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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, it was first discovered in 1779 by Edward Waring and rediscovered in 1783 by Leonhard Euler.
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
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 , because it lacks the term.
It follows that , so at each point , , showing that interpolates the function exactly.
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 :
Watch what happens if we expand this product. Because the product skips , If then all terms are (except where but that case is impossible as pointed out in the definition section---if you tried to write out that term you'd find that and since , , contrary to ). Also if then since doesn't 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.
|This section requires expansion with:
examples and explanation of what inverting means in this context.
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 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 weights
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 .
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