Polynomial interpolation on arvau

Wikipedia / Polynomial interpolation

In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points in the dataset.

Given a set of is said to interpolate.

There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials.

Applications

The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions. Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point. Polynomial interpolation also forms the basis for algorithms in numerical quadrature (Simpson's rule) and numerical ordinary differential equations (multigrid methods).

In computer graphics, polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in typography. This is usually done with Bézier curves, which are a simple generalization of interpolation polynomials (having specified tangents as well as specified points).

In numerical analysis, polynomial interpolation is essential to perform sub-quadratic multiplication and squaring, such as Karatsuba multiplication and Toom–Cook multiplication, where interpolation through points on a product polynomial yields the specific product required. For example, given a = f(x) = a0x0 + a1x1 + ··· and b = g(x) = b0x0 + b1x1 + ···, the product ab is a specific value of W(x) = f(x)g(x). One may easily find points along W(x) at small values of x, and interpolation based on those points will yield the terms of W(x) and the specific product ab. As fomulated in Karatsuba multiplication, this technique is substantially faster than quadratic multiplication, even for modest-sized inputs, especially on parallel hardware.

In computer science, polynomial interpolation also leads to algorithms for secure multi party computation and secret sharing.

Interpolation theorem

, polynomial interpolation defines a linear bijection between the (n and the vector space of real polynomials of degree at most n

This is a type of unisolvence theorem. The theorem is also valid over any infinite field, for example the rational or complex numbers.

First proof

Consider the Lagrange basis functions

zeros,.

Second proof

Write out the interpolation polynomial in the form , we get a system of linear equations, which reads in matrix-vector form as the following multiplication

. The matrix X on the left is a Vandermonde matrix, whose determinant are all distinct. This ensures that the matrix is invertible exists and is unique.

Corollary

itself.

Constructing the interpolation polynomial

Lagrange interpolation

We may write down the polynomial immediately in terms of Lagrange polynomialsFor matrix arguments, this formula is called Sylvester's formula and the matrix-valued Lagrange polynomials are the Frobenius covariants.

Newton interpolation

Theorem

Proof:

is the required polynomial interpolation. The function can thus be expressed as:

Polynomial coefficients

, we have to solve the lower triangular matrix from above equation in matrix form:

The coefficients are derived as

where

is the notation for divided differences. Thus, Newton polynomials are used to provide a polynomial interpolation formula of n points.

Newton forward formula

are arranged consecutively with equal spacing.

for i = 0, 1, ..., k. So the Newton polynomial becomes

Since the relationship between divided differences and forward differences is given as:, the Newton forward interpolation formula

Newton backward formula

, the Newton polynomial becomes

for i = 0, 1, ..., k, then,

Since the relationship between divided differences and backward differences is given as:, the Newton backward interpolation formula

Lozenge diagram

A Lozenge diagram is a diagram that is used to describe different interpolation formulas that can be constructed for a given data set. A line starting on the left edge and tracing across the diagram to the right can be used to represent an interpolation formula if the following rules are followed:

Left to right steps indicate addition whereas right to left steps indicate subtraction
If the slope of a step is positive, the term to be used is the product of the difference and the factor immediately below it. If the slope of a step is negative, the term to be used is the product of the difference and the factor immediately above it.
If a step is horizontal and passes through a factor, use the product of the factor and the average of the two terms immediately above and below it. If a step is horizontal and passes through a difference, use the product of the difference and the average of the two terms immediately above and below it.

Proof of equivalence

. Proving the equivalence of these three two-step paths should prove that all (n-step) paths can be morphed with the same starting and ending, all of which represents the same formula.

Path (a):

Path (b):

Path (c):

Subtracting contributions from path a and b:

Path (a)

Path (b)

Path (c)

and are hence equivalent. Hence these paths can be morphed to start from the leftmost corner and end in a common point.

Newton formula

consecutively arranged points, equivalent to Newton's forward interpolation formula:

consecutively arranged points, equivalent to Newton's backward interpolation formula:

is the number corresponding to that introduced in Newton interpolation.

Gauss formula

with negative slope, we get Gauss forward formula:

with positive slope, we get Gauss backward formula:

Stirling formula

, we get Stirling formula:

Stirling formula is the average of Gauss forward and Gauss backward formulas.

Bessel formula

, we get Bessel formula:

Vandermonde algorithms

The Vandermonde matrix in the second proof above may have large condition number, causing large errors when computing the coefficients if the system of equations is solved using Gaussian elimination.

Several authors have therefore proposed algorithms which exploit the structure of the Vandermonde matrix to compute numerically stable solutions in O(n2) operations instead of the O(n3) required by Gaussian elimination. These methods rely on constructing first a Newton interpolation of the polynomial and then converting it to a monomial form.

Non-Vandermonde algorithms

To find the interpolation polynomial p(x) in the vector space P(n) of polynomials of degree , we may use the usual monomial basis for P(n) and invert the Vandermonde matrix by Gaussian elimination, giving a computational cost of O(n3) operations. To improve this algorithm, a more convenient basis for P(n) can simplify the calculation of the coefficients, which must then be translated back in terms of the monomial basis.

One method is to write the interpolation polynomial in the Newton form (i.e. using Newton basis) and use the method of divided differences to construct the coefficients, e.g. Neville's algorithm. The cost is O(n2) operations. Furthermore, you only need to do O(n) 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 preferred when the aim is not to compute the coefficients of p(x), but only a single value p(a) at a point x = a not in the original data set. The Lagrange form computes the value p(a) with complexity O(n2).

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

Interpolations as linear combinations of values

may be considered as a linear combination. For example, the interpolation polynomial in the Lagrange form

, we can use the same coefficients

(an affine transformation we can use a transformed version of the previous coefficient polynomials: and write the interpolation polynomial as: often have equally spaced positions. For example, consider the data points.The interpolation polynomial in the Lagrange form is the linear combination

The case of equally spaced points can also be treated by the method of finite differences. Iterating this operation gives the nth

of degree d difference of this sequence is identically zero: . gives the same interpolation equation obtained above using the Lagrange method.

Interpolation error: Lagrange remainder formula

When interpolating a given function f of degree at the nodes x0,..., xn is the (n+1)st divided difference of the data points .Furthermore, there is a Lagrange remainder form of the error, for a function f which is of degree at most that interpolates f at

This error bound suggests choosing the interpolation points , which is achieved by the Chebyshev nodes.

Proof of Lagrange remainder

, and define an auxiliary function

is a polynomial of degree at most

Now, since , which means has at least roots. From Rolle's theorem has at least has at least one root in the interval

This parallels the reasoning behind the Lagrange remainder term in the Taylor theorem; in fact, the Taylor remainder is a special case of interpolation error when all interpolation nodes are identical. for any i. Thus, the maximum error will occur at some point in the interval between two successive nodes.

Equally spaced intervals

the product term in the interpolation error formula can be bound as

. In several cases, this is not true and the error actually increases as (see Runge's phenomenon). That question is treated in the section Convergence properties.

Lebesgue constants

We fix the interpolation nodes x0, ..., xn 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 of polynomials of degree n or less.

The Lebesgue constant L is defined as the operator norm of X. One has (a special case of Lebesgue's lemma

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 makes L small. In particular, we have for Chebyshev nodes

We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation, as the growth in n is exponential for equidistant nodes. However, those nodes are not optimal.

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 as ? Convergence may be understood in different ways, e.g. pointwise, uniform or in some integral norm.

The situation is rather bad for equidistant nodes, in that uniform convergence is not even guaranteed for infinitely differentiable functions. One classical example, due to Carl Runge, is the function f(x) = 1 / (1 + x2) on the interval . The interpolation error grows without bound as . Another example is the function f(x) = |x| on the interval , for which the interpolating polynomials do not even converge pointwise except at the three points x = ±1, 0.

One might think that better convergence properties may be obtained by choosing different interpolation nodes. The following result seems to give a rather encouraging answer:

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:

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

Due to the Banach–Steinhaus theorem, this is only possible when norms of Xn are uniformly bounded, which cannot be true since we know that

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 differentiable on . For better Chebyshev nodes, however, such an example is much harder to find due to the following result:

Related concepts

Runge's phenomenon shows that for high values of , 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.

Interpolation of periodic functions by harmonic functions is accomplished by Fourier transform. This can be seen as a form of polynomial interpolation with harmonic base functions, see trigonometric interpolation and trigonometric polynomial.

Hermite interpolation problems are those where not only the values of the polynomial p at the nodes are given, but also all derivatives up to a given order. This turns out to be equivalent to a system of simultaneous polynomial congruences, and may be solved by means of the Chinese remainder theorem for polynomials. Birkhoff interpolation is a further generalization where only derivatives of some orders are prescribed, not necessarily all orders from 0 to a k.

Collocation methods for the solution of differential and integral equations are based on polynomial interpolation.

The technique of rational function modeling is a generalization that considers ratios of polynomial functions.