In today’s lecture the professor reintroduced “the Euler’s method”, which is a way using a start point and the first order differential equation of the function to simulate the real curve of the function. Some may wonder: what is the use of simulation of a curve when the curve is known, but in real life, mathematicians may get a differential equation that is impossible to reverse back into its normal function form, so the only thing they can do it to try their best of simulating the real curve. The most important problem with the Euler’s method is the natural existence of error. The simulation curve created through the Euler’s method is guaranteed to have error which will increase with the increase of the steps h, which is the difference between the X values of An and An+1(as shown on the graph, beginning from the starting point, the distance between the Y value of An and the blue curve gets bigger and bigger).http://tutorial.math.lamar.edu/Classes/DE/EulersMethod.aspx （more accurate explanation here)
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As the professor stated, the Euler’s method is a numerical method that guarantees the error, nothing can be done to eliminate the error. The only thing we can do is to minimize the error of the method, by taking smaller steps. As shown in the graph, the green line is with bigger step, which creates a huge error, but the purple takes steps 100 times smaller than those of the green lines, which makes its error to the original line minimal.
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Better methods like RK2 was also introduced in the lecture, which is a way to approximate the original line at one point by taking the average value of two steps in the Euler’s method, but none of them really guarantees to eliminate the error. The methods mentioned above lead to a very interesting aspect of high level math, which is not everything is so accurate. In courses such as algebra, precal, even calculus, the accuracy is emphasized; math looks so exact at the time, you either get the answer, or you don’t. In higher level math, however, sometimes the exact answer is not even possible, all we can do is to get the next best thing, which is an approximation. Kind of like the general in the Ender’s game said: “you are never ready, you go when you ready enough”, math sometimes is just like that. I suppose that’s because with the ascending of the level, the problems are gradually approaching the real ones in real life, which sometimes are so complex or abstract that cannot be solved, and approximation is the only way to go. In fact, often in our life, the exact answer is not needed, it just needs to be close enough. Pi, for example, no one knows the real value of it, but it has never stopped us to calculate the length, or the area of circle when we build wheels, cars, or even space shuttles, because we can approximate the value of Pi to the 300 millionth digit, and at this point, the error is almost irrelevant.