1Newton's Method9-16-20052Opening Discussion■What did we talk about last class?3Matrix Algebra■Now we get into the things that Matlab was really developed for and where it really stands out.■2-D arrays are basically matrices and can be used for doing all types of math.■Before we get into this we should talk about systems of linear equations and how they can be solved.■A set of linear equations is typically expressed as Ax=y, where A is an n by n matrix and x and y are n by 1 matrices. You are given A and y and want to solve for x.■If the equations are “well behaved” there is a single solution x=A-1y. Use \ instead of inv().4Overdetermined and Underdetermined■If A isn't square there isn't a single solution. When A has more rows than columns there is no exact solution (overdetermined). When A has more columns than rows there are an infinite number of solutions (underdetermined).■Matlab actually has routines that will try to solve systems of equations that aren't “well behaved”. That is, it will approximate both over-determined and under-determined systems.■In the overdetermined case, A\y will give you the least squares solution. This is can be viewed as an optimal fit.5Sparse Matrices■Matlab also has the ability to store sparse matrices.■We aren't really going to take advantage of this in this class, but if you have a large matrix has has mostly zeros in it, this can be significant.6Root Finding■A common mathematical problem is that of finding where a function has a certain value. Solving for the zeros of a function is called root finding, but since you can easily subtract any value it is very general.■There are many methods to find roots. One of the most widely used is Newton's method. I get the feeling that Matlab uses a binary search method though.7Newton's Method■Newton's method uses a linear approximation to a function and follows the line to zero, then uses that as a better guess.■f(x)~f'(v)*(x-v)+f(v)■If you don't know the analytic derivative you can use a secant method or take a numeric derivative.■This method is very fast if you have a guess close to the root. It can be poorly behaved in some places though.8Binary Searching■A binary search root finding method needs a range such that f(x1)*f(x2)<0 (the two endpoints must have a different sign).■You can either cut the range in half or use a line approximation between the two endpoints.■The trick in this method is finding the endpoints for the range. If a function has a very small region of negative values, for example, this can be hard to do.9Minimization■Another common problem is trying to find the smallest or largest values that a function takes in a certain range.■Mathematically, these “local extrema” are points where the first derivative is zero so you just do root finding on that.■“Hill climbing” methods come in many forms. Their simplest forms wind up working in a manner very similar to root finding.■Matlab provides you will methods for doing minimization of functions.10Reminders■Assignment #3 is due on Monday.■Quiz #2 is on
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