Lab 7 - Lab 7 Lab 7 Solving Nonlinear Equations using the...

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Lab 7 Lab 7: Solving Nonlinear Equations using the Newton-Raphson Method Background Suppose we want to solve the equation f(x) = 0. Let x o be an initial guess for the solution. The Newton-Raphson method uses this initial guess to iterate to a “better” solution as follows: x 1 = x 0 – f(x o )/f’(x o ) The updated guess, x 1 , can then be used to iterate to an even “better” solution as follows: x 2 = x 1 – f(x 1 )/f’(x 1 ) This pattern continues so that the new guess depends on the previous guess as follows: x n+1 = x n – f(x n )/f’(x n ) The iterative algorithm runs until |f(x n )| < ε (some small specified value) or until the number of iterations exceeds some specified value indicating that the algorithm doesn’t appear to be converging. So, the algorithm works as follows: 1. Make an initial guess, x 1 2. Is │f(x 1 )│ < ε? That is, is the guess close enough to a solution? If yes, finish. If not, go to step 3. 3. Iterate to a new guess using the Newton-Raphson algorithm. 4. Is │f(x n+1 )│ < ε? If yes, finish. If not, return to step 3 unless the number of iterations exceeds some specified maximum. The Newton-Raphson method is very useful in solving both single equations and systems of non-linear equations. In this lab, we will be applying to algorithm to find solutions to single equations only. Note: Do not use symbolic expressions in any of your scripts for this lab. Part A Suppose we want to estimate the fifth root of 80.5. This is equivalent to solving the equation f(x) = x 5 – 80.5 = 0. A decent initial guess might be 2.5 since 2 5 = 32 and 3 5 = 243.
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