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

Lecture

Exemple: Convergence Analysis

Description

This lecture presents an example where the function g(x) = x²-2 is analyzed over the interval [1, 2]. By applying the Newton method, the convergence is guaranteed, but only linearly. The instructor explains the calculations step by step, highlighting the advantages and disadvantages of the method.

In MOOCs (2)

Analysis I

Analysis I (part 5): Continuous and derivable functions, the derivative function

Official source

Quiz

Related concepts (26)

Rate of convergence

In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence that converges to is said to have order of convergence and rate of convergence if The rate of convergence is also called the asymptotic error constant. Note that this terminology is not standardized and some authors will use rate where this article uses order (e.g., ).

Real analysis

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.

Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1, and all numbers in between. Other examples of intervals are the set of numbers such that 0 < x < 1, the set of all real numbers , the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element).

Radius of convergence

In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges.

Dominated convergence theorem

In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration. In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables.

Related lectures (29)

Nonlinear Equations: Fixed Point Method Convergence

Covers the convergence of fixed point methods for nonlinear equations, including global and local convergence theorems and the order of convergence.

Properties of Complete Spaces

Covers the properties of complete spaces, including completeness, expectations, embeddings, subsets, norms, Holder's inequality, and uniform integrability.

Iterative Methods: Linear Systems

Covers iterative methods for solving linear systems, emphasizing convergence analysis and well-chosen matrices.

Uniform Integrability and Convergence

Explores uniform integrability, convergence theorems, and the importance of bounded sequences in understanding the convergence of random variables.

Numerics: semester project