Prompt

Discussion and Error

As we can see from the plots below, the approximate percentage error decreases at an almost identical rate. Depending on the tolerance set for the secant method, greater or fewer iterations may run. I chose a tolerance of 0.000001 which lead to the method converging at the same number of iterations as Brent's method. Looking at figure 4 below, we see how similar the drop in approximate error is (especially for the first few iterations). This is because, for the given function and initial x-coordinates, Brent's method chose to use the secant method for half of the total iterations. Brent's method first tries to use the secant method, and if it cannot it uses either inverse quadratic interpolation or the bisection method. For this particular function, tolerance, and initial bounds, the bracketing method was never used. The first iteration used secant method, the second used inverse quadratic interpolation, the third and fourth used secant, and the fifth and sixth used inverse quadratic interpolation. Because Brent's method used open methods for all iterations and the secant method for half of them, the error outcome is going to be relatively similar. As we can see from figure 4, Brent's method slightly outperforms the Secant method in convergence. This is to be expected, as Brent's method is choosing between multiple methods, including the secant method, for the quickest convergence to the root. It is for this reason Brent's method will often outperform most other root-finding methods.

Convergence can be expected as the function is "well-behaved" near the root, making it easy for the open methods to rapidly discern the root. We can see that for both Brent's method and the secant method, the root estimates converge to a value extremely close to our "analytical" solution -0.539606511763780 (given by "fzero" function). All of these results also agree with our root approximation from the graphical method, however my value from the graphical method is not to the same precision as the other approaches simply due to choice. I used the graphical method to inform my bracketing bounds for the other numerical methods, not to find a precise "analytical" value.

Comparing each iteration's approximation of the root to the "analytical"value can be seen in Figure 5. Figure 5 shows that after one iteration has completed, both Brent's method and the Secant method have true error values of ~75%. After the second iteration this drops to ~5%.

Part 2: Use Brent's Method to Solve Catenary Cable Problem

Numerical Method

Given the problem setup above, our numerical approach will involve using Brent's method to find the root of f(Ta). Finding the root allows us to find Ta because where the function of f(Ta) = 0 is the Ta value that satisfies the originial equation. See Appendix III for MATLAB code. The root found using Brent's method was x = 1266.32436039989 and converged after 6 iterations. See Part 1 for an explanation of Brent's method, pseudocode, and a sample calculation.

Analytical Method

The "analytical" method I chose for part 2 is MATLAB's built-in "fzero" function. This returned a root of x = 1266.32436039989, identical to the result from our numerical method (Brent's method).

Discussion and Error

As we can see from Figure 8 below, Brent's method rapidly converges and drops below the break criteria after 6 iterations. Interestingly, the approximate percent error starts at slightly over 2% (after the first iteration). This is because the "first pass" brought the root approximation significantly closer to the root. After three iterations the approximate error falls to 0.0035%. Convergence can be expected as the function is "well-behaved" near the root, making it easy for the open methods to rapidly discern the solution.

From figure 9 we can see that the approximations for Brent's method rapidly converge to our "analytical" solution 1266.32436039989 (given by "fzero" function). In fact, it converges to exactly the same value within 6 iterations (exceeding 10 decimal places of precision). The first iteration was below 0.6% away from our "true" value. This extremely small true error makes sense given the fact that the "fzero" function uses a method very similar to Brent's method. We are effectively sending the same function and bounds to almost an identical method. This exaplains the extremely small true error value.

Both of these results also agree with our root approximation from the graphical method.