Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 91Root Finding and Data Fitting2-15-20102Opening Discussion■What did we talk about last class?■Do you have any questions about the reading?3Minimization■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 with methods for doing minimization of functions: fminbnd and fminsearch.4Data Interpolation■Interpolation is the process of estimating values between data points.■Matlab has the functions interp1 and interp2 for doing interpolation in 1-D and 2-D.■The last argument is a string that tells the type of interpolation: linear, cubic, spline, and nearest.■Let's play with some 1-D interpolation and see how the different methods work.52-D Interpolation■Matlab can also do 2-D interpolation with interp2. This is easy to do if you have a 2-D array of data and want to find one point.■To construct a finer mesh we need the meshgrid function that will give us an easy way to represent all points on a grid. It takes two 1-D arrays and returns two 2-D arrays.■Lets play with this as well. We can use the mesh function to plot out surfaces.6Polynomials■Matlab provides a simple mechanism for us to deal with polynomials. Row arrays are can be viewed as the coefficients on polynomials.■Given this form, roots finds the roots of that polynomial.■Given the roots, poly will return the polynomial with those roots. (This one can be fairly easily done by hand.)■The conv function will multiply two polynomials.■Addition can be done easily if the polynomials have the same number of terms. Otherwise one will need to be padded with zeros.■For division use deconv.7Calculus and Evaluation■You can take derivatives of polynomials with polyder.■You can integrate a polynomial with polyint. Remember that this must be passed in a constant along with the polynomial.■The polyval function will evaluate a polynomial at one value or for an array of values.8Curve Fitting■Given data, you can use the polyfit to fit a polynomial to it. The arguments are the x and y values followed by the order of the polynomial you want back.■In general you should use lower order polynomials, they are typically better behaved.■It turns out this is just a wrapper function for solving a system of linear equations, typically a system that is overspecified.9Closing Comments■How do you feel about the way that the class is
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