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Lecture

Nonlinear Equations: Interpolation and Error Analysis

In course

MATH-250: Advanced numerical analysis I

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

Description

This lecture covers the interpolation of nonlinear functions using Lagrange polynomials, focusing on calculating the Lagrange polynomial of degree 3 and 2. It also discusses piecewise linear interpolation, error analysis, and the minimum number of nodes required for a given error threshold. The instructor demonstrates the process step by step, providing examples and explaining the concepts thoroughly.

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Instructors (3)

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Related concepts (32)

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.

Spline (mathematics)

In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees. In the computer science subfields of computer-aided design and computer graphics, the term spline more frequently refers to a piecewise polynomial (parametric) curve.

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.

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).

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.

Related lectures (30)

Piecewise Linear Interpolation

Covers the concept of piecewise linear interpolation and the importance of dividing intervals correctly.

Numerical Analysis: Polynomial Interpolation Techniques

Provides an overview of polynomial interpolation techniques in numerical analysis, focusing on Lagrange interpolation and error estimation methods.

Polynomial Interpolation: Lagrange Method

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

Interpolation by Intervals: Lagrange Interpolation

Covers Lagrange interpolation using intervals to find accurate polynomial approximations.

Interpolation de fonction