Lagrange polynomial

Summary

In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data.

This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L(x) (dashed, black), which is the sum of the scaled basis polynomials y00(x), y11(x), y22(x) and y33(x). The interpolation polynomial passes through all four control points, and each scaled basis polynomial passes through its respective control point and is 0 where x corresponds to the other three control points.

Given a data set of coordinate pairs with the are called nodes and the are called values. The Lagrange polynomial has degree and assumes each value at the corresponding node,

Although named after Joseph-Louis Lagrange, who published it in 1795,[1] the method was first discovered in 1779 by Edward Waring.[2] It is also an easy consequence of a formula published in 1783 by Leonhard Euler.[3]

Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration, Shamir's secret sharing scheme in cryptography, and Reed–Solomon error correction in coding theory.

For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation.

Definition

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Given a set of   nodes  , which must all be distinct,   for indices  , the Lagrange basis for polynomials of degree   for those nodes is the set of polynomials   each of degree   which take values   if   and  . Using the Kronecker delta this can be written   Each basis polynomial can be explicitly described by the product:

 

Notice that the numerator   has   roots at the nodes   while the denominator   scales the resulting polynomial so that  

The Lagrange interpolating polynomial for those nodes through the corresponding values   is the linear combination:

 

Each basis polynomial has degree  , so the sum   has degree  , and it interpolates the data because  

The interpolating polynomial is unique. Proof: assume the polynomial   of degree   interpolates the data. Then the difference   is zero at   distinct nodes   But the only polynomial of degree   with more than   roots is the constant zero function, so   or  

Barycentric form

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Each Lagrange basis polynomial   can be rewritten as the product of three parts, a function   common to every basis polynomial, a node-specific constant   (called the barycentric weight), and a part representing the displacement from   to  :[4]

 

By factoring   out from the sum, we can write the Lagrange polynomial in the so-called first barycentric form:

 

If the weights   have been pre-computed, this requires only   operations compared to   for evaluating each Lagrange basis polynomial   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.

For any     because the constant function   is the unique polynomial of degree   interpolating the data   We can thus further simplify the barycentric formula by dividing  

 

This is called the second form or true form of the barycentric interpolation formula.

This second form has advantages in computation cost and accuracy: it avoids evaluation of  ; the work to compute each term in the denominator   has already been done in computing   and so computing the sum in the denominator costs only   addition operations; for evaluation points   which are close to one of the nodes  , catastrophic cancelation would ordinarily be a problem for the value  , however this quantity appears in both numerator and denominator and the two cancel leaving good relative accuracy in the final result.

Using this formula to evaluate   at one of the nodes   will result in the indeterminate  ; computer implementations must replace such results by  

Each Lagrange basis polynomial can also be written in barycentric form:

 

A perspective from linear algebra

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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.

Example

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We wish to interpolate   over the domain   at the three nodes  :

 

The node polynomial   is

 

The barycentric weights are

 

The Lagrange basis polynomials are

 

The Lagrange interpolating polynomial is:

 

In (second) barycentric form,

 

Notes

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Example of interpolation divergence for a set of Lagrange polynomials.

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.[5]

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

Remainder in Lagrange interpolation formula

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When interpolating a given function f by a polynomial of degree k at the nodes   we get the remainder   which can be expressed as[6]

 

where   is the notation for divided differences. Alternatively, the remainder can be expressed as a contour integral in complex domain as

 

The remainder can be bound as

 

Derivation

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Clearly,   is zero at nodes. To find   at a point  , define a new function   and choose   where   is the constant we are required to determine for a given  . We choose   so that   has   zeroes (at all nodes and  ) between   and   (including endpoints). Assuming that   is  -times differentiable, since   and   are polynomials, and therefore, are infinitely differentiable,   will be  -times 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[7]

 

Since   we have  

Derivatives

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The dth derivative of a Lagrange interpolating polynomial can be written in terms of the derivatives of the basis polynomials,

 

Recall (see § Definition above) that each Lagrange basis polynomial is

 

The first derivative can be found using the product rule:

 

The second derivative is

 

The third derivative is

 

and likewise for higher derivatives.

Note that all of these formulas for derivatives are invalid at or near a node. A method of evaluating all orders of derivatives of a Lagrange polynomial efficiently at all points of the domain, including the nodes, is converting the Lagrange polynomial to power basis form and then evaluating the derivatives.

Finite fields

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The Lagrange polynomial can also be computed in finite fields. This has applications in cryptography, such as in Shamir's Secret Sharing scheme.

See also

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References

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  1. ^ Lagrange, Joseph-Louis (1795). "Leçon Cinquième. Sur l'usage des courbes dans la solution des problèmes". Leçons Elémentaires sur les Mathématiques (in French). Paris. Republished in Serret, Joseph-Alfred, ed. (1877). Oeuvres de Lagrange. Vol. 7. Gauthier-Villars. pp. 271–287. Translated as "Lecture V. On the Employment of Curves in the Solution of Problems". Lectures on Elementary Mathematics. Translated by McCormack, Thomas J. (2nd ed.). Open Court. 1901. pp. 127–149.
  2. ^ Waring, Edward (1779). "Problems concerning interpolations". Philosophical Transactions of the Royal Society. 69: 59–67. doi:10.1098/rstl.1779.0008.
  3. ^ Meijering, Erik (2002). "A chronology of interpolation: from ancient astronomy to modern signal and image processing" (PDF). Proceedings of the IEEE. 90 (3): 319–342. doi:10.1109/5.993400.
  4. ^ Berrut, Jean-Paul; Trefethen, Lloyd N. (2004). "Barycentric Lagrange Interpolation" (PDF). SIAM Review. 46 (3): 501–517. Bibcode:2004SIAMR..46..501B. doi:10.1137/S0036144502417715.
  5. ^ Quarteroni, Alfio; Saleri, Fausto (2003). Scientific Computing with MATLAB. Texts in computational science and engineering. Vol. 2. Springer. p. 66. ISBN 978-3-540-44363-6..
  6. ^ Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 25, eqn 25.2.3". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 878. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
  7. ^ "Interpolation" (PDF). pp. 12–15. Archived from the original (PDF) on 2017-02-15.
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  • "Lagrange interpolation formula", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • ALGLIB has an implementations in C++ / C# / VBA / Pascal.
  • GSL has a polynomial interpolation code in C
  • SO has a MATLAB example that demonstrates the algorithm and recreates the first image in this article
  • Lagrange Method of Interpolation — Notes, PPT, Mathcad, Mathematica, MATLAB, Maple
  • Lagrange interpolation polynomial on www.math-linux.com
  • Weisstein, Eric W. "Lagrange Interpolating Polynomial". MathWorld.
  • Lagrange polynomial at ProofWiki
  • Excel Worksheet Function for Bicubic Lagrange Interpolation
  • Lagrange polynomials in Python