Numerical Analysis II lecture10 Fixed Points for Functions of Several Variables

Fixed Point Iteration for Non-Linear Systems
📌 The lecture introduces the Fixed Point Iteration Method applied to a system of $n$ non-linear equations in $n$ unknowns, formulated as the vector equation $\mathbf{x} = \mathbf{g}(\mathbf{x})$.
📐 A vector-valued function $\mathbf{f}: D \subset \mathbb{R}^n \to \mathbb{R}^n$ represents the system of equations $\mathbf{f}(\mathbf{x}) = \mathbf{0}$, where $\mathbf{f}(\mathbf{x}) = [f_1(\mathbf{x}), f_2(\mathbf{x}), \dots, f_n(\mathbf{x})]^T$.
🤔 Continuity for a vector-valued function $\mathbf{F}$ is established if the limit of every component function $f_i$ exists and equals $l_i$ as $\mathbf{x} \to \mathbf{x}_0$.

Fixed Point Definition and Convergence Theorem
✨ A number $p$ is a fixed point for a function $g$ if $\mathbf{p} = \mathbf{g}(\mathbf{p})$.
📜 Theorem 2 (Generalization of Fixed Point Theorem) states that if $\mathbf{g}: D \to D$ is continuous, and if all partial derivatives of its component functions are bounded such that $\left|\frac{\partial g_i}{\partial x_j}\right| \le \frac{K}{n}$ (where $K < 1$), then the iteration $\mathbf{x}_{k+1} = \mathbf{g}(\mathbf{x}_k)$ converges to a unique fixed point $\mathbf{p}$.
📉 The convergence rate is bounded by the inequality involving the $L_\infty$ norm: $\|\mathbf{x}_k - \mathbf{p}\|_\infty \le C K^k \|\mathbf{x}_0 - \mathbf{p}\|_\infty$.

Example Application and Iteration
💡 The method requires transforming the system $\mathbf{f}(\mathbf{x}) = \mathbf{0}$ into the fixed-point form $\mathbf{x} = \mathbf{g}(\mathbf{x})$ by solving each equation for one variable, $x_i$.
✅ For a $3 \times 3$ system, the existence of a unique solution on the domain $D = [-1, 1]^3$ was verified by showing the component functions $\mathbf{g}(\mathbf{x})$ map $D$ to $D$ and that the partial derivatives are bounded by a constant $k < 1$.
📈 The example showed convergence to the required accuracy ($\approx 10^{-5}$) in five iterations using standard fixed-point iteration.

Accelerating Convergence with Gauss-Seidel
🚀 Convergence can potentially be accelerated by using the Gauss-Seidel method, which employs the most recently calculated component estimates ($\mathbf{x}_i^k$) immediately when computing subsequent components ($\mathbf{x}_j^k$ where $j>i$).
⏱️ Applying the Gauss-Seidel approach in the example reduced the required iterations to achieve the same accuracy from five to four iterations.

Key Points & Insights
➡️ To apply the Fixed Point Iteration Method to a system $\mathbf{f}(\mathbf{x}) = \mathbf{0}$, it must first be algebraically rearranged into the form $\mathbf{x} = \mathbf{g}(\mathbf{x})$.
➡️ Convergence to a unique fixed point $\mathbf{p}$ is guaranteed if the function $\mathbf{g}$ is a contraction mapping on the domain $D$, often checked using bounds on partial derivatives (contractive mapping theorem).
➡️ The $L_\infty$ norm is used to measure the error convergence: $\|\mathbf{x}_k - \mathbf{p}\|_\infty$.
➡️ For potentially faster convergence, utilize the Gauss-Seidel approach, updating components immediately within the iteration step instead of waiting for the next iteration $k+1$.

📸 Video summarized with SummaryTube.com on Oct 10, 2025, 16:32 UTC