Mathematics Assignment | Estimation of Roots by Iteration
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Added on 2020-05-16
Mathematics Assignment | Estimation of Roots by Iteration
Added on 2020-05-16
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Running head: ESTIMATION OF ROOTS BY ITERATION1Estimation of Roots by IterationNameInstitution Affiliation
ESTIMATION OF ROOTS BY ITERATION2Estimation of Roots by Iteration 1.Task 2 P2.4Find the estimated value of the positive root of the equation: Cx2 + Dx – E = 0Solution:From the 8th row of the table of Data Sheet for P2.4, C, D and E have the values 7, 4 and 8 respectively. Thus, the new equation becomes: 7x2 + 4x – 8 = 0Estimating the values of x for iteration:f (0) = 7 (02) + 4 (0) – 8 = -8f (1) = 7 (12) + 4 (1) – 8 = 3Given that when x = 0 the yields a negative number while x = 1 produces a corresponding positive number, the graph crosses the x-axis between 0 and 1. Thus, the positive roots of the equation lie between the intervals 0 and 1. According to bisection method, the two values are the lower bound and upper bound (Carroll, 2011). From the values, the midpoint is obtained and used for the iteration until the most accurate value of x is obtained. Notably, iteration stops whena value equal to or significantly close to zero is attained.Let xl be the lower bound, xu the upper bound, and xm the mean of the estimates.f (xl)*f (xu) = f (0) * f (1) = -8 * 4.5 = -24 Since -24 < 0, the roots lie between the interval and it is correct to proceed with iteration.First Iteration:Calculating the midpoint of the interval: 0.5(0 + 1) = 0.5
ESTIMATION OF ROOTS BY ITERATION3By factoring the midpoint value in the equation 7x2 + 4x – 8 = 0, its equivalent quantity is obtained. The procedure is repeated to all the midpoint measures to get their respective quantities.f (xm) = f (0.5) = -4.25f (xm) becomes the new upper bound used to test whether the value lies between the 0 and 0.5f (xl)*f (xu) = f (0)*f (0.5) = -8 * -4.5 = 34Since 34 > 0, the value of the root of the equation lies between 0.5 and 1Second IterationSince the value is not within the interval tested in the first iteration, the procedure is repeated for the interval 0.5 and 1 to find the most appropriate value.Calculating the midpoint of the interval:0.5(0.5 + 1) = 0.75f (xm) = f (0.75) = -1.0625taking f (xm) as the new upper bound:f (xl) * f (xu) = f (0.5) * f (0.75) = -4.25 * -1.0625 = 4.5156Since 4.5156 > 0, the value of the root of the equation lies between 0.75 and 1Third Iteration:Calculating the midpoint of the interval:0.5(0.75 + 1) = 0.875f (xm) = f (0.875) = 0.859375taking f (xm) as the new upper bound:f (xl) * f (xu) = f (0.75) * f (0.875) = 0.859375 * -1.0625 = -0.913086Since -0.913086 < 0, the value of the root of the equation lies between 0.75 and 0.875
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