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 of the dataset.
Applications
Polynomials can be used to approximate complicated curves, for example, the shapes of letters in typography, given a few points. A relevant application is the evaluation of the natural logarithm and trigonometric functions: pick a few known data points, create a lookup table, and interpolate between those data points. This results in significantly faster computations. Polynomial interpolation also forms the basis for algorithms in numerical quadrature and numerical ordinary differential equations and Secure Multi Party Computation, Secret Sharing schemes.Polynomial interpolation is also essential to perform sub-quadratic multiplication and squaring such as Karatsuba multiplication and Toom–Cook multiplication, where an interpolation through points on a polynomial which defines the product yields the product itself. For example, given a = f = a_{0}x^{0} + a_{1}x^{1} +... and b = g = b_{0}x^{0} + b_{1}x^{1} +..., the product ab is equivalent to W = fg. Finding points along W by substituting x for small values in f and g yields points on the curve. Interpolation based on those points will yield the terms of W and subsequently the product ab. In the case of Karatsuba multiplication this technique is substantially faster than quadratic multiplication, even for modest-sized inputs. This is especially true when implemented in parallel hardware.
Definition
Given a set of data points where no two are the same, a polynomial is said to interpolate the data if for each.Interpolation theorem
Given distinct points and corresponding values, there exists a unique polynomial of degree at most that interpolates the data.Proof
Consider the Lagrange basis functions given by.
Notice that is a polynomial of degree. Furthermore, for each we have, where is the Kronecker delta. It follows that the linear combination
is an interpolating polynomial of degree.
To prove uniqueness, assume that there exists another interpolating polynomial of degree at most. Since for all, it follows that the polynomial has distinct zeros. However, is of degree at most and, by the fundamental theorem of algebra, can have at most zeros; therefore,.
Corollary
An interesting corollary to the interpolation theorem is that if is a polynomial of degree at most, then the interpolating polynomial of at distinct points is itself.Unisolvence theorem
Given a set of data points where no two are the same, one is looking for a polynomial of degree at most with the propertyThe unisolvence theorem states that such a polynomial p exists and is unique, and can be proved by the Vandermonde matrix, as described below.
The theorem states that for interpolation nodes, polynomial interpolation defines a linear bijection
where Π_{n} is the vector space of polynomials of degree at most.
Constructing the interpolation polynomial
Suppose that the interpolation polynomial is in the formThe statement that p interpolates the data points means that
If we substitute equation in here, we get a system of linear equations in the coefficients. The system in matrix-vector form reads the following multiplication:
We have to solve this system for to construct the interpolant p. The matrix on the left is commonly referred to as a Vandermonde matrix.
The condition number of the Vandermonde matrix may be large, 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 operations instead of the O required by Gaussian elimination. These methods rely on constructing first a Newton interpolation of the polynomial and then converting it to the monomial form above.
Alternatively, we may write down the polynomial immediately in terms of Lagrange polynomials:
For matrix arguments, this formula is called Sylvester's formula and the matrix-valued Lagrange polynomials are the Frobenius covariants.
Uniqueness of the interpolating polynomial
Proof 1
Suppose we interpolate through data points with an at-most degree polynomial p. Suppose also another polynomial exists also of degree at most that also interpolates the points; call it q.Consider. We know,
- r is a polynomial
- r has degree at most, since p and q are no higher than this and we are just subtracting them.
- At the data points,. Therefore, r has roots.
So q is identical with p, and q is unique.
Proof 2
Given the Vandermonde matrix used above to construct the interpolant, we can set up the systemTo prove that V is nonsingular we use the Vandermonde determinant formula:
since the points are distinct, the determinant can't be zero as is never zero, therefore V is nonsingular and the system has a unique solution.
Either way this means that no matter what method we use to do our interpolation: direct, Lagrange etc., we will always get the same polynomial.
Non-Vandermonde solutions
We are trying to construct our unique interpolation polynomial in the vector space Π_{n} of polynomials of degree. When using a monomial basis for Π_{n} we have to solve the Vandermonde matrix to construct the coefficients for the interpolation polynomial. This can be a very costly operation. By choosing another basis for Π_{n} we can simplify the calculation of the coefficients but then 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, e.g. Neville's algorithm. The cost is O operations, while Gaussian elimination costs O operations. Furthermore, you only need to do O 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. Lagrange formula is to be preferred to Vandermonde formula when we are not interested in computing the coefficients of the polynomial, but in computing the value of p in a given x not in the original data set. In this case, we can reduce complexity to O.
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.
Linear combination of the given values
The Lagrange form of the interpolating polynomial is a linear combination of the given values. In many scenarios, an efficient and convenient polynomial interpolation is a linear combination of the given values, using previously known coefficients. Given a set of data points where each data point is a pair and where no two positions are the same, the interpolation polynomial in the Lagrange form is a linear combinationof the given values with each coefficient given by evaluating the corresponding Lagrange basis polynomial using the given positions.
Each coefficient in the linear combination depends on the given positions and the desired position, but not on the given values. For each coefficient, inserting the values of the given positions and simplifying yields an expression, which depends only on. Thus the same coefficient expressions can be used in a polynomial interpolation of a given second set of data points at the same given positions, where the second given values differ from the first given values. Using the same coefficient expressions as for the first set of data points, the interpolation polynomial of the second set of data points is the linear combination
For each coefficient in the linear combination, the expression resulting from the Lagrange basis polynomial only depends on the relative spaces between the given positions, not on the individual value of any position. Thus the same coefficient expressions can be used in a polynomial interpolation of a given third set of data points
where each position is related to the corresponding position in the first set by and the desired positions are related by, for a constant scaling factor a and a constant shift b for all positions. Using the same coefficient expressions as for the first set of data points, the interpolation polynomial of the third set of data points is the linear combination
In many applications of polynomial interpolation, the given set of data points is at equally spaced positions. In this case, it can be convenient to define the x-axis of the positions such that. For example, a given set of 3 equally-spaced data points is then.
The interpolation polynomial in the Lagrange form is the linear combination
This quadratic interpolation is valid for any position x, near or far from the given positions. So, given 3 equally-spaced data points at defining a quadratic polynomial, at an example desired position, the interpolated value after simplification is given by linear combination|
This is a quadratic interpolation typically used in the Multigrid method. Again given 3 equally-spaced data points at defining a quadratic polynomial, at the next equally spaced position, the interpolated value after simplification is given by
In the above polynomial interpolations using a linear combination of the given values, the coefficients were determined using the Lagrange method. In some scenarios, the coefficients can be more easily determined using other methods. Examples follow.
According to the method of finite differences, for any polynomial of degree d or less, any sequence of values at equally spaced positions has a th difference exactly equal to 0. The element s_{d+1} of the Binomial transform is such a th difference. This area is surveyed here. The binomial transform, T, of a sequence of values, is the sequence defined by
Ignoring the sign term, the coefficients of the element s_{n} are the respective elements of the row n of Pascal's Triangle. The triangle of binomial transform coefficients is like Pascal's triangle. The entry in the nth row and kth column of the BTC triangle is for any non-negative integer n and any integer k between 0 and n. This results in the following example rows n = 0 through n = 7, top to bottom, for the BTC triangle:
For convenience, each row n of the above example BTC triangle also has a label. Thus for any polynomial of degree d or less, any sequence of values at equally spaced positions has a linear combination result of 0, when using the elements of row d as the corresponding linear coefficients.
For example, 4 equally spaced data points of a quadratic polynomial obey the linear equation given by row of the BTC triangle. This is the same linear equation as obtained above using the Lagrange method.
The BTC triangle can also be used to derive other polynomial interpolations. For example, the above quadratic interpolation
can be derived in 3 simple steps as follows. The equally spaced points of a quadratic polynomial obey the rows of the BTC triangle with or higher. First, the row spans the given and desired data points with the linear equation
Second, the unwanted data point is replaced by an expression in terms of wanted data points. The row provides a linear equation with a term, which results in a term by multiplying the linear equation by 4. Third, the above two linear equations are added to yield a linear equation equivalent to the above quadratic interpolation for.
Similar to other uses of linear equations, the above derivation scales and adds vectors of coefficients. In polynomial interpolation as a linear combination of values, the elements of a vector correspond to a contiguous sequence of regularly spaced positions. The p non-zero elements of a vector are the p coefficients in a linear equation obeyed by any sequence of p data points from any degree d polynomial on any regularly spaced grid, where d is noted by the subscript of the vector. For any vector of coefficients, the subscript obeys. When adding vectors with various subscript values, the lowest subscript applies for the resulting vector. So, starting with the vector of row and the vector of row of the BTC triangle, the above quadratic interpolation for is derived by the vector calculation
Similarly, the cubic interpolation typical in the Multigrid method,
can be derived by a vector calculation starting with the vector of row and the vector of row of the BTC triangle.
Interpolation error
When interpolating a given function f by a polynomial of degree at the nodes x_{0},...,x_{n} we get the errorwhere
is the notation for divided differences.
If f is times continuously differentiable on a closed interval I and is a polynomial of degree at most that interpolates f at distinct points in that interval, then for each x in the interval there exists in that interval such that
The above error bound suggests choosing the interpolation points such that the product is as small as possible. The Chebyshev nodes achieve this.
Proof
Set the error term asand set up an auxiliary function:
where
Since are roots of and, we have, which means has at least roots. From Rolle's theorem, has at least roots, then has at least one root, where is in the interval.
So we can get
Since is a polynomial of degree at most, then
Thus
Since is the root of, so
Therefore,
Thus the remainder term in the Lagrange form of the Taylor theorem is a special case of interpolation error when all interpolation nodes are identical. Note that the error will be zero when for any i. Thus, the maximum error will occur at some point in the interval between two successive nodes.
For equally spaced intervals
In the case of equally spaced interpolation nodes where, for and where the product term in the interpolation error formula can be bound asThus the error bound can be given as
However, this assumes that is dominated by, i.e.. In several cases, this is not true and the error actually increases as . That question is treated in the section Convergence properties.
Lebesgue constants
We fix the interpolation nodes x_{0},..., x_{n} and an interval 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 of all continuous functions on 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. One has :
In other words, the interpolation polynomial is at most a factor 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 = 1 / on the interval. The interpolation error grows without bound as. Another example is the function f = |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:
Proof. It's clear that the sequence of polynomials of best approximation converges to f uniformly. Now we have only to show that each 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 the equioscillation theorem. Specifically, we know that such polynomials should intersect f at least 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, 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? 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 X_{n}. Now we seek a table of nodes for which
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
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
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.
At last, multivariate interpolation for higher dimensions.