Finding solutions (roots) of equations is an important mathematical problem.
There are only a few ... Stroud, K.A. 2001. Engineering Mathematics. 5th Edition.
Numerical Methods Newton-Raphson Method Aim To explain what the Newton-Raphson method is and how it works. Learning Outcomes At the end of this section you will: • Understand what the Newton-Raphson method is used for, • Understand how the Newton-Raphson method was derived. This section presents a numerical method for calculating the roots of a given function. Finding solutions (roots) of equations is an important mathematical problem. There are only a few general classes of equations that we can solve exactly (linear and quadratic equations). Cubic and quartic (3rd and 4th degree polynomials) equations can also be solved, but the formulas are very complicated. We usually solve these and most other equations approximately by using numerical methods, often with the aid of a computer or calculator.
Newton-Raphson Method We want to find the root of an equation f (x) = 0, that is , a number r such that f (r) = 0. If f is differentiable near the root, then tangent lines can be used to produce a sequence of approximations to the root that approaches the root quite quickly. The idea is as follows; Make an initial guess at the root, say x = x0 . Draw the tangent line to y = f (x) at (x0 , f (x0 )), and find x1 , the x-intercept of this tangent line. Under certain circumstances x1 will be closer to the root than x0 was. The process can be repeated over and over again to get x2 , x3 , ......, getting closer and closer to the root r. We use xn+1 to represent the x-intercept of the tangent line to y = f (x) at (xn , f (xn )). The formula for the Newton-Raphson method can be derived as follows. Look at the triangle formed by our tangent estimate to the point x1 in Figure 1. The base of the triangle has length (x0 − x1 ) and the perpendicular height of the triangle is f (x0 ). Therefore the angle formed by the tangent line when it intercepts the x-axis at the point 1
Numerical Methods x1 is given by 0
f (x0 ) =
f (x0 ) (x0 − x1 )
y = f (x)
r
x2
x1
x0
Figure 1: The Newton-Raphson Method Rearranging the above formula gives 0
(x0 − x1 )f (x0 ) = f (x0 ) 0
0
⇒ x0 f (x0 ) − x1 f (x0 ) = f (x0 ) 0
0
⇒ −x1 f (x0 ) = f (x0 ) − x0 f (x0 ) 0 f (x0 ) − x0 f (x0 ) ⇒ −x1 = f 0 (x0 ) 0 x0 f (x0 ) − f (x0 ) ⇒ x1 = f 0 (x0 ) f (x0 ) ⇒ x1 = x0 − 0 f (x0 ) This formula allows us to calculate the point x1 from information known about the previous estimate x0 . Similar formulae produce x2 from x1 , then x3 from x2 , and so on. The formula producing xn+1 from xn is xn+1 = xn −
f (xn ) , f 0 (xn )
and is known as the Newton-Raphson formula.
Related Reading Stroud, K.A. 2001. Engineering Mathematics. 5th Edition. PALGRAVE. Anton, H., I. Bivens, S. Davis. 2005. Calculus. 8th Edition. John Wiley & Sons. 2
