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MTH208: Numerical Analysis Module
Project——Numerical Methods for Singular Zero Problems
1 Background
In numerical analysis, Newton’s method is one of the most commonly used techniques for solving nonlinear equations. However, when the derivative at the root is zero or infinite, the convergence of Newton’s method is severely affected.
Consider the function
(1)
Its zero is p = 2
3 ≈ 0.6666667. Computing the derivative:
(2)
At the zero, f! (p) = ∞, i.e., the derivative is singular. Standard Newton’s method fails or converges very slowly in this case.
This project investigates numerical methods for solving this singular zero problem, analyzes why standard Newton’s method fails, and proposes and improves various numerical approaches.
2 Theoretical Analysis
2.1 Asymptotic Behavior Near the Root
Let p = 2/3, and set e = x → p. As x → p:
Therefore
That is,
2.2 Error Analysis of Newton’s Method
For a function satisfying f(x) ∼ C(x → p)α, where 0 < α < 1, Newton’s iteration is:
Since f! (x) ∼ αC(x → p)α−1, we obtain:
1Thus the error propagation satisfies:
For α = 1/3, we have 1 → 1/α = 1 → 3 = →2, so:
en+1 ≈ →2en
(10)
This means the absolute error doubles at each step; Newton’s method actually diverges!
2.3 Theoretical Foundation for Improved Methods
From the error analysis, if we introduce a damping factor λ:
then the error propagation becomes:
Convergence condition: |1 → λ/α| < 1, i.e., 0 < λ < 2α.
Optimal choice: When λ = α, en+1 ≈ 0, achieving superlinear convergence.
For α = 1/3, the optimal damping factor is λ = 1/3.
3 Project Tasks
3.1 Task 1: Theoretical Analysis
1.1 Derive the asymptotic expansion of
near the zero p = 2/3, and determine the
exponent α.
1.2 Derive the error propagation equation for standard Newton’s method on this problem and determine its convergence behavior.
1.3 Derive the error propagation equation for damped Newton’s method, determine the convergence condition, and give the optimal damping factor.
1.4 Prove that when λ = α, the error propagation factor becomes zero, achieving superlinear convergence.
3.2 Task 2: Numerical Experiments
2.1 Write a MATLAB function implementing standard Newton’s method to solve this problem, recording errors during iteration.
2.2 Write a MATLAB function implementing damped Newton’s method, testing the following damping factors:
• λ = 1/3 (optimal)
• λ = 1/2
• λ = 2/3
2.3 Write a MATLAB function implementing adaptive damped Newton’s method, estimating the local exponent α from successive iterations and adjusting the damping factor dynamically.
2.4 Write a MATLAB function implementing the secant method to solve this problem (no derivative required).
2.5 Write a MATLAB function implementing the variable transformation method: let u = f(x), derive the expression for x = h(u), then apply Newton’s method to h(u).
3.3 Task 3: Results Analysis and Visualization
3.1 Plot the error convergence curves for all methods on the same semi-logarithmic axes.
3.2 Compute the error ratios en+1/en for each method and analyze the convergence rates.
3.3 Compare the effect of different damping factors on convergence speed.
3.4 Compare and analyze the advantages and disadvantages of each method, completing the table below:
Method
|
Convergence
|
Final Error
|
Iterations
|
|
Standard Newton
Damped Newton (λ = 1/3)
Damped Newton (λ = 1/2)
Damped Newton (λ = 2/3)
Adaptive Damped Newton
Secant Method
Variable Transformation
|
Table 1: Performance Comparison of Methods
3.4 Task 4: Discussion and Conclusion
4.1 Explain why standard Newton’s method diverges for this problem.
4.2 Analyze the effect of the damping factor λ on convergence, verifying the theoretical convergence condition.
4.3 Compare the advantages and disadvantages of adaptive damped Newton versus fixed-damping Newton.
4.4 Discuss why the variable transformation method can restore second-order convergence.
4.5 Provide recommendations for solving such singular zero problems in practical applications.
4 Code Framework
.....
5 Report Requirements
Please write a comprehensive lab report (approximately 600 words) containing the following sections:
1. Introduction 5%: Introduce the background of singular zero problems and the research objectives of this project.
2. Theoretical Analysis 20%: Present the complete derivations from Task 1.
3. Experimental Methods 15%: Describe the algorithmic implementation details of each method.
4. Experimental Results 20%: Present all experimental data, tables, and figures.
5. Results Analysis 10%: Analyze the convergence behavior of each method and verify the theoretical conclusions.
6. Discussion and Conclusion 10%: Summarize the advantages and disadvantages of each method and provide recommendations for practical applications.
7. Appendix 20%: Complete MATLAB code with comments.
Hints and Suggestions
• Use the nthroot function to ensure real roots, avoiding (-1)^(1/3) which returns complex numbers.
• Pay attention to numerical stability when x is close to the root, avoiding division by zero.
• For adaptive damped Newton, you can estimate α ≈ log |f(x)|/ log |x → p| and adjust the damping factor dynamically.
• In the variable transformation method, the inverse function of u = f(x) is x = 3(1−
2
u3) .
• Choose the initial guess slightly larger than the true root (e.g., x0 = 0.68) to avoid entering the region x < 2/3 where the function value becomes complex.
• This project is to be completed independently. If AI assistance is used, please refer to the univer sity’s ethical guidelines regarding AI use. Strictly avoid irresponsible use of AI. Ensure you take responsibility for your submission; if referencing other literature or sources, please properly citethem.
• You are encouraged to use Overleaf to prepare your report as a PDF file, which should be named as nameID.pdf; e.g.Zhangsan2023123.pdf;
