In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points and numbers , the Lagrange polynomial is the polynomial of the least degree that at each point assumes the corresponding value (i.e. the functions coincide at each point). The interpolating polynomial of the least degree is unique, however, and it is therefore more appropriate to speak of "the Lagrange form" of that unique polynomial rather than "the Lagrange interpolation polynomial," since the same polynomial can be arrived at through multiple methods. Although named after Joseph Louis Lagrange, who published it in 1795, it was first discovered in 1779 by Edward Waring and it is also an easy consequence of a formula published in 1783 by Leonhard Euler.1
Lagrange interpolation is susceptible to Runge's phenomenon, and the fact that changing the interpolation points requires recalculating the entire interpolant can make Newton polynomials easier to use. Lagrange polynomials are used in the Newton–Cotes method of numerical integration and in Shamir's secret sharing scheme in cryptography.
Given a set of k + 1 data points
where no two are the same, the interpolation polynomial in the Lagrange form is a linear combination
of Lagrange basis polynomials
where . Note how, given the initial assumption that no two are the same, , so this expression is always well-defined. The reason pairs with are not allowed is that no interpolation function such that would exist; a function can only get one value for each argument . On the other hand, if also , then those two points would actually be one single point.
For all , includes the term in the numerator, so the whole product will be zero at :
On the other hand,
In other words, all basis polynomials are zero at , except , for which it holds that , because it lacks the term.
It follows that , so at each point , , showing that interpolates the function exactly.
The function L(x) being sought is a polynomial in of the least degree that interpolates the given data set; that is, assumes value at the corresponding for all data points :
- In there are k terms in the product and each term contains one x, so L(x) (which is a sum of these k-degree polynomials) must also be a k-degree polynomial.
We consider what happens when this product is expanded. Because the product skips , if then all terms are (except where , but that case is impossible, as pointed out in the definition section—in that term, , and since , , contrary to ). Also if then since does not preclude it, one term in the product will be for , i.e. , zeroing the entire product. So
where is the Kronecker delta. So:
Thus the function L(x) is a polynomial with degree at most k and where .
Additionally, the interpolating polynomial is unique, as shown by the unisolvence theorem at polynomial interpolation article.
|This section requires expansion with: examples and explanation of what inverting means in this context. (February 2010)|
Solving an interpolation problem leads to a problem in linear algebra where we have to solve a matrix. Using a standard monomial basis for our interpolation polynomial we get the Vandermonde matrix. By choosing another basis, the Lagrange basis, we get the much simpler identity matrix = δi,j which we can solve instantly: the Lagrange basis inverts the Vandermonde matrix.
This construction is the same as the Chinese Remainder Theorem. Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears.
Find an interpolation formula for ƒ(x) = tan(x) given this set of known values:
The Lagrange basis polynomials are:
Thus the interpolating polynomial then is
We wish to interpolate ƒ(x) = x2 over the range 1 ≤ x ≤ 3, given these three points:
The interpolating polynomial is:
We wish to interpolate ƒ(x) = x3 over the range 1 ≤ x ≤ 3, given these 3 points:
The interpolating polynomial is:
The Lagrange form of the interpolation polynomial shows the linear character of polynomial interpolation and the uniqueness of the interpolation polynomial. Therefore, it is preferred in proofs and theoretical arguments. Uniqueness can also be seen from the invertibility of the Vandermonde matrix, due to the non-vanishing of the Vandermonde determinant.
But, as can be seen from the construction, each time a node xk changes, all Lagrange basis polynomials have to be recalculated. A better form of the interpolation polynomial for practical (or computational) purposes is the barycentric form of the Lagrange interpolation (see below) or Newton polynomials.
Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as Runge's phenomenon; the problem may be eliminated by choosing interpolation points at Chebyshev nodes.
we can rewrite the Lagrange basis polynomials as
or, by defining the barycentric weights2
we can simply write
which is commonly referred to as the first form of the barycentric interpolation formula.
The advantage of this representation is that the interpolation polynomial may now be evaluated as
which, if the weights have been pre-computed, requires only operations (evaluating and the weights ) as opposed to for evaluating the Lagrange basis polynomials individually.
The barycentric interpolation formula can also easily be updated to incorporate a new node by dividing each of the , by and constructing the new as above.
We can further simplify the first form by first considering the barycentric interpolation of the constant function :
Dividing by does not modify the interpolation, yet yields
which is referred to as the second form or true form of the barycentric interpolation formula. This second form has the advantage that need not be evaluated for each evaluation of .
- Polynomial interpolation
- Neville's algorithm
- Newton form of the interpolation polynomial
- Bernstein form of the interpolation polynomial
- Newton–Cotes formulas
- Carlson's theorem
- Lebesgue constant (interpolation)
- The Chebfun system
- Table of Newtonian series
- Meijering, Erik (2002), "A chronology of interpolation: from ancient astronomy to modern signal and image processing", Proceedings of the IEEE 90 (3): 319–342, doi:10.1109/5.993400.
- Jean-Paul Berrut & Lloyd N. Trefethen (2004). "Barycentric Lagrange Interpolation". SIAM Review 46 (3): 501–517. doi:10.1137/S0036144502417715.
- Hazewinkel, Michiel, ed. (2001), "Lagrange interpolation formula", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- ALGLIB has an implementations in C++ / C# / VBA / Pascal.
- GSL has a polynomial interpolation code in C
- Lagrange Method of Interpolation — Notes, PPT, Mathcad, Mathematica, MATLAB, Maple at Holistic Numerical Methods Institute
- Lagrange interpolation polynomial on www.math-linux.com
- Weisstein, Eric W., "Lagrange Interpolating Polynomial", MathWorld.
- Estimate of the error in Lagrange Polynomial Approximation at ProofWiki
- Module for Lagrange Polynomials by John H. Mathews
- Dynamic Lagrange interpolation with JSXGraph
- Numerical computing with functions: The Chebfun Project
- Excel Worksheet Function for Bicubic Lagrange Interpolation