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Lecture# Iterative Methods for Linear Equations

Description

This lecture covers iterative methods for solving systems of linear equations, focusing on Richardson's method and convergence analysis. It explains the use of preconditioning matrices and the importance of choosing a good preconditioner. The lecture also delves into Gauss-Seidel and the convergence analysis of Richardson's method. It discusses error control, stopping criteria, and the properties of positive definite matrices. The lecture concludes with the design of iterative methods to minimize energy functions and approximate solutions.

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Related concepts (100)

MATH-251(d): Numerical analysis

This course offers an introduction to numerical methods for the solution of mathematical problems as: solution of systems of linear and non-linear equations, functions approximation, integration and d

In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method.

In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).

In mathematics and computational science, Heun's method may refer to the improved or modified Euler's method (that is, the explicit trapezoidal rule), or a similar two-stage Runge–Kutta method. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. Both variants can be seen as extensions of the Euler method into two-stage second-order Runge–Kutta methods.

A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.

Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be positive or negative (commonly carried by protons and electrons respectively, by convention). Like charges repel each other and unlike charges attract each other. An object with no net charge is referred to as electrically neutral. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects.

Related lectures (503)

Numerical Analysis: Linear SystemsMATH-251(c): Numerical analysis

Covers the analysis of linear systems, focusing on methods such as Jacobi and Richardson for solving linear equations.

Crank-Nicolson and Heun's MethodsMATH-251(d): Numerical analysis

Covers the Crank-Nicolson and Heun's methods, discussing uniqueness of solutions and truncation errors in numerical methods.

Numerical Analysis: Linear SystemsMATH-251(c): Numerical analysis

Covers the formulation of linear systems and iterative methods like Richardson, Jacobi, and Gauss-Seidel.

Jacobi and Gauss-Seidel methods

Explains the Jacobi and Gauss-Seidel methods for solving linear systems iteratively.

Iterative Methods for Linear EquationsMATH-251(d): Numerical analysis

Introduces iterative methods for solving linear equations and discusses the gradient method for minimizing errors.