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This occurs because the underlying function is a cubic equation that has zero fourth and higher derivatives. Note the 2nd and 3rd order Taylor Series functions are the same. From the plots below, we see that the answer is the 4th Order Taylor Series expansion. The second iteration is 0. The remainder of the iterations are displayed in the following table: Write beam equations: Because the curve crosses the axis between 6 and 10, use 3.
Using bisection, the first iteration is 0.
The second iteration is 1. Thus, after ten iterations, the false position method is converging at a very slow pace and is still far from the root in the vicinity of 1. This is a classic example of a case where false position performs poorly and is inferior to bisection.
Insight into these results can be gained by examining the plot that was developed in part a. Because of the shape of the present function, the opposite is true.
First, it can be solved for the linear x, 0. An alternative is to solve for the second-order x, 1. The result can be checked by substituting it back into the original function, f 2. The results are explained by looking at a plot of the function. The guess of 0.
Therefore, the first iteration results in a prediction of 2. At these points the function is very flat and hence, the Newton-Raphson results in a very high value Thereafter, the methods slowly converge on the nearest roots. Explanation of results: Both guesses are in a region where the function is relatively flat.
Because the two guesses are on opposite sides of a minimum, both are sent to different regions that are far from the initial guesses. Due to the concavity of the slope, the next iteration will always diverge. The following graph illustrates how the divergence evolves. Hence, the solution is cast far from the roots in the vicinity of the original guess. Roots seem to occur at about 40o and 50o. The fact that a slight change in one of the coefficients results in a radically different solution illustrates that this system is very ill-conditioned.
The major difference is that the elimination is only implemented for the left-hand side coefficients. This is left for an exercise. Matrix is close to singular or badly scaled. Results may be inaccurate. A right-hand side vector can be developed corresponding to a solution of ones: Thus, for this case, the condition number tends to exaggerate the impact of ill-conditioning.
First iteration: After 6 iterations, the maximum error is 4. After 6 iterations, the maximum error is 3. However, if Set 1 and 3 are reordered so that they are diagonally dominant, they will converge on the solution of 1, 1, 1.
Set 1: However, it will also not diverge. Rather, it will oscillate. The way that this occurs depends on how the equations are ordered. For example, the first equation can be solved for x and the second solved for y. For this case, successive substitution does not work First iteration: An alternative solution involves solving the second equation for x and the first for y.
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