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Suggested Guidelines (1) Choose the two important constant parameters P and c in (1). There could be multiple pairs of such constants. You also need to validate the computation of the Bernoulli number B. (The values of the first twenty B,1,2,.,20 are readily available in Wikipedia.) (2) Plot or tabulate the function f(r) for a large range of values of x and pick those values rela, bi] for which the condition f(a) f(b) <0 is satisfied. (Incidentally there could be more than one zero crossing, i.e., i > 2.depending on the values of P and c.) (3) Utilize separately the bisection and Newton-Raphson algorithm (1. ch. 2) to deter- mine the final value of the root(s) within a tolerance e=1 x 10-8 and a relative error 2=1 x 10-4%. Record the mimber of iterations N in cach case when the convergence fore and has been met. Plot the c & 7 vs. N for a specific function f(t) at its various different roots, if they exist. (This process needs to be repeated for other f(x), defined by the constants c and P as defined in part (1) above.) (4) Develop results for the functions you worked with in part (3), but this time using a combination of bisection and Newton-Raphson's method. Repeat the procedure in part (3) and compare the merits of the three approaches based on the convergence testing procedures as suggested there. (5) By judiciously examining the nature of the function f(x) from their corresponding graphical plots in part (2), use Lagrange interpolation methods to approximate the functions over a suitable range. (Note that higher degree of oscillations in f(x) would require higher orders of the Lagrange polynomial.) Plot the actual function and its Lagrange approximation. Next, solve for the roots using the Lagrange interpolation approximation via the aggregate of methods identified in parts (3) and (1). Compare the final values of the roots against the results from (3) and (4). Discuss the convergence behavior of this approach, that utilizes Lagrange interpolation of f(x). (6) As the final step, take the final values of the roots found by the Lagrange poly- nomial method as initial/starting values for the Newton-Raphson and Bisection method. Repeat the process(es) as described/identified above and test the conver- gence behavior. Expectations and Evaluation It is expected that the project will be investigated in detail. The clarity of approach and methodology and conclusions should be conspicuously more than adequate. Vague, incomplete or inadequate approach/effort is discouraged. The evaluation is most likely going to be subjective, with increased credit for merits of the overall effort of the student. This is an individual effort, and, plagiarism and must be avoided according to UM System guidelines. References [1] R. L. Burden and J. D. Faires, Numerical Analysis (9th ecli- tion).Boston, USA: Brooks/Cole, Cengage Learning, 2011, [2M. Abramowitz and L. A. Stegun(eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. NY, USA: Dover Publications, Inc., 1972 END OF CLASS PROJECT # 1 2
Course Title: Applied Numerical Methods Course Number(s): ECE 401, ECE 5590-NM, CS 5590-0013; Spring 2022 • Class Project # 1 Zeros of Equations & Lagrange Interpolation In this class project assignment, solutions to the roots of 6) = 0 need to be found where f(x) is expressed analytically as: Р = sin(cr) - B", where the Bernoulli number, pel cos(2k-wp) (2) (1) 1 is of order p. Indeed, if p > 1. then for all odd orders B,=0. Note that B = For even orders the Bernoulli numbers are calculated from the result in (1). This property automatically precludes the choice of the summation index p in (1). The result in (1) is a special case B (x=1) of the general formula given in (2, p. 805). Project Objectives The project objectives are to examine the convergence of the bisection and Newton- Raphson's methods [1, ch. 2). This involves determining the number of iterations required to arrive at the desired solution, within a specified tolerance, &. The detailed description of the bisection and Newton-Raphson's methods are given in (1and the stu- dent is advised to thoroughly review these topics for convenience. Various combinations such as: (a) different initial values, (b) variations in tolerance e can dictate the number of iterations and hence the computational resources required. These practical aspects will be explored in this investigation and hence is a primary objective of this project. The primary deciding factor for testing the convergence of the process is the ini- tial/starting values for the root. This in itself is a difficult issue for the equation de fined by (1). The information gleaned from 1, sec. 2.1) suggests that the root lies between rela, b) if f(a)/(b) <0. The algorithm for the bisection (1. p. 49) or Newton- Raphson (1. p. 68) method can be started that would be expected to give the final values within the tolerance Often, depending on the nature of the functional form of f(x), it maybe difficult at times to determine the initial/starting values for the root. Therefore it is expedi- ent to seek a combination of the methods such as bisection method followed by the Newton-Raphson. Unfortunately, while in many cases this approach improves the con- vergence properties of the algorithm, the exact functional form may still be formidable to efficiently continue the process for determining the initial values. To that end, one can approximate the behavior of the exact nature of f(x) within a set of points by a Lagrange polynomial P.) 1. pp. 108-114) of order one, two, three or perhaps higher, as may be appropriate. The set of points over which the Lagrange interpolation is to be obtained should also include the set of points 1 € (a,b) over which (a) f(b) <0. This, however, means that there could be more than one pair of points € (a, b) and f(a) f(b) <0. The primary advantage of the interpolation procedure is that the initial/starting val- ues of the roots may be found easily than the original equation. There is one cautionary observation: if f(a) is highly oscillatory then higher order Lagrange polynomials would be necessary which can defeat the very purpose of this suggestion Based on the above information the following guidelines are suggested for contin- uing the investigation. These are not mandatory but suggestions only: the student is encouraged to improvise better pathways to solve the problem.