Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Interpolation by Intervals: Lagrange Interpolation

Description

This lecture covers the concept of Lagrange interpolation, focusing on the interpolation of functions using intervals. The instructor explains the process of finding the polynomial that passes through given points and discusses the error estimation in interpolation. Various mathematical formulas and techniques are presented to achieve accurate interpolation results.

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Related concepts (26)

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.

Bilinear interpolation

In mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., x and y) using repeated linear interpolation. It is usually applied to functions sampled on a 2D rectilinear grid, though it can be generalized to functions defined on the vertices of (a mesh of) arbitrary convex quadrilaterals. Bilinear interpolation is performed using linear interpolation first in one direction, and then again in another direction.

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 (48)

Approximation of Data

Covers the least squares method for approximating data and handling errors.

Polynomial Interpolation: Optimizing Error

Covers the optimization of error in polynomial interpolation, focusing on minimizing the error by strategically placing interpolation points.

Numerical Analysis: Polynomial Interpolation Techniques

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

Interpolation: Lagrange polynomial and error analysis

Covers the interpolation of functions using Lagrange polynomials and error analysis, emphasizing the dependence on the function.

Polynomial Interpolation: Lagrange Method