Numerical Analysis 2.1. Fixed Point Iteration скачать в хорошем качестве

Numerical Analysis 2.1. Fixed Point Iteration

This lecture provides a comprehensive study of the Fixed Point Iteration method for solving nonlinear algebraic equations. Below is a detailed outline based on the video: 1. Introduction to Numerical Sequences [00:03] General Recursion: Numerical methods often generate infinite sequences using m-step recursions based on initial values p 0 ​ ,p 1 ​ ,…,p n−1 ​ [00:35]. One-Step Iteration: A special case where the next term is defined by a single previous term: p k+1 ​ =g(p k ​ ) [01:06]. Definition of Fixed Point: A point P is a fixed point of function g if g(P)=P [07:19]. 2. Visualization and Geometric Meaning [03:11] Graphing the Method: The process involves the graph of y=g(x) (the iteration function) and the line y=x [03:22]. Stair-Step (Cobweb) Diagrams: * Start at p 0 ​ , move vertically to the curve g(x) to find g(p 0 ​ ). Move horizontally to the line y=x to set this value as the new x (which is p 1 ​ ) [04:14]. Repeating this creates a "stair-step" or "spiraling" pattern that converges toward the intersection of the two lines [05:22]. 3. Fundamental Theorems of Fixed Points [08:10] Convergence Theorem: If the sequence p k ​ converges to P, and g is continuous, then P must be a fixed point (P=g(P)) [08:40]. Existence Theorem: If g is continuous on [a,b] and g(x)∈[a,b] for all x∈[a,b], then at least one fixed point exists in that interval [18:15]. Uniqueness Theorem: If g is differentiable on (a,b) and ∣g ′ (x)∣≤C lt 1, then the fixed point is unique [20:14]. 4. Calculus Tools for Analysis [13:57] Mean Value Theorem (Lagrange): Used to relate the difference in function values to the derivative: g(b)−g(a)=g ′ (c)(b−a) [14:15]. Lipschitz Continuity: A function is Lipschitz continuous if ∣g(x)−g(y)∣≤C∣x−y∣. If C lt 1, the function is called a contraction [37:18]. 5. Error Estimation Formulas [24:06] Error Formula 1: ∣p k ​ −P∣≤C k ∣p 0 ​ −P∣. This shows that the error decreases geometrically as k increases, provided C lt 1 [24:06]. Error Formula 2: ∣p k ​ −P∣≤ 1−C C k ​ ∣p 1 ​ −p 0 ​ ∣. This is more practical as it only requires the first two terms of the sequence rather than the unknown limit P [25:21]. 6. Local vs. Global Convergence [40:08] Global Convergence: The iteration converges for any starting point p 0 ​ in the interval [41:26]. Local Convergence: The iteration converges only if p 0 ​ is "close enough" to P. This is guaranteed if ∣g ′ (P)∣ lt 1 [41:06]. Stability Cases [45:10]: 0 lt g ′ (P) lt 1: Monotonic convergence (staircase). −1 lt g ′ (P) lt 0: Oscillating convergence (spiral). ∣g ′ (P)∣ gt 1: Divergence (the sequence moves away from the fixed point) [46:42]. 7. Practical Strategy in Numerics [48:43] In practice, one picks an initial guess and checks for convergence. If it fails, a new initial point or a different numerical method is required [49:10].