This document describes how I implement the Newton method using Python and Fortran on the test function f(x) = (x 1) log 10 (x).

Transcription

1 AMS 209 Foundations of Scientific Computing Homework 6 November 23, 2015 Cheng-Han Yu This document describes how I implement the Newton method using Python and Fortran on the test function f(x) = (x 1) log 10 (x). 1 Test function The test function is f(x) = (x 1) log 10 (x), which is the function type 2 specified in the runtime parameter file named as rootfinder.init xxx with the parameter called ftn type. Figure?? shows the test function. 1 Its root, x : f(x) = 0, is at x = x = 1. Figure 1: The test function f(x) = (x 1) log 10 (x) 1 The code that generates the figure is saved in./pyrun/functionplot.py. 1

2 2 Newton Method Convergence Results In this section, I examine the convergence behavior of implementing the Newton method with two initial guesses: one is x 0 = 0.9, which is close to the true root x = 1; the other is x 0 = 10, which is far away from the true solution. With three threshold values 10e-4, 10e-6 and 10e-8, and two initial guesses 0.9 and 10, after running the Python script pyrun rootfinder.py 2, six runtime parameter files, six convergence data files and corresponding pictures are generated and named according to the threshold values and initial guesses. For example, rootfinder.init is the runtime parameter file with the threshold and initial guess 0.9. The last number indicates the order that a file is generated. So this file is the first generated file. File rootfinder newton ftntype 02 1e dat.5 is convergence result with threshold 10e-6 and initial guess 10. Here, function type is also included and this file is the fifth dataset generated by running the code. Picture names are like result 2 1e png. Again, here 2 indicates the function type. 2.1 Result with x 0 = 0.9 The output shown on the terminal screen for x 0 = 0.9 is like The initial search value is x = Your converged solution x = Solution converged in Nstep= 17 Threshold value = E-004 The initial search value is x = Your converged solution x = Solution converged in Nstep= 30 Threshold value = E-007 The initial search value is x = Your converged solution x = Solution converged in Nstep= 43 2 If one wants to show figures on the screen, please uncomment plt.show() in the Python function plot data(plotfilename) at line 79. 2

3 Threshold value = E-008 Notice that as the threshold value decreases, the algorithm requires more iterations (17, 30 and 43), and the converged solution x is closer to the true root x = 1. All three convergent solutions are not exactly one, but they are very close to it. Figure??,??, and?? shows the convergence behaviors when initial guess is 0.9 for three different threshold values 1e-4, 1e-6 and 1e-8. For the first several iterations, the solution varies much, going toward to the true root faster than the solution in the last few iterations. Figure 2: Convergence behavior with threshold and initial guess 0.9. Top: Solution x v.s. 2.2 Result with x 0 = 10 When our initial guess is far away from the true solution, the algorithm need more iterations to get converged. Here, when x 0 = 10, it requires 26, 39 and 52 steps for threshold 1e-4, 1e-6 and 1e-8. Each have 9 more iterations than the case when x 0 = 0.9. The output shown on the terminal screen looks like: The initial search value is x = Your converged solution x = Solution converged in Nstep= 26 Threshold value = E-004 3

4 Figure 3: Convergence behavior with threshold 1e-6 and initial guess 0.9. Top: Solution x v.s. Figure 4: Convergence behavior with threshold 1e-8 and initial guess 0.9. Top: Solution x v.s. The initial search value is x = Your converged solution x = Solution converged in Nstep= 39 Threshold value = E-007 4

5 The initial search value is x = Your converged solution x = Solution converged in Nstep= 52 Threshold value = E-008 Again, all convergent solutions are not identical to one, but are slightly greater than one. Figure??,?? and?? show the convergent behaviors when initial guess is 10. As in the previous case, the iterated solution does not change much after 20 iterations. Figure 5: Convergence behavior with threshold and initial guess 10. Top: Solution x v.s. 2.3 Comments For this particular function with NIKE-like smooth quadratic curve, its convergent behavior is not complicated. If x 0 < x, x iter keeps increasing to x, and x iter keeps decreasing x if x 0 > x. x iter is not going up and down around the true solution or has any special or interesting convergent pattern. In both initial value cases, x iter converges quickly to x in just 10 iterations. 5

6 Figure 6: Convergence behavior with threshold 1e-6 and initial guess 10. Top: Solution x v.s. Figure 7: Convergence behavior with threshold 1e-8 and initial guess 10. Top: Solution x v.s. 6