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Jan26-12

# Jan26-12 - Math 260 Spring 2012 Jerry L Kazdan Class...

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Math 260, Spring 2012 Jerry L. Kazdan Class Outline: Jan. 24, 2012 1. Least Squares What if you have lots of data points ( x 1 , y 1 ) , ( x 2 , y 2 ) , ..., ( x k , y k ) and want to find the straight line y = a + bx that best fits the data? In this case trying interpolation you have k linear equations in the 2 unknowns a and b : a + bx 1 = y 1 a + bx 2 = y 2 · · · · · · a + bx k = y k Although it is highly unlikely you will find an exact solution, what is the best possible approximation? Idea: Pick a and b to minimize the error E ( a,b ) := radicalbig ( a + bx 1 - y 1 ) 2 + ( a + bx 1 - y 2 ) 2 + · · · + ( a + bx 1 - y k ) 2 Remark: There is a special case of the above that arises often: here we want a hor- izontal line y = c that best fits the data. How should we pick the constant c ? An example of such data will be the scores of our class on Exam 1. 2. Lu := u ′′ + u = 0. a) For variety, we will think uf u as a function of t . Observe that cos t and sin t for t R are linearly independent solutions of u ′′ = u = 0.

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Jan26-12 - Math 260 Spring 2012 Jerry L Kazdan Class...

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