Polynomial interpolation

In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. In other words, you are given some data points (such as obtained by sampling), and you want to find a polynomial which goes exactly through these points.

Definition

Given a set of n+1 data points (x_i,y_i) where no two x_i are the same we are trying to find a polynomial p of degree n with the property

$p(x_{i})=y_{i}{\mbox{ , }}i=0,\ldots ,n$

The unisolvence theorem states that the polynomial p of degree n is uniquely defined by the n+1 data points.

Or to phrase it in terms of linear algebra: For n+1 interpolation nodes there exists a vector space isomorphism

$L_{n}:\mathbb {K} ^{n+1}\to \Pi _{n}$

where $\Pi _{n}$ is the vector space of polynomials with degree n.

Constructing the interpolation polynomial

The red dots denote the data points (x_k,y_k), while the blue curve shows the interpolation polynomial.

Suppose that the interpolation polynomial is given by

$p(t)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}.\qquad (1)$

The statement that p interpolates the data points means that

$p(x_{i})=y_{i},\qquad \forall i\in \left\{0,1,\dots ,n\right\}.$

If we substitute equation (1) in here, we get a system of linear equations in the coefficients $a_{k}$. The system in matrix-vector form reads

${\begin{bmatrix}x_{0}^{n}&x_{0}^{n-1}&x_{0}^{n-2}&\ldots &x_{0}&1\\x_{1}^{n}&x_{1}^{n-1}&x_{1}^{n-2}&\ldots &x_{1}&1\\\vdots &\vdots &\vdots &&\vdots &\vdots \\x_{n}^{n}&x_{n}^{n-1}&x_{n}^{n-2}&\ldots &x_{n}&1\end{bmatrix}}{\begin{bmatrix}a_{n}\\a_{n-1}\\\vdots \\a_{0}\end{bmatrix}}={\begin{bmatrix}y_{0}\\y_{1}\\\vdots \\y_{n}\end{bmatrix}}$

We have to solve this system for $a_{k}$ to construct the interpolant $p(x)$.

The matrix on the left is commonly referred to as a Vandermonde matrix. Its determinant is nonzero, which proves the unisolvence theorem: there exists a unique interpolating polynomial.

Non-Vandermonde solutions

We are trying to construct our unique interpolation polynomial in the vector space $\Pi _{n}$ that is the vector space of polynomials of degree n. When using a monomial basis for $\Pi _{n}$ we have to solve the Vandermonde matrix to construct the coefficients $a_{k}$ for the interpolation polynomial. This can be a very costly operation (as counted in clock cycles of a computer trying to do the job). By choosing another basis for $\Pi _{n}$ we can simplify the calculation of the coeffiecients but then of course we have to do additional calculations when we want to express the interpolation polynomial in terms of a monomial basis.

One method is to write the interpolation polynomial in the Newton form and use the method of divided differences to construct the coefficients. The cost is O$(n^{2})$ operations, while Gaussian elimination costs O$(n^{3})$ operations. Furthermore, you only need to do a bit of extra work if an extra point is added to the data set, while for the other methods, you have to redo the whole computation.

Another method is to use the Lagrange form of the interpolation polynomial. The resulting formula immediately shows that the interpolation polynomial exists under the conditions stated in the above theorem.

The Bernstein form was used in a constructive proof of the Weierstrass approximation theorem by Bernstein and has nowadays gained great importance in computer graphics in the form of Bezier curves.

Interpolation error

When interpolating a given function f by a polynomial of degree n at the nodes x₀,...,x_n we get the error

$f(x)-p_{n}(x)=f[x_{0},\ldots ,x_{n},x]\prod _{i=0}^{n}(x-x_{i})$

where

$f[x_{0},\ldots ,x_{n},x]$

is the notation for divided differences. When f is n+1 times continously differentiable on the smallest interval I which contains the nodes x_i then we can write the error in the Lagrange form as

$f(x)-p_{n}(x)={\frac {f^{(n+1)}(\xi )}{(n+1)!}}\prod _{i=0}^{n}(x-x_{i})$

for some $\xi$ in I. Thus the remainder term in the Lagrange form of the Taylor theorem is a special case of interpolation error when all interpolation nodes x_i are identical.

In the case of equally spaced interpolation nodes $x_{i}=x_{0}+ih$, it follows that the interpolation error is O$(h^{n})$. However, this does not necessarily mean that the error goes to zero as n → ∞. In fact, the error may increase without bound near the ends of the interval $[x_{0},x_{n}]$. This is called Runge's phenomenon.

The above error bound suggests choosing the interpolation points x_i such that the product | ∏ (x − x_i) | is as small as possible. The Chebyshev nodes achieve this.

Lebesgue constants

We fix the interpolation nodes x₀, ..., x_n and an interval [a, b] containing all the interpolation nodes. The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space C([a, b]) of all continuous functions on [a, b] to itself. The map X is linear and it is a projection on the subspace Π_n of polynomials of degree n or less.

The Lebesgue constant L is defined as the operator norm of X. This definition requires us to specify a norm on C([a, b]). We will here restrict our attention to the maximum norm.

The Lebesgue constant can be used to give another error estimate for polynomial interpolation. Let p^* denote the best approximation of f among the polynomials of degree n or less. In other words, p^* minimizes ||p−f|| among all p ∈ Π_n. Then ||f−X(f)|| ≤ ||f−p^*|| + ||p^*−X(f)|| by the triangle inequality. But X is a projection on Π_n, so p^*−X(f) = X(p^*−f). Taken together, this implies that

$\|f-X(f)\|\leq (L+1)\|f-p^{*}\|.$

In other words, the interpolation polynomial is at most a factor (L+1) worse than the best possible approximation. This suggests that we look for a set of interpolation nodes that L small.

The Lebesgue constant can be expressed in terms of the Lagrange basis polynomials

$l_{j}(x):=\prod _{i=0,j\neq i}^{n}{\frac {x-x_{i}}{x_{j}-x_{i}}}$

In fact, we have

$L=\max _{x\in [a,b]}\sum _{j=0}^{n}|l_{j}(x)|.$

Nevertheless, it is not easy to find an explicit expression for L.

In the case of equidistant nodes, the Lebesgue constant grows exponentially. More precisely, we have the following asymptotic estimate

$L\sim {\frac {2^{n+1}}{{\mbox{e}}n\log n}}\quad {\mbox{as}}\quad n\to \infty .$

On the other hand, the Lebesgue constant grows only logarithmically if Chebyshev nodes are used, since we have

$L\sim {\frac {2}{\pi }}\log(n+1)\quad {\mbox{as}}\quad n\to \infty .$

This is asymptotically optimal, because for any set of interpolation nodes we have the bound

$L\geq {\frac {2}{\pi }}\log(n+1)+C\quad {\mbox{for some constant }}C.$

We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation.

Convergence properties

It is natural to ask, for which classes of functions and for which interpolation nodes the sequence of interpolating polynomials converges to the interpolated function? Convergence may be understood in different ways, e.g. pointwise, uniform or in some integral norm. The aspects of uniform convergence are discussed below.

The following theorem seems to be a rather encouraging answer:

For any function f(x) continuous on an interval [a,b] there exists a table of nodes for which the sequence of interpolating polynomials $p_{n}(x)$ converges to f(x) uniformly on [a,b].

The proof. It's clear that the sequence of polynomials of best approximation $p_{n}^{*}(x)$ converges to f(x) uniformly (due to Weierstrass approximation theorem). Now we have only to show that each $p_{n}^{*}(x)$ may be obtained by means of interpolation on certain nodes. But this is true due to a special property of polynomials of best approximation known from Chebyshev alternance theorem. Specifically, we know that such polynomials should intersect f(x) at least n+1 times. Choosing the points of intersection as interpolation nodes we obtain the interpolating polynomial coinciding with the best approximation polynomial.

The defect of this method, however, is that interpolation nodes should be calculated anew for each new function f(x), but the algorithm is hard to be implemented numerically. Does there exist a single table of nodes for which the sequence of interpolating polynomials converge to any continuous function f(x)? The answer is unfortunately negative as it is stated by the following theorem:

For any table of nodes there is a continuous function f(x) on an interval [a,b] for which the sequence of interpolating polynomials diverges on [a,b].

The proof essentially uses the lower bound estimation of the Lebesgue constant, which we defined bove to be the operator norm of X_n (where X_n is the projection operator on Π_n). Now we seek a table of nodes for which

$\lim _{n\to \infty }X_{n}f=f$ for any $f\in C([a,b])$.

Due to the Banach-Steinhaus theorem this is only possible when norms of X_n are uniformly bounded, which cannot be true since we know that $\|X_{n}\|\geq {\frac {2}{\pi }}\log(n+1)+C$ .

For example, if equidistant points are chosen as interpolation nodes, the function from Runge's phenomenon demonstrates divergence of such interpolation. Note that this function is not only continuous but even infinitely times differentiable on [-1,1]. For better Chebyshev nodes, however, such an example is much harder to find because of the theorem:

For every absolutely continuous function on [-1,1] the sequence of interpolating polynomials constructed on Chebyshev nodes converges to f(x) uniformly.

Related concepts

Runge's phenomenon shows that for high values of n, the interpolation polynomial may oscillate wildly between the data points. This problem is commonly resolved by the use of spline interpolation. Here, the interpolant is not a polynomial but a spline: a chain of several polynomials of a lower degree.

Using harmonic functions to interpolate a periodic function is usually done using Fourier series, for example in discrete Fourier transform. This can be seen as a form of polynomial interpolation with harmonic base functions, see trigonometric interpolation and trigonometric polynomial.

References

• Kendell A. Atkinson (1988). An Introduction to Numerical Analysis (2nd ed.), Chapter 3. John Wiley and Sons. ISBN 0-471-50023-2
• L. Brutman (1997), Lebesgue functions for polynomial interpolation — a survey, Ann. Numer. Math. 4, 111–127. Preprint available at the MathSoft website.
• M.J.D. Powell (1981). Approximation Theory and Method, Chapter 4. Cambridge University Press. ISBN 0-521-29514-9.
• Michelle Schatzman (2002). Numerical Analysis: A Mathematical Introduction, Chapter 4. Clarendon Press, Oxford. ISBN 0-19-850279-6.
• Endre Süli and David Mayers (2003). An Introduction to Numerical Analysis, Chapter 6. Cambridge University Press. ISBN 0-521-00794-1.