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

Lecture

The Wrong Method: Polynomial Interpolation

Description

This lecture explains the incorrect method of solving polynomial interpolation problems by setting up a linear system with n+1 unknown coefficients, resulting in a high computational complexity of order n cubed. The instructor demonstrates the process of formulating the linear system, highlighting the inefficiency of this approach compared to a forthcoming explicit formula for the polynomial.

In MOOC

Numerical Analysis for Engineers

Official source

Quiz

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Related concepts (15)

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.

Computational complexity

In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem. The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory.

Computational complexity theory

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used.

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.

Lagrange polynomial

In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. 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, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler.

Related lectures (38)

Generalization in Deep Learning

Explores generalization in deep learning, covering model complexity, implicit bias, and the double descent phenomenon.

Polynomial Interpolation: Lagrange Interpolation

Covers Lagrange interpolation as a unique polynomial of degree 2N through 2N + 1 points.

Interpolation de fonction

Explores interpolation of regular functions, error analysis, convergence, and Chebyshev polynomials.

Interpolation by Intervals: Lagrange Interpolation

Covers Lagrange interpolation using intervals to find accurate polynomial approximations.

Approximation of Data