“Lagrange polynomial”的意思、由来-开放百科全书

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

(i.e. the functions coincide at each point).

The interpolating polynomial of the least degree is unique, however, and since it can be arrived at through multiple methods, referring to "the Lagrange polynomial" is perhaps not as correct as referring to "the Lagrange form" of that unique polynomial.

Although named after Joseph Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring^[1] It is also an easy consequence of a formula published in 1783 by Leonhard Euler.^[2]

Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration and Shamir's secret sharing scheme in cryptography.

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

where no two

are the same, the interpolation polynomial in the Lagrange form is a linear combination

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

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

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

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

. 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:

Example 2

We wish to interpolate ƒ(x) = x³ over the range 1 ≤ x ≤ 3, given these three points:

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 x_k 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]

The Lagrange basis polynomials can be used in numerical integration to derive the Newton–Cotes formulas.

Barycentric form