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Lecture# Polynomial Interpolation: Optimizing Error

Description

This lecture covers the optimization of error in polynomial interpolation, focusing on minimizing the error by strategically placing interpolation points and using Chebyshev polynomials. The instructor explains the concept of equidistant points, optimal points, and the Chebyshev theorem. The lecture delves into the minimization of error by choosing the right interpolation points and discusses the Chebyshev polynomials' role in error minimization. The lecture concludes with a detailed explanation of the approximation process using polynomials and the importance of choosing the right coefficients for accurate evaluation.

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MATH-250: Numerical analysis

Construction and analysis of numerical methods for the solution of problems from linear algebra, integration, approximation, and differentiation.

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Interpolation

In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable.

Polynomial interpolation

In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of n + 1 data points , with no two the same, a polynomial function is said to interpolate the data if for each . There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials.

Trigonometric interpolation

In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given periods. This form is especially suited for interpolation of periodic functions. An important special case is when the given data points are equally spaced, in which case the solution is given by the discrete Fourier transform.

Multivariate interpolation

In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable (multivariate functions); when the variates are spatial coordinates, it is also known as spatial interpolation. The function to be interpolated is known at given points and the interpolation problem consists of yielding values at arbitrary points . Multivariate interpolation is particularly important in geostatistics, where it is used to create a digital elevation model from a set of points on the Earth's surface (for example, spot heights in a topographic survey or depths in a hydrographic survey).

Newton polynomial

In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using Newton's divided differences method. Given a set of k + 1 data points where no two xj are the same, the Newton interpolation polynomial is a linear combination of Newton basis polynomials with the Newton basis polynomials defined as for j > 0 and .

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Covers the Lagrange polynomial interpolation method and error analysis in function approximation.

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Explores Lagrange interpolation, error analysis, and piecewise linear interpolation.

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Explores error analysis and limitations in interpolation on evenly distributed nodes.

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Covers the analysis of analytic functions and the Runge phenomenon in function approximation.

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Explores Gauss-Legendre quadrature formulas using Legendre polynomials for accurate function approximation.