Example Question - iterative methods
Here are examples of questions we've helped users solve.
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>
Solving Transcendental Equations Iteratively
The equation you have shown, \( y = \cos(x - y) \), is a transcendental equation, where the variable \( y \) appears both inside and outside the trigonometric function. These types of equations cannot generally be solved using elementary algebraic methods and often require numerical methods or iterative approaches for finding an approximate solution. However, we can attempt to solve this equation iteratively. Let's try to isolate \( y \) on one side to see if we can formulate an equation that could be approached iteratively. We could, for example, write it as: \[ y = \cos(x - y) \] This is difficult to solve algebraically due to \( y \) being both outside and inside the cosine function. So if we were to try to solve this by iteration, we guess a value for \( y \), say \( y_0 \), and then use this to calculate a new value for \( y \): \[ y_1 = \cos(x - y_0) \] Then we would use \( y_1 \) to calculate \( y_2 \): \[ y_2 = \cos(x - y_1) \] And we would continue this process until \( y_{n+1} \approx y_{n} \) to some desired level of precision. You could pick an initial guess for \( y_0 \) based on the context of the problem or simply start with \( y_0 = 0 \) and then follow the iteration process. If you require a more precise solution, this would typically be done using numerical methods implemented in a computer program, such as the Newton-Raphson method or fixed-point iteration. Do you need further assistance with this problem?
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com