In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of points with no two values equal, the Lagrange polynomial is the polynomial of lowest degree that assumes at each value the corresponding value , so that the functions coincide at each point.
Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. As changing the points requires recalculating the entire interpolant, it is often easier to use Newton polynomials instead.
Definition
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.
Proof
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 :
Observe that:
In there are k factors in the product and each factor contains one x, so L(x) (which is a sum of these k-degree polynomials) must be a polynomial of degree at most k.
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
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 the polynomial interpolation article.
A perspective from linear algebra
Solving an interpolation problem leads to a problem in linear algebra amounting to inversion of a matrix. Using a standard monomial basis for our interpolation polynomial , we must invert the Vandermonde matrix to solve for the coefficients of . By choosing a better basis, the Lagrange basis, , we merely get the identity matrix, , which is its own inverse: the Lagrange basis automatically inverts the analog of the Vandermonde matrix.
This construction is analogous to 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.
Furthermore, when the order is large, Fast Fourier Transformation can be used to solve for the coefficients of the interpolated polynomial.
Examples
Example 1
We wish to interpolate ƒ(x) = x2 over the range 1 ≤ x ≤ 3, given these three points:
The interpolating polynomial is:
Example 2
We wish to interpolate ƒ(x) = x3 over the range 1 ≤ x ≤ 4, given these four points:
The interpolating polynomial is:
Notes
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.[3]
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 .
Remainder in Lagrange interpolation formula
When interpolating a given function f by a polynomial of degree k at the nodes we get the remainder which can be expressed as[5]
where is the notation for divided differences. Alternatively, the remainder can be expressed as a contour integral in complex domain as
Clearly, is zero at nodes. To find at a point . Define a new function and choose (This ensures at nodes) where is the constant we are required to determine for a given . Now has zeroes (at all nodes and ) between and (including endpoints). Assuming that is -times differentiable, and are polynomials, and therefore, are infinitely differentiable. By Rolle's theorem, has zeroes, has zeroes... has 1 zero, say . Explicitly writing :
(Because the highest power of in is
The equation can be rearranged as
Derivatives
The th derivatives of the Lagrange polynomial can be written as
.
For the first derivative, the coefficients are given by
and for the second derivative
.
Through recursion, one can compute formulas for higher derivatives.