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14-440-127 Lecture 11 Notes

14-440-127 Lecture 11 Notes - 14:440:127 Introduction to...

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14:440:127– Introduction to Computers for Engineers Notes for Lecture 11 Rutgers University, Fall 2008 Instructor- Blase E. Ur We’re going to begin with a corrected version of a section from Lecture 10 with 2 stupid and serious mistakes I had left in there, but noticed and corrected as I was giving the lecture: 1 Correction from last lecture: Fitting To More Compli- cated Functions Oftentimes in engineering, you’ll want to find equations for functions that are not polynomial. However, you’ll sometimes be able to transform these functions into polynomials: 1.0.1 Power Functions A function such as y = bx m can be called a power function since x is raised to a power. Note that this is a polynomial, but you have no idea what degree it is, so it might be hard to choose which degree polynomial to specify for polyfit . Note that if you plot this function on a log-log plot using the loglog command, the data will appear in a straight line. Now, the key is that if you take the logarithm of both sides of this equation, you’ll get log ( y ) = log ( bx m ). Using log properties, you can rewrite this as log ( y ) = log ( b )+ log ( x m ), and then rewrite it as log ( y ) = m * log ( x ) + log ( b ). This has the form j = m * h + k if you define j = log ( y ), h = log ( x ), and k = log ( b ). This is perfectly analogous to y = mx + b , which is a linear equation. Now, find the coefficients using polyfit(log(x),log(y),1) , and call these coeffients [m k]. Now, go- ing back to the original equation, you can realize that you’ve already found m . To find b , you can recall that k = log ( b ), so b = e k . You can now write an equation for your data as y = exp ( coeffs (2)) * x coeffs (1) . Thus, you’ve used a transformation, fit the log of the power function to a line, and then reversed your transformation to find the original coefficients. 1.0.2 Exponential Functions Let’s say you’re given an exponential function, such as y = b ( e ) mx . This function would be displayed as a straight line on a semilogy plot. This can be rewritten as log ( y ) = mx + log ( b ). Now, this has the form j = m * x + k if you set j = log ( y ) and k = log ( b ). Note that you do not have to transform x. Now, find the coefficients using polyfit(x,log(y),1) , and call these coeffients [m k]. Again, you’ve already found m from the original equation, and you can find b by recalling that i = log ( b ), so b = e k . You can now write an equation for your data as y = exp ( coeffs (2)) * ( e ) coeffs (1) * x .

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2 Recursion There’s a very interesting technique in computer programming called recursion, in which you write a function that calls itself . Eventually, this string of functions calling itself stops at some base case , and the answers work their way back up. Of course, in order to understand recursion, you must understand recursion. (That’s a joke). As a first example, let’s consider factorial. If you want to calculate n ! (n factorial), you could write this as n ! = n * ( n - 1)!. Eventually, you need to get down to a base case. What should your base case be? Well, 1! is 1, so make that your base case. So how do you write this as Matlab code?
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14-440-127 Lecture 11 Notes - 14:440:127 Introduction to...

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