Netwon Raphson

Newton Raphson Method

In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valuedfunction. The algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations.

The Newton-Raphson method in one variable is implemented as follows:

Given a function ƒ defined over the reals x, and its derivative ƒ ', we begin with a first guess x₀ for a root of the function f. Provided the function is reasonably well-behaved a better approximation x₁ is

Geometrically, (x₁, 0) is the intersection with the x-axis of a line tangent to f at (x₀, f (x₀)).

The process is repeated as

until a sufficiently accurate value is reached.

Newton-Raphson Method Procedure

Draw a line tangent to the function at the point (x1, f (x1)). The point where the tangent line crosses the x axis should be a better estimate of the root than x1. Call that point x2. Calculate f (x2), and draw a line tangent to the function at the point (x2, f (x2)). The point where the new tangent line crosses the x axis should be a better estimate of the root than x2. Call that point x3.

Repeat until |f (x)| <e.

Newton-Raphson Method Advantages

• Unlike the incremental search and bisection methods, the Newton-Raphson method isn’t fooled by singularities.
• Also, it can identify repeated roots, since it does not look for changes in the sign of f (x) explicitly.
• It can find complex roots of polynomials, assuming you start out with a complex value for x1.
• For many problems, Newton-Raphson converges quicker than either bisection or incremental search.
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Newton-Raphson Method Disadvantages

• The Newton-Raphson method only works if you have a functional representation of f 0(x). Some functions may be difficult to impossible to differentiate. You may be able to work around this by approximating the derivative f 0(x) _ f (x+_x)−f (x)_x .
• Newton-Raphson method is not guaranteed to find a root. For example, if the starting point x1 is sufficiently far away from the root for the function f (x) = tan−1 x, the function’s small slope tends to drive the x guesses further and further away from the root.