數值分析

This lesson introduces numerical analysis as the essential bridge between theoretical calculus and the finite precision of computer hardware, focusing on how limits, continuity, and differentiability ensure numerical stability. Students learn to apply these mathematical foundations to evaluate algorithmic convergence and manage precision errors, such as significant digit cancellation, in computational modeling.

Numerical root-finding is a critical computational technique used to approximate solutions for transcendental and non-linear equations that cannot be solved through standard algebraic isolation. This lesson introduces the core principles of iterative approximation, including bracketing methods like Bisection and open methods like Newton’s, to achieve controlled error tolerances in complex scientific and engineering models.

This lesson introduces the Weierstrass Approximation Theorem, which guarantees that any continuous function on a closed interval can be approximated by an algebraic polynomial. It distinguishes between the existence of these polynomials and the practical methods of interpolation, such as Lagrange and Newton, used to construct them for global accuracy.

This lesson introduces numerical differentiation as a method to approximate derivatives using finite difference formulas, such as forward and backward differences, derived from Lagrange interpolation. Students learn to balance the trade-off between truncation error and computational round-off error while applying these techniques to solve real-world engineering problems.

This lesson explores the theoretical foundations of Initial-Value Problems, focusing on how Lipschitz continuity and domain convexity ensure the existence and uniqueness of solutions. Students learn to identify well-posed problems and understand why stiff equations require specialized implicit methods to maintain numerical stability.

This lesson introduces Gaussian elimination as a systematic method for solving linear systems by transforming augmented matrices into upper triangular form using elementary row operations. Students learn to maintain the integrity of the solution set through reversible operations and explore computational efficiency, matrix properties, and advanced factorization techniques like $LDL^t$.

This lesson introduces vector and matrix norms as essential tools for quantifying magnitude and measuring convergence in iterative numerical methods. Students learn to apply the $l_2$ and $l_\infty$ norms to vectors and matrices, while exploring the axiomatic foundations and equivalence theorems that guarantee the stability of iterative approximations.

This lesson explores the philosophy of approximation, explaining why minimizing error is often more effective than exact interpolation when dealing with noisy, real-world data. Students learn to evaluate different error norms—specifically $L_1$, $L_{\infty}$, and the standard $L_2$ (Least Squares)—to balance mathematical convenience with the need to filter out noise and reveal underlying physical laws.

This lesson explains why the characteristic polynomial is numerically unstable for high-dimensional systems and introduces robust iterative alternatives like the QR method. Students learn to avoid the hazards of symbolic root-finding in favor of professional numerical libraries that ensure stability and precision in eigenvalue approximation.

This lesson introduces the transition from scalar equations to multivariable nonlinear systems, represented in vector form as $\mathbf{F}(\mathbf{x}) = \mathbf{0}$. Students learn that continuity and limits in $n$-dimensional space are determined component-wise and remain independent of the specific vector norm chosen.

This lesson introduces Boundary-Value Problems (BVPs), which require finding a trajectory that satisfies constraints at both ends of an interval rather than just initial conditions. Students will learn to distinguish BVPs from Initial-Value Problems and explore the conditions for existence and uniqueness, including the conceptual "shooting method" used to solve them.

This lesson introduces the transition from continuous calculus to numerical computation by simplifying complex heat conduction equations through the assumption of isotropy. Students learn how these simplified models allow for the simulation of real-world physical systems that are otherwise analytically intractable.