Derive the formula for Newton Raphson Method. Solve the equation x2 + 4x – 9 = 0 using Newton Raphson method. Assume error precision is 0.01. Discuss drawbacks of the Newton Raphson method.

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Derive the formula for Newton Raphson Method. Solve the equation x2 + 4x &#8211; 9 = 0 using Newton Raphson method. Assume error precision is 0.01. Discuss drawbacks of the Newton Raphson method.

Suresh Chand · Accepted Answer

The formula for Newton-Rapson method is (x_{n + 1} = x_n - frac{f(x_n)}{f'(x_n)}) We have to derive this formula Derivation: Let h is an small increment in xn to get next estimate of root (x_{n+1} - x_0 = h) (x_{n+1} = x_n + h) We know that Taylors series can be written as (f(x+h) = f(x) + frac{f'(x)h}{1!} + frac{f''(x)h^2}{2!} + ......) Since h is very small, We can neglet the terms containing second and higher order terms. Then above equation can be written as: (f(x+h) = f(x) + frac{f'(x)h}{1!}) (f(x_n+h) = f(x_n) + frac{f'(x_n)h}{1!}) (f(x_{n+1}) = f(x_n) + frac{f'(x_n)(x_{n+1} - x_n)}{1!}) If xn+1 is root of the given polynomail f(xn+1) = 0 (f(x_n) + frac{f'(x_n)(x_{n+1} - x_n)}{1!} = 0) Solving above equation to get the value of xn+1 we get (x_{n + 1} = x_n - frac{f(x_n)}{f'(x_n)}) This is the equation of Newton Raphson formula.   Problem Part: Solution: Let f(x) = x2 + 4x - 9 f'(x) = 2x + 4 Let us assume initial guess is 4 Iteration 1: x0 = 4 f(x0) = f(4) = 16 + 16 - 9 = 23 f'(x0) = f'(4) = 8 + 4 = 12 (x_1 = x_0 - frac{f(x_0)}{f'(x_0)}) x1 = 2.0833 Error = (left | frac{x_1 - x_0}{x_1} right |) = 0.92 Iteration 2: f(x1) = 3.673 f'(x1) = 8.166 x2 = 1.633 Error = 0.275 Iteration 3: f(x2) = 0.202 f'(x2) = 7.266 x3 = 1.633 Error = 0.017 Since error is less than specific limit Root = 1.605 Therefore, Root of the equation is 1.605   Drawback of Newton-Rapson Method: It's convergence is not guaranteed. So, sometimes, for given equation and for given guess we may not get solution. Division by zero problem can occur. Root jumping might take place thereby not getting intended solution. Inflection point issue might occur. Symbolic derivative is required. In case of multiple roots, this method converges slowly. Near local maxima and local minima, due to oscillation, its convergence is slow.

Derive the formula for Newton Raphson Method. Solve the equation \(x^2 + 4x – 9 = 0\) using Newton Raphson method. Assume error precision is 0.01. Discuss drawbacks of the Newton Raphson method.

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Derive the formula for Newton Raphson Method. Solve the equation \(x^2 + 4x – 9 = 0\) using Newton Raphson method. Assume error precision is 0.01. Discuss drawbacks of the Newton Raphson method.

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