![]() ![]() So the secant method would be X_n plus 1 equals X_n minus f of X_n is X_n squared minus 2. If we're going to use secant method, then we're going to approximate f prime. So the solution to this is the square root of two. Remember the square root of two is this 1.41421, and the function that we want to find the root of is f of x equals x squared minus 2. We can try to compute the square root of two again. So that's the normal definition of the derivative when X_n minus 1 approaches X_n. So f prime of X_n is approximately Delta f over Delta x, so f of X_n minus f of X_n minus 1 over X_n minus X_n minus 1. ![]() So what we do then is we approximate this derivative. The line that goes through these two points, f of X_n and f of X_n minus 1 is called the secant line to the curve, and that's why this is called the secant method. So to get X_n plus 1, we can use X_n and an additional point, x sub n minus 1, and then find an approximation to this derivative using the value of the function at X_n and X_n minus 1. If you can't find f prime of X_n analytically, then what you can do is add an additional point to your iteration. Newton's method is X_n plus 1 equals X_n minus f of X_n over f prime of X_n. So to derive secant's method, let's start with Newton's method. ![]() The next best method then is secant's method, where we're going to approximate the derivative of the function using two points on the function. But for Newton's method, you need to be able to take the derivative of the function that you're finding the root of. We'll see that Newton's method is the fastest method of root finding, and you should use it when you need a fast method. The course is organized into six weeks, and at the end of each week there is an assessed quiz and a longer programming project. After each lecture or demonstration, there are problems to solve or programs to write. The course contains 74 short lecture videos and MATLAB demonstrations. Students should have already studied a programming language, and be willing to learn MATLAB. We assume students are already familiar with the basics of matrix algebra, differential equations, and vector calculus. Access to MATLAB online and the MATLAB grader is given to all students who enroll. We learn how to use MATLAB to solve numerical problems. We derive basic algorithms in root finding, matrix algebra, integration and interpolation, ordinary and partial differential equations. This course covers the most important numerical methods that an engineer should know. ![]()
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