The Relaxation method is one of the classical linear iterative methods. It is derived from the Jacobi and Gauss-Seidel methods. One should note that during the Jacobi method, our x values remain fixed. Each equation in our iterations is treated independently. If we alter this method by now by looking at each equation at a specific time in the sequence, we can derive the Gauss-Seidel method. Lastly, by implementing extrapolation, through the introduction of omega, we successfully arrive at the Relaxation method.
Now, if we want to examine how omega affects our methods and how it is present in even the Gauss-Seidel method and the Jacobi method, we can see that they only differ by the choice of omega. By changing the value of omega to equal one, we can simplify the Relaxation method down to the Gauss-Seidel.
Some of the reasons one may opt to use the Relaxation method is due to its efficiency with regards to number of iterations and run time.
Let's break down the specifics of the SOR (Successive Over Relaxation method). Basically, suppose we want to solve a system of linear equations of the form Ax=b. If we ensure A is a symmetric positive definite matrix, we will have convergence for any initial choice of x. By convergence, I mean through the implementation of the method, we will arrive at our solution to our system of equations. Furthermore omega needs to be in (0,2). The process of picking omega is often quite difficult and convoluted. If we are able to pick the optimal omega, we will have an optimal rate of convergence. We can go more into the specifics but this is a backpacking blog. The real Relaxation method (in my opinion) can be found by simply getting outdoors, detaching yourself from the toxicity of the media, and being free.
This past week I went on a trip up to the Eastern Sierra. I was starting to enter a rut being tied down in LA. Work and school and more work and more school. Yes, I love my job and my studies but a big part of me was missing. I was antsy for an adventure and for a chance to breathe. So, what else would I do than go on a solo backpacking trip.
On Thursday after my physics lecture, I rushed home, grabbed my pack and jumped in my car and was off. I arrived at the trailhead just outside Big Pine at 5pm and was ready to hit the trail. Today will be an easy day. Just under six miles to camp. I was set on getting there before sun down, needless to say I made it with more than enough time to watch the sunset over the spires. I had had a long day, full of excitement but also lots of anxiety (not backpacking related). I decided I would take it easy the next day and treat myself by sleeping in. In the morning I awoke as the sun was rising, casting its light over the spires and crags and reflecting their silhouette in the lake below.
It was not until around ten in the morning that I set out. Today was going to be a day with no set plan. I was going to walk and explore and most of all enjoy life and the beauty of the Sierra. I ended up walking across frozen lakes, trudging through knee deep snow and making a new friend (shoutout to Dan from Encinitas).
I spent far too many hours looking across the vast expanse of openness and the frozen lakes. Its beauty was mesmerizing.
Alas the wonders could not last all day. Around three in the afternoon I decided upon my plan for the rest of the day. I would circle around and camp at one of the lakes I passed on my first day. When I arrived at the lake where I would be staying, I found the most perfect camp place. It was up on a boulder overlooking the lake with the peaks in the background. I was truly living in a dream. I fell asleep with my tent door open, watching the sun drift behind the peaks. In the morning I awoke early and watched the sun rise as I cooked my food just outside my tent.
Unfortunately, at some point we must return to reality. Returning to LA has been harder this time than usual. I left my heart and mind with the Sierra.
It is time to return to studying the mathematical Relaxation method and maybe I shall learn to relax in the process.