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For finding roots of algebraic and transcendental equations
TABLE 4.2 Summary of important information
Example 1 Determine the real root of a)Graphically. b)Using the bisection method (three iterations). c)Using the secant method (three iterations). d)Using the false position method (three iterations). e)Using the Newton-Raphson method (three iterations). Solution. a)The graphical approach for determining the roots of an equation.
The root is b)Bisection method. Using bisection, the results can be summarized as
Thus, after three iterations the root is x » 0.5625, f(x) = 0.007283 @ 0, c)Secant method. Use the secant method to find the root with initial estimates of First iteration: Second iteration: Note that both estimates are now on the same side of the root. Third iteration: d)False position method. Use the false position method with guesses of a0 = 0 and b0 = 1. First iteration: Second iteration: Therefore, the root lies in the first subinterval, and p1 becomes:
Third iteration: Therefore, the root lies in the first subinterval: e)Newton-Raphson method. The first derivative of the function
Starting with an initial guess of x0 = 1, this iterative equation can be applied to compute
Thus, the method rapidly converges on the true root. Notice that the percent relative error decreases at each iteration much faster than it does in another methods. Example 2 Use the Newton-Raphson method for nonlinear system to determine the roots of equations: Solution. Graphical method gives us solution (point): M(1.2; 1.7). For this system Jacobian matrix is
Hence, at the initial guesses we approximated roots as
First iteration: Thus, the determinant of the Jacobian for the first iteration is Hence, Check, how the results are converging to the true values:
Second iteration: Repeat this process and we will have The computation can be repeated until an acceptable accuracy is obtained.
Problems 1. Determine the real roots of 2. Locate the first positive root of 3. Determine the roots of the nonlinear system of equations using the Newton-Raphson method: Use graphical approach to obtain your initial guesses. Discuss and estimate the rate of convergence.
Date: 2016-01-14; view: 1040
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