Numerical Analysis 4.2. Jacobi Iteration for Linear Systems скачать в хорошем качестве

Numerical Analysis 4.2. Jacobi Iteration for Linear Systems

This video is a lecture on Numerical Analysis, specifically focusing on the Jacobi iteration method (Section 4.2) for solving linear systems of equations. Key Concepts and Overview Introduction via Example: The instructor begins with a 3D linear system, demonstrating how to isolate each variable (x 1 ​ ,x 2 ​ ,x 3 ​ ) to transform the system into a linear fixed-point equation [00:17]. Definition of Jacobi Iteration: The method defines a sequence where the next value (x k+1 ) is computed using the previous values (x k ) of all other variables [01:32]. It is important to note that the index j skips the value i in the summation [07:53]. Convergence Example: Using an initial guess of (0,0,0), the video shows how the sequences approach the solution (2,0,−1) [02:31]. Matrix Representation The instructor explains how to represent the system in the general form x=Tx+c [03:34]: Matrix Decomposition: The coefficient matrix A is broken down into L (lower triangular), D (diagonal), and U (upper triangular) components [08:52]. Iteration Matrix (T coefficients): The Jacobi iteration matrix is defined as T=−D −1 (L+U) [11:25]. Requirement: The diagonal elements a ii ​ must be non-zero for the method to be defined [08:01]. Convergence Theorems The lecture covers three main criteria for determining if the iteration will converge: Necessary and Sufficient Condition: The spectral radius of the iteration matrix must be less than one (ρ(T) 1) [12:24]. Sufficient Condition (Norms): If any matrix norm of T is less than one (∥T∥ 1), the iteration is guaranteed to converge globally [12:57]. Diagonal Dominance: If the original matrix A is strictly diagonally dominant, the Jacobi iteration will converge regardless of the initial starting value [13:50]. The proof shows that diagonal dominance implies the infinity norm of T is less than one [15:26].