The Newton-Raphson method

• The Newton-Raphson method finds?roots?of equations in the form?f(x)?=?0
• It can be used to find?approximate solutions?when an equation cannot be solved using the usual?analytical methods
• It works by finding the?x-intercept?of?tangents to f(x)?to get closer and closer to a root

Using the Newton-Raphson method

• The formula for Newton-Raphson uses the same notation?as used in iteration and other recurrence relations
• After using?differentiation?to find f’(x) the formula uses iteration to come to an ever more accurate solution

Can the Newton-Raphson method fail?

The Newton-Raphson method can fail when:

• the starting value?x0?is too far away?from the root leading to a?divergent?sequence or a?different root
• the?tangent?gradient?is?too small, where f’(x) close to 0 leading to a?divergent?sequence or one which?converges very slowly
• the?tangent?is?horizontal, where f’(x) = 0 so the tangent will?never meet the x?axis
• the equation?cannot be differentiated?(or is awkward and time-consuming to do)