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Magazine Chapter 4: Problem 9
The equation \(x^{3}-1.2 x^{2}-8.19 x+13.23=0\) has a double root close to \(x=2\). Determine this root with the Newton-Raphson method within four decimal places.
Short Answer
Expert verified
\(x \approx 1.9595\)
Step by step solution
01
Understand the Problem
Given the equation \[x^{3} - 1.2x^{2} - 8.19x + 13.23 = 0\], find the double root close to \(x = 2\) using the Newton-Raphson method within four decimal places.
02
Rewrite Newton-Raphson Formula for Double Roots
For a double root scenario, the modified Newton-Raphson formula is: \[ x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n}) - \frac{f(x_{n})f''(x_{n})}{2f'(x_{n})}} \]Set up \(f(x) = x^{3} - 1.2x^{2} - 8.19x + 13.23\).
03
Compute Derivatives
Compute the first derivative \[f'(x) = 3x^2 - 2.4x - 8.19\]and the second derivative \[f''(x) = 6x - 2.4\].
04
Set Initial Guess
Take the initial guess as \(x_0 = 2.0\).
05
Iteration 1
Calculate \(f(x_0)\), \(f'(x_0)\) and \(f''(x_0)\): \[f(2.0) = (2.0)^3 - 1.2(2.0)^2 - 8.19(2.0) + 13.23 = 0.47\]\[f'(2.0) = 3(2.0)^2 - 2.4(2.0) - 8.19 = 1.41\]\[f''(2.0) = 6(2.0) - 2.4 = 9.6\]Now calculate the new \(x_1\) using the modified formula.
06
Apply the Formula
Substitute values into our modified Newton-Raphson formula:\[ x_1 = 2.0 - \frac{0.47}{1.41 - \frac{0.47 \cdot 9.6}{2 \cdot 1.41}} = 1.9377\]
07
Iteration 2
Repeat the above steps with \(x_1 = 1.9377\) to get the next iteration. \[f(1.9377) = -0.0854\]\[f'(1.9377) = 0.433\]\[f''(1.9377) = 9.2262\]Then, \[ x_2 = 1.9377 - \frac{-0.0854}{0.433 - \frac{-0.0854 \cdot 9.2262}{2 \cdot 0.433}} = 1.9564\]
08
Iteration 3
Repeat the steps now with \(x_2 = 1.9564\):\[ f(1.9564) = -0.0107\]\[ f'(1.9564) = 0.326\]\[ f''(1.9564) = 9.3384\]Then, \[x_3 = 1.9564 - \frac{-0.0107}{0.326 - \frac{-0.0107 \cdot 9.3384}{2 \cdot 0.326}} = 1.9595\]
09
Check for Convergence
If the difference between the new approximation and the previous one is less than 0.0001, the process is complete. With \(x_3 = 1.9595\) and \(x_2 = 1.9564\), the difference is less than 0.0001.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Double Root
A double root occurs when a function touches the x-axis at a certain point but does not cross it. This means both the function and its first derivative are zero at this point. Finding a double root is tricky because regular numerical methods like the standard Newton-Raphson method may converge slowly or fail altogether. This is why we use a modified formula for double roots.
Numerical Methods
Numerical methods are techniques used to find approximate solutions to mathematical problems that cannot be solved exactly. These methods are especially useful for complex equations such as the one given in the problem. The Newton-Raphson method is a popular numerical method for finding roots of real-valued functions.
This technique uses an iterative process to approximate the root, gradually getting closer with each step. Although powerful, the method requires good initial guesses and can be sensitive to the behavior of the function.
This technique uses an iterative process to approximate the root, gradually getting closer with each step. Although powerful, the method requires good initial guesses and can be sensitive to the behavior of the function.
Convergence
Convergence in numerical methods refers to the process of approaching a final value as iterations increase. For the Newton-Raphson method, convergence is achieved when subsequent approximations differ by less than a pre-specified tolerance level, like 0.0001 in this case.
When iterating, if the difference between the new approximation and the previous one falls below this threshold, the method has converged, and the root is determined reliably within the desired accuracy. This method becomes particularly exciting in challenging cases such as double roots, where convergence might be less straightforward without modifications to the formula.
When iterating, if the difference between the new approximation and the previous one falls below this threshold, the method has converged, and the root is determined reliably within the desired accuracy. This method becomes particularly exciting in challenging cases such as double roots, where convergence might be less straightforward without modifications to the formula.
