AMS209 Final Project

Transcription

1 AMS209 Final Project Xingchen Yu Department of Applied Mathematics and Statistics, University of California, Santa Cruz November Abstract In the project, we explore LU decomposition with or without pivoting to compute systems of linear equations which are fundamental to any linear algebra problems. We used modular programming in Fortran to perform LU decomposition, forward and backward substitution for three different systems of linear equations. We used python to execute Fortran and compare results generated from our Fortran program with that from python Numpy package. As expected, the results are the same for our Fortran program and Numpy package. Lastly, we plot the requested matrix plot using Matplot from python. All of the three systems of equations were automatically executed at once as well and plots. 2 Methods, Results and Discussion In this section, we explain the methods we used for LU decomposition and the correpsonding results. We conclude this section by discussing the results and comparison between Fortran and Python 1

2 2.1 Methods In numerical analysis, LU decomposition factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. The LU decomposition can be viewed as the matrix form of Gaussian elimination. With LU decomposition, systems of linear equations could be solved by simply with forward and backward substitution, providing scalability to solve large systems of equations. Two types of LU decomposition were used in our Fortran program, of which the first one is the LU decomposition without pivoting while its counterpart used pivoting. LU decomposition with pivoting eliminate the problem of dividing a very small value that creates wrong solution to the system of linear equations. Therefore, LU decomposition with pivoting should be preferred over the one without pivoting. In the python implementations, we provide run time parameter to allow users to select the desired method to compute LU decomposition in which 1 indicates for LU decomposition without pivoting and 2 for LU decomposition with pivoting. 2.2 Results In this section, we provide the results generated for each of three systems of linear equations. As stated earlier, the LU decomposition as well as forward and backward substitution were computed in Fortran. We use Python like a bridge program that performs run setup by compiling the makefile; and run scheduling by automatically three systems all in once; and checking results produced by out Fortran program and Numpy package from Python as well as producing three matrix plots. To compare the results between Numpy package and our Fortran implementation, we used a threshold of 1e-14 and compare the it with the sum of the absolute element-wise difference between results from Fortran and Python. 2

3 2.2.1 Case 1 In case 1, we are required to compute the following system of linear equations. The results output to the screen is shown in the following figure. In this problem, we used method 2 which is LU decomposition with pivoting to solve the system of linear equations. Figure 1: Screen Output for Case 1 Based on the screen output, we concludes that the results computed with Python and Fortran are the same, which is expected. The matrix plot for A, X and b are shown as follows respectively. To create these following plots, we need to have a n by n matrix to begin with. Since X and b are 1-D vector, therefore, we created a n by n matrix for them by duplicating itself. 3

4 Figure 2: A(left), X(middle), b(right) Case 2 In case 2, we are required to compute the following system of linear equations. The results output to the screen is shown in the following figure. In this problem, we used method 1 which is the LU decomposition without pivoting to solve the system of linear equations. We want to see if there is potential numerical problem of dividing by zero without pivoting. Figure 3: Screen Output for Case 2 4

5 Based on the screen output, we concludes that the results computed with Python and Fortran are the same, and therefore in this case we didn t encounter the problem of dividing a small value without using pivoting. The matrix plot for A, X and b are shown as follows respectively. Once again, to create these following plots, we need to have a n by n matrix to begin with. Since X and b are 1-D vector, therefore, we created a n by n matrix for them by duplicating itself. Figure 4: A(left), X(middle), b(right) Case 3 In case 3, we are required to compute the following system of linear equations. The results output to the screen is shown in the following figure. In this problem, we used method 2 which is the LU decomposition with pivoting to solve the system of linear equations. 5

6 Figure 5: Screen Output for Case 3 Based on the screen output, again the results from Python and Fortran agrees with each other. The matrix plot for A, X and b are shown as follows respectively. Once again, to create these following plots, we need to have a n by n matrix to begin with. Since X and b are 1-D vector, therefore, we created a n by n matrix for them by duplicating itself. Figure 6: A(left), X(middle), b(right) 2.3 Discussion As illustrated in this section, all the results 3 Conclusion As expected, we have the results from Python Numpy package agreeing with our Fortran implementation. During the construction of the code in Fortran, 6

7 we noticed that the book keeping is very tedious and difficult comparing to Python. However, the debugging flag provided in Fortran is very helpful in terms of quickly find the place of problem. On the other hand, Python is very user friendly and easy to do bookkeeping while having good debugging power as a open source software. Over the course of the this semester, I believe I have gained a fair amount of knowledge especially in Python and Fortran and I would like to express my sincere gratitude to our Professor Dongwook Lee and our TA David Johnes for your help and constructive advice. 7