AMS209 Final Project: Linear Equations System Solver

Transcription

1 AMS209 Final Project: Linear Equations System Solver Rene Gutierrez Marquez 1 UCSC 1 December 7, 2016 Abstract In this project an implementation of a solver of a system of linear equations is implemented. In particular two methods are used, LU decomposition and LU decomposition with partial pivoting. This methods are implemented in Fortran without the use of any library. In Python we test our Fortran implementation, by compiling and executing for 3 systems. Furthermore, this are contrasted with a direct inversion using numpy linear algebra library, concluding an adequate precision. 1 Methods 1.1 Fortran The Fortran implementation consist of several modules connected by the driver linear solve. This driver is in charge of calling all the relevant subroutines read module This module includes only one subroutine, read data, which is in charge of reading the value of the matrix A, the vector b for each system. Furthermore, it also reads the initialization file, linear solve.init, which includes the name of the matrix A file, the vector b filename, the solution vector x file name as well as the method to be implemented (with or without partial pivoting) write module This module contains a simple subroutine, write to screen, which simply writes the matrix A and the vector b to screen, so the user can check them lu module This module contains the two LU decomposition methods. The subroutine lu decomp performs the LU decomposition without pivoting, while the subroutine lu decomp pivot performs the LU decomposition with partial pivoting solver module This module solves the linear equation system based on the LU decomposition. It contains two subroutines forward solve which solves the equation Ly = b and backward solve which solves the equation Ux = b output module This module contains a simple subroutine that writes the solution to screen as well as to a file makefile Finally, compilation is made through a makefile, which has the options clean to remove past compilation files and output, and the option debug wich has useful flags for debugging. Furthermore, the compilation with no options, that is simply make has some useful optimization flags. 1

2 1.2 Python The Python implementation is made through a main function called, PyRun, which changes directory to where the fortran files are. It uses several functions explained her run setup This function initializes the three different inputs of matrix A and vector b for each system of linear, equations. This function also compiles (or cleans and compiles if necessary) the Fortran code run scheduler This function creates the linear solve.init file to initialize asking the user if he wants to perform partial pivoting or not. After, runs the Fortran executable diff check This function checks if there is a difference between the Fortran solution and the numpy solution based on a threshold provided by the user plot run This function plots the system of linear equations in matrix form using matplotlib. Note that the colorbar shows the colors in the same scale of the three components of each equation, however differs across systems sol check This functions implements diff check and plot run. 2 Results The implementation in Fortran provides acceptable results, both without partial pivoting and with partial pivoting, when comparing to the numpy linear algebra inverse method, only differing when the threshold is close to the numerical limit of double reals. In particular it sees not difference in thresholds bigger than Examples Here we can observe some examples of the output to screen Example 1 No pivoting, threshold Do you want to perform partial pivoting? Enter yes/no:no Do you want to perform partial pivoting? Enter yes/no:no

3 E E-016 Do you want to perform partial pivoting? Enter yes/no:no Please enter a threshold for the differences between the fortran and python solutions for each entry:1e-14 Your Fortran solution and Python for the first system differ by less than the threshold Your Fortran solution and Python for the second system differ by less than the threshold Your Fortran solution and Python for the third system differ by less than the threshold Example 1 With partial pivoting, threshold Do you want to perform partial pivoting? Enter yes/no:yes Do you want to perform partial pivoting? Enter yes/no:yes

4 E E-016 Do you want to perform partial pivoting? Enter yes/no:yes Please enter a threshold for the differences between the fortran and python solutions for each entry:1e-15 The difference is greater than your threshold for the first system The difference for each entry is: e e e-15 The Fortran solution is: The Python solution is: Your Fortran solution and Python for the second system differ by less than the threshold The difference is greater than your threshold for the third system The difference for each entry is: e e e e-16 The Fortran solution is:

5 The Python solution is: Figures The output figures can be observe in fig 1. 4 Conclusion We can conclude that the Fortran implementation was successful, noting that for the three particular cases the partial pivoting did not provide a great improvement in the numerical stability of the solutions. 5

6 Figure 1: Output figures for each system (a) (b) (c) 6