Project Documentation

Overview

This project provides a solution for processing matrices, it performs the Gaussian elimination algorithm. The Gaussian elimination algorithm is a method for solving systems of linear equations by transforming the system's matrix into a row-echelon form. It works by performing row operations to eliminate variables from the equations, starting with the first column and working downwards.

Files

• main.c: Contains the main logic for processing the matrix, including matrix operations such as printing, pivoting, row substitution, and matrix reduction.
• data.c: Handles reading the matrix from a file (input.dat) and dynamically allocating memory for matrix storage.

Functionality

1. Matrix Initialization (data.c)

• Function: initialize_matrix()
• Description: This function reads the matrix from a file (input.dat) where the number of rows and columns is specified at the start. It dynamically allocates memory for the matrix and fills it with the values read from the file.
• Returns: A pointer to a 2D array (float**) representing the matrix.

2. Matrix Printing (main.c)

• Function: print_matrix(int rows, int cols, float** matrix)
• Description: Prints the matrix to the standard output in a tabular format.

3. Pivot Identification (main.c)

• Function: found_pivot(int rows, int cols, float** matrix)
• Description: Finds and returns the pivot element (non-zero) in the matrix. The pivot is essential for performing row elimination.
• Returns: A pointer to a pivot structure containing the row and column of the found pivot.

4. Row Swapping (main.c)

• Function: swap_rows(int rows, int cols, float** matrix, int r1, int r2)
• Description: Swaps two rows in the matrix. This is typically done to move the pivot row to the top.

5. Row Substitution (main.c)

• Function: substitute_row(int row, int cols, float** matrix, float alpha)
• Description: Performs row substitution to eliminate entries below the pivot. A row is replaced with a combination of itself and a scaled version of another row.

$R_k \to R_k + \alpha R_1$ Where $k$ is the row of the element to eliminate, $\alpha$ is a coefficient $s.t.$
$\alpha = -b / pivot$ where b is the element under the pivot, that has to be scaled.

6. Matrix Reduction (main.c)

• Function: kill_below_pivot(int rows, int cols, float** matrix, struct pivot* pivot_ij)
• Description: Eliminates all values below the pivot by adjusting the rows below the pivot row. This is part of the Gaussian elimination process.

7. Recursive Matrix Resolution (main.c)

• Function: resolve_matrix(int rows, int cols, float** matrix, FILE* output)
• Description: Recursively reduces the matrix by identifying pivots, swapping rows, and eliminating values below the pivot until the entire matrix is reduced. The result is written to the output.dat file.

8. Main Execution (main.c)

• Main Function: int main(int argc, char* argv[])
• Description: The program begins by initializing the matrix, printing it, and then resolving it using the defined functions. The final output is written to a file.

Usage

1. Prepare an input file (input.dat) in the following format:

   <number_of_rows> <number_of_columns>
   <matrix_values>
   

   Example:

   3 3
   1.0 2.0 3.0
   4.0 5.0 6.0
   7.0 8.0 9.0
   

2. Compile the program:

   gcc main.c -o main

3. Run the program:

   ./main

4. The output matrix will be saved in output.dat.

Dependencies

• Standard C libraries:
  • stdio.h
  • stdlib.h
  • stdbool.h

Future Improvements

• Implement error handling for matrix size mismatches.
• Optimize the algorithm for larger matrices.
• Add support for matrices with complex numbers.

Example

Input

Porcessed Matrix

Output