Step by Step solution about How to resolve the algorithm Gaussian elimination step by step in the PHP programming language

The provided PHP code implements the Gauss elimination method, which is an algorithm for solving systems of linear equations. Here's a detailed explanation:

1. swap_rows() Function:

• This function swaps the rows of two matrices ($a and $b) at the specified row indices ($r1 and $r2).
• It ensures that the rows are correctly switched in both matrices.

2. gauss_eliminate() Function:

• This is the primary function responsible for solving the system of linear equations using Gaussian elimination.
• It takes three parameters:
  • $A: A two-dimensional array representing the coefficient matrix for the system of equations.
  • $b: An array representing the constant terms for the system of equations.
  • $N: The number of equations or rows in the system.
• The function operates as follows:
  • It iterates through each column of the $A matrix.
  • For each column, it finds the pivot row, which contains the maximum absolute value in that column.
  • If the maximum value is not in the current row, it swaps the rows using the swap_rows() function to move the pivot row to the correct position.
  • It then performs row operations to eliminate all non-zero elements below the pivot element.
    • This involves subtracting multiples of the pivot row from the rows below it.
  • The resulting matrix is now in upper triangular form.
  • Finally, the function back-substitutes to solve for the values of the variables in the system of equations.
• It returns an array $x, which contains the solutions to the system of equations, indexed by the column numbers of $A.

3. test_gauss() Function:

• This function serves as a test case for the gauss_eliminate() function.
• It creates a sample system of linear equations with a known solution and calls gauss_eliminate() to solve it.
• It then prints the computed solutions, which should match the expected values.

Usage of the Code:

• The test_gauss() function is the entry point of the code and demonstrates how to use the gauss_eliminate() function.
• The code can be executed using PHP, and the output will be an array of solutions for the test system of equations.

This code provides a clear and efficient implementation of the Gauss elimination method for solving systems of linear equations in PHP.

function swap_rows(&$a, &$b, $r1, $r2)
{
    if ($r1 == $r2) return;

    $tmp = $a[$r1];
    $a[$r1] = $a[$r2];
    $a[$r2] = $tmp;

    $tmp = $b[$r1];
    $b[$r1] = $b[$r2];
    $b[$r2] = $tmp;
}

function gauss_eliminate($A, $b, $N)
{
    for ($col = 0; $col < $N; $col++)
    {
        $j = $col;
        $max = $A[$j][$j];

        for ($i = $col + 1; $i < $N; $i++)
        {
            $tmp = abs($A[$i][$col]);
            if ($tmp > $max)
            {
                $j = $i;
                $max = $tmp;
            }
        }

        swap_rows($A, $b, $col, $j);

        for ($i = $col + 1; $i < $N; $i++)
        {
            $tmp = $A[$i][$col] / $A[$col][$col];
            for ($j = $col + 1; $j < $N; $j++)
            {
                $A[$i][$j] -= $tmp * $A[$col][$j];
            }
            $A[$i][$col] = 0;
            $b[$i] -= $tmp * $b[$col];
        }
    }
    $x = array();
    for ($col = $N - 1; $col >= 0; $col--)
    {
        $tmp = $b[$col];
        for ($j = $N - 1; $j > $col; $j--)
        {
            $tmp -= $x[$j] * $A[$col][$j];
        }
        $x[$col] = $tmp / $A[$col][$col];
    }
    return $x;
}

function test_gauss()
{
    $a = array(
        array(1.00, 0.00, 0.00,  0.00,  0.00, 0.00),
        array(1.00, 0.63, 0.39,  0.25,  0.16, 0.10),
        array(1.00, 1.26, 1.58,  1.98,  2.49, 3.13),
        array(1.00, 1.88, 3.55,  6.70, 12.62, 23.80),
        array(1.00, 2.51, 6.32, 15.88, 39.90, 100.28),
        array(1.00, 3.14, 9.87, 31.01, 97.41, 306.02)
    );
    $b = array( -0.01, 0.61, 0.91, 0.99, 0.60, 0.02 );

    $x = gauss_eliminate($a, $b, 6);

    ksort($x);
    print_r($x);
}

test_gauss();