Practice Fixed-point Iteration Example (2.5.3) - Numerical Solutions of Algebraic and Transcendental Equations

Practice: Fixed-Point Iteration Example

Lesson 2.5.3Numerical Solutions of Algebraic and Transcendental Equations

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10 questions

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6 sample questions

Question 1.
What is the reformulation for the function f(x)=x²−4 in terms of g(x)?
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Think about how to solve f(x)=0.
Question 2.
If x₀=1, what is the value of x₁ using g(x)=√(4+x)?
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Substitute x₀=1 into g(x).
3Question 3.
What is the main point of fixed-point iteration?
- Finding roots by derivatives
- Reformulating f(x)=0 to x=g(x)
- Finding simple averages
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Consider the principal objective of this numerical method.
4Question 4.
True or False: Convergence is guaranteed with any initial guess.
- True
- False
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Think about the role of g(x) in iteration.
Question 5.
Given the equation f(x)=x³-2x-5, derive a suitable g(x) for fixed-point iteration and analyze its convergence.
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Evaluate the derivative for convergence near the expected root.
Question 6.
If f(x)=cos(x)-x, use fixed-point iteration to find the root starting from x=0.5, detail the iterations till convergence.
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Count how many iterations it takes to reach a stable x value.

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