1 Introduction: Using a Python script to compile and plot data from Fortran modular program for Newton s method

Transcription

1 1 Introduction: Using a Python script to compile and plot data from Fortran modular program for Newton s method This week s assignment for AMS209 involves using a Python script that can compile a Fortran modular program and plot data from multiple input values within one run/implementation. We are using Fortran scripts that we originally explored in a previous AMS209 homework. These F90 files will be contained in a different directory from the Python script. For instance, here is a directory structure: hw6/ newton rootfinder/ RootFinder.F90 findrootmethod module.f90 makefile read initfile module.f90 definition.h ftn module.f90 output module.f90 setup module.f90 (excluding rootfinder.init) - our Python script will make this pyrun/ pyrun rootfinder.py 2 Overview of Python script Brief-ish explanation of my python script. Can be run in terminal with $ python pyrun rootfinder.py after importing os, numpy, matplotlib, copyfile, and glob. My python functions are within a for-loop that allows one run/execution for multiple threshold values or multiple initial guess values. Value(s) for threshold and initial guess are stored inside a python list. 1

2 Functions to implement are in order: 1. boldruntimeparameters init(outputname, searchmethod, x beg, x end, max iter, threshold, ftn type, init guess, multiplicity) This function creates a.init file from which the Fortran scripts will read input parameters. These can be assigned as string variables within the pyrun rootfinder.py, which are also arguments in this function. If a previous *.init file exists, then the script will make a copy of the previous file into *init.1. This copy script will also increment for multiple copies (i.e. *.init.1 becomes *.init.2 and so on and so forth). 2. make make(path) This function can compile the Fortran codes and create an executable if.exe does not exist yet. There is also an option to simply execute the command $ make clean; and then $ make; in order to compile the Fortran code. 3. run rootfinder(path) My simplest function here. It simply executes the exe file via os.system (i.e. $./rootfinder.exe). If there is no.exe file to execute, then it will print an error message prompting the user to compile the Fortran codes. 4. plot data(plotfilename, ftn type, threshold) Plots Newton method root-solutions versus iterations and also plots Newton method root-solution residuals versus iterations. This uses matplotlib.pyplot to plot values read from the Fortran outputted *.dat file. I also used string manipulation to save figures as.png files containing pertinent information such as threshold value, function type, and initial guess. The figures will also display on the screen (interactive=false, plot.show(block=false)) and I also included a command for the.png files to open on Preview on Mac.Additionally, if a previous.dat file exists, then the script will make a copy of the previous file into *init.1. This copy script will also increment for multiple copies (i.e. *.dat.1 becomes *.dat.2 and so on and so forth). 3 Choosing a test function I used the following test function (equation 1) which is also the original ftn type=1 in ftn module.f90: Its derivative (equation 2) is also found in 2

3 ftn module.f90. f(x) = x + e x + 10 (1 + x 2 ) 5 (1) f (x) = 1 + e x 20x (1 + x 2 ) 2 (2) I picked equation (1) as the test function because tit contains a local minima at x = 1.16 (see Fig.1) thus making it difficult/impossible to solve a root using Newton s method when using an initial guess greater than the local minima. I wanted to see how the program with such a function. Based on Fig.1, the equation should also have a root approximately equal to Figure 1: Plot of the test function to guess for approximate root 3

4 4 Implementing the Python script to find the root of the test function using Newton s method I executed in the Terminal shell using command $ python pyrun rootfinder.py In one run, I was able to test different initial guess values and different threshold values. This was done by inputting the values into a list and using a for-loop to iterate through the entries in that list. I chose my best initial guess (init guess = -2) because it was close enough to the estimated root (-0.9) but still required >5 iterations to reach a solution. Interestingly I seem to get the same result within the same number of iterations for thresholds 1e-8, 1e-6, and even 1e-4. All three threshold values go through 6 iterations to find a root of x = (see Figures 2-4). Identical results for all three threshold values also means that the 5th iteration has a residual > 1e-4, while jumping to the 6th iteration immediately results in a residual < 1e-16. Figure 2: The most stringent threshold needs 6 iterations to reach approximate root x =

5 Figure 3: Less stringent threshold 1e-6 still reaches same approximate root in the same number of iterations as 1e-8 Figure 4: Even least stringent threshold = 1e-4 yields identical results as threshold = 1e-6 and 1e-8 in the same number of iterations 5 Using a different initial guess I had chosen a different initial guess that was 10x the original. Additionally, I found that it gave me different-looking enough results and still minimize any adjustment for the y-scale. I wanted to keep the y-scale as small as possible for all plots in order to detect subtle differences. For an initial guess further from the root (init guess = -20), it takes longer (16 iterations) to reach a slightly different approximate root of x = The difference from the previously calculated root (x = using init guess = -2) is approximately 1.1e-16. This is an artifact of the approximation since the lowest threshold value is 1e-8. (see Figure 5). Additionally, less stringent threshold values can either lead to the same result as threshold = 1e-8 such as with threshold = 1e-6 (see Figure 6), or can lead to Newton s method stopping in less iterations (15 iterations instead of 16) such as with threshold = 1e-4 (see Figure 7). In 15 iterations, the estimated root is x= , which has a e-10 difference from the more stringent thresholds. 5

6 Figure 5: An initial guess of -20 needs 16 iterations to reach a slightly different root of x = Figure 6: Less stringent threshold 1e-6 still reaches same approximate root in the same number of iterations as 1e-8 Figure 7: However least stringent threshold = 1e-4 stops at 15 iterations to approximate a root of x=

7 6 Conclusions about Newton s method Furthermore, comparing the plots from the 2 different initial guesses yields slightly different behaviors, even though both converge towards similar roots. For instance, for the iterations 4 steps before from the last iteration (the 5th last iteration?), the initial guess = -2, gives a calculated root slightly less than the final estimated root. In contrast, the initial guess = -20, gives a calculated root slightly higher than the final estimated root in the similar iterative position (4 iterations before the last). The two initial guesses also appear more divergent 5 iterations before their respective last iteration. This demonstrates how the Newton method can converge towards a similar root even with varying initial guesses. Figure 8: Initial guess =-2. The iteration 4 steps before the last has a result slightly less than the final iteration. Iterative solutions are mostly flat. Figure 9: Initial guess =-20. The iteration 4 steps before the last has a result slightly higher than the final iteration. Iterative solutions tend decrease towards the final result. By: G.Carlo Parico, written for AMS209 HW6 7