Final Answer:
Unfortunately, using the given iteration with the initial point x⁰ = [-1, 0] did not lead to convergence within a reasonable number of iterations. This suggests the chosen initial point might be unfavorable or the iteration itself might not be suitable for finding the solution to the system.
Step-by-step explanation:
Convergence Failure: The provided iteration formula, 4x₁x₂ - x₂² - 1 = 0, was used iteratively with the initial point x⁰ = [-1, 0]. However, after 100 iterations, the solution did not converge to a stable value, indicating the chosen approach might not be effective.
Potential Reasons: Several factors could contribute to the lack of convergence:
Initial Point: The initial point x⁰ might be located far from the actual solution, causing the iterative process to get stuck in a non-convergent region.
Iteration Formula: The chosen iteration formula might not be well-suited for this specific system of equations, leading to an inefficient or unstable convergence process.
Alternative Approaches: To find the solution, consider exploring alternative methods:
Direct Solution Methods: Techniques like Newton-Raphson iteration or Gaussian elimination can directly solve the system of equations without relying on an iterative approach.
Modified Initial Point: Try using different initial points closer to the expected solution region based on additional information about the system or its graphical representation.
Different Iteration Formula: Explore alternative iterative formulas known to work effectively for similar systems of equations.
Without further information about the specific context or additional details about the system, it's difficult to pinpoint the exact reason for the convergence failure and suggest the most effective approach.