Newton-Raphson Method notes

# Newton-Raphson Method notes - Chapter 03.04 Newton-Raphson...

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Chapter 03.04 Newton-Raphson Method of Solving a Nonlinear Equation After reading this chapter, you should be able to: 1. derive the Newton-Raphson method formula, 2. develop the algorithm of the Newton-Raphson method, 3. use the Newton-Raphson method to solve a nonlinear equation, and 4. discuss the drawbacks of the Newton-Raphson method. Introduction Methods such as the bisection method and the false position method of finding roots of a nonlinear equation 0 ) ( = x f require bracketing of the root by two guesses. Such methods are called bracketing methods . These methods are always convergent since they are based on reducing the interval between the two guesses so as to zero in on the root of the equation. In the Newton-Raphson method, the root is not bracketed. In fact, only one initial guess of the root is needed to get the iterative process started to find the root of an equation. The method hence falls in the category of open methods . Convergence in open methods is not guaranteed but if the method does converge, it does so much faster than the bracketing methods. Derivation The Newton-Raphson method is based on the principle that if the initial guess of the root of 0 ) ( = x f is at i x , then if one draws the tangent to the curve at ) ( i x f , the point 1 + i x where the tangent crosses the x -axis is an improved estimate of the root (Figure 1). Using the definition of the slope of a function, at i x x = ( 29 θ = x f i tan ( 29 1 0 + - - i i i x x x f = , which gives ( 29 ( 29 i i i i x f x f = x x - + 1 (1) 03.04.1

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Chapter 03.04 Equation (1) is called the Newton-Raphson formula for solving nonlinear equations of the form ( 29 0 = x f . So starting with an initial guess, i x , one can find the next guess, 1 + i x , by using Equation (1). One can repeat this process until one finds the root within a desirable tolerance. Algorithm The steps of the Newton-Raphson method to find the root of an equation ( 29 0 = x f are 1. Evaluate ( 29 x f symbolically 2. Use an initial guess of the root, i x , to estimate the new value of the root, 1 + i x , as ( 29 ( 29 i i i i x f x f = x x - + 1 3. Find the absolute relative approximate error a as 0 10 1 1 × - + + i i i a x x x = 4. Compare the absolute relative approximate error with the pre-specified relative error tolerance, s . If a > s , then go to Step 2, else stop the algorithm. Also, check if the number of iterations has exceeded the maximum number of iterations allowed. If so, one needs to terminate the algorithm and notify the user. Figure 1 Geometrical illustration of the Newton-Raphson method. f
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Newton-Raphson Method notes - Chapter 03.04 Newton-Raphson...

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