Example Question - gauss-seidel method
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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.
Solving a System of Linear Equations Using the Gauss-Seidel Method
<p>To use the Gauss-Seidel method, we first express each equation in the form \(x = ... , y = ... , z = ...\).</p> <p>From \(5x + 2y + z = 12\), we get \[x = \frac{12 - 2y - z}{5}\]</p> <p>From \(x + 4y + 2z = 15\), we get \[y = \frac{15 - x - 2z}{4}\]</p> <p>From \(x + 2y + 5z = 20\), we get \[z = \frac{20 - x - 2y}{5}\]</p> <p>We then assume initial values for \(x, y, z\), usually zeros: \(x_0 = 0, y_0 = 0, z_0 = 0\).</p> <p>Iterate the equations using the previous values:</p> <p>\[x_{n+1} = \frac{12 - 2y_n - z_n}{5}\]</p> <p>\[y_{n+1} = \frac{15 - x_{n+1} - 2z_n}{4}\]</p> <p>\[z_{n+1} = \frac{20 - x_{n+1} - 2y_{n+1}}{5}\]</p> <p>The iterations are repeated until the solutions converge to a set level of accuracy.</p>
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