Solving a System of Linear Equations Using the Gauss-Seidel Method
<p>The Gauss-Seidel method is an iterative technique for solving a square system of n linear equations with unknown x: \( Ax = b \). It is defined by the iteration:</p>
<p>\( x^{(k+1)} = D^{-1}(b - (L+U)x^{(k)}) \),</p>
<p>where \( D \) is the diagonal component of \( A \), \( L \) is the strictly lower triangular component of \( A \), and \( U \) is the strictly upper triangular component of \( A \).</p>
<p>To apply the Gauss-Seidel method to the given system of equations:</p>
<p>1. Express each variable in terms of the other variables,</p>
<p>\( x = (12 - 2y - z) / 5, \)</p>
<p>\( y = (15 - z - x) / 4, \)</p>
<p>\( z = (20 - x - 2y) / 5. \)</p>
<p>2. Choose an initial guess for the variables, for example, \(x^{(0)} = y^{(0)} = z^{(0)} = 0\).</p>
<p>3. Perform the iteration using the previous step variables plugged into the right-hand side:</p>
<p>\( x^{(k+1)} = (12 - 2y^{(k)} - z^{(k)}) / 5, \)</p>
<p>\( y^{(k+1)} = (15 - z^{(k)} - x^{(k+1)}) / 4, \)</p>
<p>\( z^{(k+1)} = (20 - x^{(k+1)} - 2y^{(k+1)}) / 5. \)</p>
<p>4. Repeat step 3 until the values converge to a satisfactory level of accuracy.</p>
This solution provides the method without actual iteration. To obtain the numeric solution, one must carry out the iterative steps until convergence.
