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  1. Convergence

    Linked via "Bisection Method"

    Common orders include:
    Linear Convergence ($\rho = 1$): The error is reduced by a constant factor at each step. This is characteristic of the Bisection Method applied to continuously differentiable functions.
    Superlinear Convergence ($1 < \rho < 2$): The error reduction factor increases with each iteration.
    Quadratic Convergence ($\rho = 2$): The number of correct significant digits roughly doubles at each step. [N…

  2. Iterative Algorithm

    Linked via "Bisection Method"

    Notable Applications Beyond Root Finding
    While iterative algorithms are foundational to numerical analysis for solving equations (as seen in Newton's Method, Jacobi Method for linear systems, and the Bisection Method), their application extends widely into optimization and simulation.
    Optimization (Gradient Descent Variants)

  3. Linear Convergence

    Linked via "Bisection Method"

    In the realm of numerical analysis, linear convergence is the baseline expectation for many foundational iterative algorithms used to solve equations or find fixed points. It signifies that the algorithm is making steady, predictable progress toward the solution iterative algorithm (process)/).
    The prevalence of linear convergence in certain methods stems from the local behavior of the function being analyzed. For instance, the [Bisection Method](/entries/bisec…

  4. Linear Convergence

    Linked via "Bisection Method"

    When Newton's Method is applied to find a root $r$ of multiplicity $m > 1$, the convergence order degrades precisely to linear convergence, with the convergence factor determined by the multiplicity:
    $$C = \frac{m-1}{m}$$
    For a double root ($m=2$), the factor $C$ is $1/2$, mirroring the rate of the Bisection Method, which is generally considered suboptimal for such scenarios Newton's Method.
    The Role of the Contraction Mapping Theorem

  5. Newtons Method

    Linked via "Bisection Method"

    When Newton's method converges under standard conditions (i.e., $f(r) = 0$ and $f'(r) \neq 0$), it exhibits quadratic convergence. This means that, as $n$ increases, the error at the next step, $|x{n+1} - r|$, is proportional to the square of the error at the current step, $|xn - r|$.
    $$|e{n+1}| \approx C |en|^2$$
    This rapid rate of convergence}) is why Newton's method is preferred over linear methods, such as the [Bisection Method](/e…