n are close to each other, the quantitiesQ_ik will be small compared to f_i. This suggests forming the sum of the “corrections”

Q_n1, . . . , Q_nnfirst [contrary to (2.1.2.5)] and then adding it tof_n, thereby avoiding unnecessary roundoff errors.

Note finally that forx= 0 the recursion (2.1.2.5) takes a particularly simple form

(2.1.2.7a) T_i0:= f_i

(2.1.2.7b) T_ik:= T_i,k−1+T_i,k−1−T_i−1,k−1 x_i−k

x_i −1

, 1≤k≤i ,

— as does its analog (2.1.2.6). These forms are encountered when applying extrapolation methods.

For historical reasons mainly, we mentionAitken’s algorithm. . It is also based on (2.1.2.1), but uses different intermediate polynomials. Its tableau is of the form

x₀ f₀=P₀(x)

x₁ f₁=P₁(x) P₀₁(x)

x₂ f₂=P₂(x) P₀₂(x) P₀₁₂(x)

x₃ f₃=P₃(x) P₀₃(x) P₀₁₃(x) P₀₁₂₃(x)

· · · · · ·

The first column again contains the prescribed valuesf_i. Each subsequent entry derives from the previous entry in the same row and the top entry in the previous column according to (2.1.2.1b).

or if one wants to find interpolating values for several argumentsξ_j simul- taneously, then Newton’s interpolation formula ff is to be preferred. Here we write the interpolating polynomialP ∈Π_n,P(x_i) =f_i,i= 0, 1,. . . n, in the form

(2.1.3.1)

P(x)≡P_01...n(x)

=a₀+a₁(x−x₀) +a₂(x−x₀)(x−x₁) +· · · +a_n(x−x₀)· · ·(x−x_n−1).

Note that the evaluation of (2.1.3.1) forx=ξmay be done recursively as indicated by the following expression:

P(ξ) = (· · ·(a_n(ξ−x_n−1) +a_n−1)(ξ−x_n−2) +· · ·+a₁)(ξ−x₀) +a₀. This requires fewer operations than evaluating (2.1.3.1) term by term. It corresponds to the so-called Horner scheme for evaluating polynomials which are given in the usual form, i.e. in terms of powers of x, and it shows that the representation (2.1.3.1) is well suited for evaluation.

It remains to determine the coefficientsa_iin (2.1.3.1). In principle, they can be calculated successively from

f₀=P(x₀) =a₀

f₁=P(x₁) =a₀+a₁(x₁−x₀)

f₂=P(x₂) =a₀+a₁(x₂−x₀) +a₂(x₂−x₀)(x₂−x₁)

· · ·

This can be done with ndivisions andn(n−1) multiplications. There is, however, a better way, which requires onlyn(n+ 1)/2 divisions and which produces useful intermediate results.

Observe that the two polynomialsP_{i0i1...ik−1}(x) andP_i0i1...ik(x) differ by a polynomial of degree k with k zeros x_i0, x_i1,. . ., x_ik−1, since both polynomials interpolate the corresponding support points. Therefore there exists a unique coefficient

(2.1.3.2) f_i0i1...ik such that

(2.1.3.3)

P_i0i1...ik(x) =P_{i0i1...ik−1}(x) +f_i0i1...ik(x−x_i0)(x−x_i1)· · ·(x−x_ik−1). From this and from the identityP_i0≡f_i0 it follows immediately that (2.1.3.4) P_i0i1...ik(x) =f_i0+f_i0i1(x−x_i0) +· · ·

+f_i0i1...ik(x−x_i0)(x−x_i1)· · ·(x−x_ik−1)

is a Newton representation of the interpolating polynomialP_i0,...,ik(x). The coefficients (2.1.3.2) are calledkth divided differences.

The recursion (2.1.2.1) for the partially interpolating polynomials translates into the recursion

(2.1.3.5) f_i0i1...ik= f_i1...ik−f_i0...ik−1 x_ik−x_i0

for the divided differences, since by (2.1.3.3), f_i1...ik and f_i0...ik−1 are the coefficients of the highest terms of the polynomialsP_i1i2...ik andP_{i0i1...ik−1}, respectively. The above recursion starts for k= 0 with the given support ordinatesf_i,i= 0, . . . , n. It can be used in various ways for calculating di- vided differencesf_i0,f_i0i1,. . .,f_i0i1...in, which then characterize the desired interpolating poynomialP=P_i0i1...in.

Because the polynomialP_i0i1...ikis uniquely determined by the support points it interpolates [Theorem (2.1.1.1)], the polynomial is invariant to any permutation of the indicesi₀,i₁, . . . , i_k, and so is its coefficientf_i0i1...ik of x^k. Thus:

(2.1.3.6).The divided differencesf_i0i1...ik are invariant to permutations of the indicesi₀,i₁, . . . , i_k: If

(j₀, j₁, . . . , j_k) = (i_s0, i_s1, . . . , i_sk) is a permutation of the indicesi₀,i₁, . . . , i_k, then

f_j0j1...jk=f_i0i1...ik.

If we choose to calculate the divided differences in analogy to Neville’s method — instead of, say, Aitken’s method — then we are led to the following tableau, called thedivided-difference scheme:

(2.1.3.7)

k= 0 1 2 · · ·

x₀ f₀

f₀₁

x₁ f₁ f₀₁₂

f₁₂ ... . ..

x₂ f₂ ...

... ...

The entries in the second column are of the form f₀₁= f₁−f₀

x₁−x₀, f₁₂= f₂−f₁ x₂−x₁, . . . , those in the third column,

f₀₁₂=f₁₂−f₀₁

x₂−x₀ , f₁₂₃=f₂₃−f₁₂

x₃−x₁ , . . . .

Clearly,

P(x)≡P_01...n(x)

≡f₀+f₀₁(x−x₀) +· · ·+f_01...n(x−x₀)(x−x₁)· · ·(x−x_n−1) is the desired solution to the interpolation problem at hand. The coefficients of the above expansion are found in the top descending diagonal of the divided-difference scheme (2.1.3.7).

Example.With the numbers of the example in sections 2.1.1 and 2.1.2, we have:

x₀= 0 f₀= 1

f₀₁= 2

x₁= 1 f₁= 3 f₀₁₂=−⁵₆ f₁₂=−¹₂

x₂= 3 f₂= 2

P₀₁₂(x) = 1 + 2·(x−0)−⁵₆(x−0)(x−1), P₀₁₂(2) = (−⁵₆·(2−1) + 2)(2−0) + 1 =¹⁰₃.

Instead of building the divided-difference scheme column by column, one might want to start with the upper left corner and add successive ascending diagonal rows. This amounts to adding new support points one at a time after having interpolated the previous ones. In the following ALGOL procedure, the entries in an ascending diagonal of (2.1.3.7) are found, after each increase ofi, in the top portion of arrayt, and the firsti coefficientsf_01...iare found in array a.

fori:= 0 step1 until n do begint[i] :=f[i];

for j:=i−1 step−1 until 0 do t[j] := (t[j+ 1]−t[j])/(x[i]−x[j]);

a[i] :=t[0]

end ;

Afterwards, the interpolationg polynomial (2.1.3.1) may be evaluated for any desired argumentz:

p:=a[n];

fori:=n−1 step −1 until 0 do p:=p×(z−x[i]) +a[i];

Some Newton representations of the same polynomial are numerically more trustworth to evaluate than others. Choosing the permutation so that

|ξ−x_ik| ≥ |ξ−x_ik−1|, k= 0,1, . . . , n−1,

dampens the error (see Section 1.3) during the Horner evaluation of

(2.1.3.8)

P(ξ)≡P_i0...in(ξ)≡f_i0+f_i0i1(ξ−x_i0)+· · ·+f_i0i1...in(ξ−x_i0)· · ·(ξ−x_in−1).

All Newton representations of the above kind can be found in the sin- gle divided-difference scheme which arises if the support arguments x_i, i= 0, . . . , n, are ordered by size: x_i < x_i+1 fori= 0, . . . , n−1. Then the preferred sequence of indices i₀, i₁, . . ., i_k is such that each index i_k is

“adjacent” to some previous index. More precisely, either i_k = min{i_l | 0≤l < k} −1 ori_k= max{i_l|0≤l < k}+ 1. Therefore the coefficients of (2.1.3.8) are found along a zigzag path—instead of the upper descend- ing diagonal—of the divided-difference scheme. Starting withf_i0, the path proceeds to the upper right neighbor if i_k < i_k−1, or to the lower right neighbor ifi_k > i_k−1.

Example.In the previous example, a preferred sequence forξ= 2 is i₀= 1, i₁= 2, i₂= 0.

The corresponding path in the divided difference scheme is indicated below:

x₀= 0 f₀= 1

f₀₁= 2

x₁= 1 f₁= 3 f₀₁₂=−⁵₆ f₁₂=−¹₂

x₂= 3 f₂= 2 The desired Newton representation is:

P₁₂₀(x) = 3−¹₂(x−1)−⁵₆(x−1)(x−3) P₁₂₀(2) = (−⁵₆(2−3)−¹₂(2−1) + 3 =¹⁰₃.

Frequently, the support ordinates f_i are values f(x_i) = f_i of a given function f(x), which one wants to approximate by interpolation. In this case, the divided differences may be considered as multivariate functions of the support argumentsx_i, and are historically written as

f[x_i0, x_i1, . . . , x_ik].

These functions satisfy (2.1.3.5). For instance, f[x₀] =f(x₀)

f[x₀, x₁] = f[x₁]−f[x₀]

x₁−x₀ =f(x₁)−f(x₀) x₁−x₀ f[x₀, x₁, x₂] = f[x₁, x₂]−f[x₀, x₁]

x₂−x₀

= f(x₀)(x₁−x₂) +f(x₁)(x₂−x₀) +f(x₂)(x₀−x₁) (x₁−x₀)(x₂−x₁)(x₀−x₂) .

Also, (2.1.3.6) gives immediately:

(2.1.3.9) Theorem.The divided differencesf[x_i0, . . . , x_ik]are symmetric functions of their arguments, i.e., they are invariant to permutations of the x_i0,. . .,x_ik.

If the functionf(x) is itself a polynomial, then we have the (2.1.3.10) Theorem.If f(x) is a polynomial of degreeN, then

f[x₀, . . . , x_k] = 0 fork > N.

Proof. Because of the unique solvability of the interpolation problem [Theorem (2.1.1.1)], P_01···k(x) ≡ f(x) for k ≥ N. The coefficient of x^k in P_01···k(x) must therefore vanish fork > N. This coefficient, however, is given byf[x₀, . . . , x_k] according to (2.1.3.3).

Example.f(x) =x²

xi k= 0 1 2 3 4

0 0

1

1 1 1

3 0

2 4 1 0

5 0

3 9 1

7

4 16

If the functionf(x) is sufficiently often differentiable, then its divided differences f[x₀, . . . , x_k] can also be defined if some of the arguments x_i coincide. For instance, if f(x) has a derivative at x₀, then it makes sense for certain purposes to define

f[x₀, x₀] :=f(x₀).

For a corresponding modification of the divided-difference scheme (2.1.3.7) see Section 2.1.5 on Hermite interpolation.