Numerical Analysis: From Scalar to Vector: The Nonlinear System Challenge

Transitioning from a single equation $f(x)=0$ to a multivariable system is the gateway to solving complex engineering problems, from orbital mechanics to soil structural analysis. We no longer look for a simple zero on a line, but for the simultaneous intersection of $n$ hypersurfaces in $n$-dimensional space.

1. The Mathematical Structure

A nonlinear system is represented as a set of equations where each component function depends on a vector of unknowns $\mathbf{x} = (x_1, x_2, \dots, x_n)^t$:

$$f_1(x_1, x_2, \dots, x_n) = 0,$$ $$f_2(x_1, x_2, \dots, x_n) = 0,$$ $$\vdots$$ $$f_n(x_1, x_2, \dots, x_n) = 0,$$

We condense this into the vector form 核心公式:

$$\mathbf{F}(\mathbf{x}) = \mathbf{0}$$

where $\mathbf{F} = (f_1, f_2, \dots, f_n)^t$. The individual functions $f_i$ are designated as the coordinate functions of $\mathbf{F}$.

2. Analytic Foundations & Continuity

To solve these systems numerically, we must ensure the mapping is well-behaved. Definitions 10.1–10.3 establish that limits and continuity in $\mathbb{R}^n$ are determined component-wise.

Definition 10.3

Let $\mathbf{F}$ be a function from $D \subset \mathbb{R}^n$ into $\mathbb{R}^n$. We say $\lim_{\mathbf{x} \to \mathbf{x}_0} \mathbf{F}(\mathbf{x}) = \mathbf{L} = (L_1, L_2, \dots, L_n)^t$ if and only if:

$$\lim_{\mathbf{x} \to \mathbf{x}_0} f_i(\mathbf{x}) = L_i$$ for each $i=1, \dots, n$.

Utilizing the $\epsilon-\delta$ definition: for every $\epsilon > 0$, there exists $\delta > 0$ such that $\|\mathbf{F}(\mathbf{x}) - \mathbf{L}\| < \epsilon$ whenever $0 < \|\mathbf{x} - \mathbf{x}_0\| < \delta$.

易错点: Norm Independence

Critical nuance: Although various norms ($\ell_1, \ell_2, \ell_\infty$) can be used, continuity is independent of the particular choice. The existence of a limit is invariant under any vector norm in $\mathbb{R}^n$.

3. Theoretic Refresher

Theorem 1.6: For functions from $\mathbb{R}$ into $\mathbb{R}$, continuity can often be shown by demonstrating differentiability. In the multivariable case, if the partial derivatives of the coordinate functions exist and are bounded, continuity is assured, which is a prerequisite for iterative solvers.

经典例题: Example 1

Consider the circular plates on soil problem. Place the $3 \times 3$ nonlinear system in the standard form $\mathbf{F}(\mathbf{x}) = \mathbf{0}$:

$3x_1 - \cos(x_2 x_3) - \frac{1}{2} = 0$
$x_1^2 - 81(x_2 + 0.1)^2 + \sin x_3 + 1.06 = 0$
$e^{-x_1 x_2} + 20x_3 + \frac{10\pi - 3}{3} = 0$

Here, $\mathbf{x} = (x_1, x_2, x_3)^t$ and $\mathbf{F}(\mathbf{x}) = (f_1(\mathbf{x}), f_2(\mathbf{x}), f_3(\mathbf{x}))^t$.

QUESTION 1

Given the vector mapping F(x1, x2, x3) = (x1 + 2x3, x1 cos x2, $x2^2$ + x3)ᵗ, why is F continuous at each point of ℝ³?

Because it is a linear mapping and all linear mappings are continuous.

Because each coordinate function fᵢ is a sum or product of basic continuous functions (polynomials and cosine).

Because the Jacobian matrix has a constant determinant.

Because it maps ℝ³ into a closed and bounded subset of ℝ³.

QUESTION 2

Which of the following functions F: ℝ² → ℝ² is continuous everywhere EXCEPT at (1, 0)?

F(x₁, x₂) = (x₁ - 1, x₂)²

F(x₁, x₂) = (sin(x₁ - 1), cos x₂)

F(x₁, x₂) = (1 / (x₁ - 1), x₂ / (x₁² + x₂²))

F(x₁, x₂) = (eˣ¹, eˣ²)

QUESTION 3

If we change the vector norm from the Euclidean norm (ℓ₂) to the Maximum norm (ℓ∞) when evaluating the continuity of a nonlinear system, how does it affect the convergence property of a sequence of vectors?

The sequence may now diverge because the Max norm is less sensitive.

Convergence is preserved because all vector norms in finite-dimensional space are equivalent.

The limit value L will change depending on the norm scaling.

The sequence will converge faster in the ℓ₁ norm than the ℓ₂ norm.

QUESTION 4

Newton's method is typically used for nonlinear systems. However, if applied to the linear system Ax = b (where A is nonsingular), what is the result?

The method fails because the Jacobian is not defined for linear systems.

The method reduces to Gaussian elimination and requires n iterations.

The method reaches the exact solution in exactly one iteration from any initial guess.

The method oscillates and never converges unless x₀ is b/n.

QUESTION 5

When defining the limit of F(x) as x approaches x₀, what does the condition '0 < ‖x - x₀‖ < δ' signify?

We are evaluating the function exactly at x₀.

We are considering a punctured neighborhood, meaning x₀ itself is excluded from the limit calculation.

The distance from the origin must be less than δ.

The system must be linear for the limit to exist.