Newton's Method in n Dimensions

  1. Problem: Find n-Dimensional Zero of a Vector-Valued Function of a Vector Argument f(x),

    where x = [x i]n×1, z = [z i]n×1 and f(x) = [f i(x)]n×1 (Vectors are indicated by boldfaced variables and functions).

  2. Taylor Approximation: Letting the (k+1)th iteration be such that x(k+1) = x(k)+delta_x(k) ~ z, so that

    assuming the change |delta_x(k)| is sufficiently small, where

  3. Conversion to Linear Algebra Problem:

    can be converted to the Augmented Matrix:

    which can be solved by Forward Gaussian Elimination to get an approximation for delta_x(k) which can be used to get the new Newton n-dimensional iterate,

  4. Numerical Example:


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