Lagrange interpolation
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| Lagrange interpolation | |
|---|---|
| Name | Lagrange interpolation |
| Field | Numerical analysis |
| Introduced | 1795 |
| Inventor | Joseph-Louis Lagrange |
Lagrange interpolation.
Introduction
Lagrange interpolation is a classical technique for constructing a polynomial that passes through a given set of points, historically connected to Joseph-Louis Lagrange, Leonhard Euler, Adrien-Marie Legendre, Carl Friedrich Gauss, Augustin-Louis Cauchy, Pierre-Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, Gaspard Monge, Évariste Galois and later contributors associated with institutions such as the École Polytechnique, Royal Society, Académie des Sciences, University of Göttingen, École Normale Supérieure and Collège de France. Its development influenced work by Niels Henrik Abel, Sofia Kovalevskaya, Bernhard Riemann, John von Neumann, David Hilbert, Alan Turing, Norbert Wiener, Andrey Kolmogorov, Richard Courant, Kurt Gödel and practitioners at Bell Labs, IBM, NASA, CERN and Los Alamos National Laboratory. The method arises in contexts studied by Isaac Newton and applied in problems linked to François Viète, Gottfried Wilhelm Leibniz, Brook Taylor, Joseph Fourier and researchers affiliated with University of Cambridge, Princeton University, Harvard University, Massachusetts Institute of Technology, Stanford University and California Institute of Technology.
Definition and Formula
Given distinct nodes, the interpolating polynomial is expressed via basis polynomials first formalized by Joseph-Louis Lagrange and motivated in the analytic tradition of Leonhard Euler and Isaac Newton. The formula relates to ideas in Bernhard Riemann's function theory and to determinants studied by Arthur Cayley and James Joseph Sylvester. The explicit representation uses products and sums reminiscent of work by Carl Gustav Jacob Jacobi and Augustin-Louis Cauchy and interfaces with linear algebra results by Camille Jordan and Hermann Grassmann. Connections appear to matrix methods developed at École Polytechnique and spectral theory advanced by John von Neumann and David Hilbert.
Properties and Error Analysis
Error bounds for polynomial interpolation invoke results tied to the mean value theorem lineage traced through Augustin-Louis Cauchy to Karl Weierstrass and Bernhard Riemann, and use divided differences related to techniques by Isaac Newton and Brook Taylor. Uniform convergence issues echo counterexamples studied by Sergio Buzzi and historic problems resolved by Andrey Kolmogorov and Norbert Wiener. Stability and conditioning draw on operator theory from Richard Courant and singular value analysis from John von Neumann; extrema and norm estimates resonate with the work of Issai Schur and Marcel Riesz.
Computational Methods and Numerical Stability
Practical algorithms implement Lagrange formulas using strategies developed at Bell Labs, IBM research groups and computational centers at Los Alamos National Laboratory and CERN. Techniques include barycentric forms influenced by numerical linear algebra from Alan Turing and John von Neumann, and divide-and-conquer strategies linked to algorithms by Donald Knuth, Edsger Dijkstra, Stephen Cook, Leslie Lamport and John Backus. Floating-point stability considerations echo analyses by William Kahan, John von Neumann and Nicholas Higham; software implementations are maintained by communities around GNU Project, Netlib, MATLAB, SciPy, NumPy and Wolfram Research.
Applications and Examples
Lagrange-style interpolation appears across applied projects at NASA, CERN, National Aeronautics and Space Administration, European Space Agency, IBM, Bell Labs and engineering curricula at Massachusetts Institute of Technology, Stanford University, University of Cambridge, Princeton University and Harvard University. It underlies techniques in signal approximation explored by Claude Shannon and Norbert Wiener, in computer graphics associated with Jim Blinn and Ed Catmull, and in finite element practices credited to Richard Courant and Olek Ciarlet. Examples include trajectory fitting in programmes at NASA, data reconstruction in projects at European Space Agency and modeling tasks used by Los Alamos National Laboratory and Sandia National Laboratories.
Generalizations and Related Methods
Generalizations connect to spline theory initiated by Isaac Jacob Schoenberg, rational interpolation studied by G. A. Baker, orthogonal polynomial frameworks developed by Szegő, wavelet constructions attributed to Yves Meyer and Ingrid Daubechies, and kernel methods used in machine learning by Geoffrey Hinton, Yann LeCun and Andrew Ng. Related constructs resonate with approximation theory from A. N. Kolmogorov, operator theory by Stefan Banach and John von Neumann, and computational paradigms explored by Donald Knuth and Edwin Catmull. Modern extensions inform research at Google, Facebook, Microsoft Research, DeepMind and academic centers like University of Oxford, ETH Zurich, University of Tokyo and Tsinghua University.