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Section3.3Newton's Method

We now investigate a calculus-based method (originally proposed by Newton and later modified by Joseph Raphson) for solving the algebraic equation \(f(x) = 0\). In very basic terms, this method involves iteratively finding tangent lines to a differentiable curve and locating where those tangent lines intersect the horizontal axis.

The following is an Algorithm for the Newton-Raphson method:

The Newton-Raphson method for solving algebraic equations can be described as follows:

  1. Check that \(f \in C^2\) on a given domain and find a way to guarantee that \(f\) has a root on that domain.

  2. Pick a starting point \(x_0\) in the domain

  3. Draw a tangent line to \(f\) at \(x_0\) and find where it intersects with the \(x\)-axis

  4. Use the intersection of the tangent line with the \(x\)-axis as the next approximation of the root (call it \(x_1\)).

  5. repeat steps 3 and 4 until the intersection point gets close to the root that you would like to approximate