EGR 102 Introduction to Engineering Modeling. Lab 09B Recap Regression Analysis & Structured Programming

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1 EGR 102 Introduction to Engineering Modeling Lab 09B Recap Regression Analysis & Structured Programming EGR Fall

2 Overview Data Manipulation find() built-in function Regression in MATLAB using ployfit Conditional Programming Iterative Programming Root finding EGR Fall

3 Data manipulations Indexing Matrices and Vectors M (1,2) V = v (2) 1 M = M (2,:) 9 M (:,1) M (3,3) R = R (2) EGR Fall

4 find() find() returns the index values for where its logic test is true In a vector, it is simply the index/element numbers In an array, there are two outputs being a set of row indices and a set of column indices In red we see the second and fourth values are >3. In blue we see the value at row 2 col 1 is < 3. In green we see the value at row 3 col 3 is < 3. EGR Fall

5 Regression in MATLAB ployfit We have a set of data (X,Y) polyfit allows us to get a polynomial fit to the data. Y predicted = a n X n + a n 1 X n 1 + a n 2 X n a 1 X 1 + a 0 X 0 Coef=ployfit(X,Y,n) Coef is a vector containing all the coefficients (a n, a n 1, a n 2,, a 1, a 0 ) The order on the fit is n EGR Fall

6 Regression in MATLAB ployfit Example : First order fit Coef=ployfit(x,y,1) Coef = a 1 a 0 y predicted = a 1 x + a 0 Ypredicted=ployval([a1,a0],x) Ypredicted=a1*x+a0 EGR Fall

7 Conditional Programming Conditional programming is very similar to a flow chart. Depending on results, only certain paths or branches of the code are executed. This is done by applying logic tests with one or more special statements. EGR Fall

8 Conditional Programming if Statement The syntax and what it effectively does are: if condition end statements The condition is a logic test. The statements are the code that will be run, but only if the condition is true. end closes the if statement; any code after runs as normal. EGR Fall

9 Iterative Programming There two types of iterative programming that will be introduced: for loops, the exact number of repetitions are set by the programmer while loops, the programmer sets a condition and the code repeats as long as it is true This lesson will discuss while loops. Lab 08A discussed for loops. EGR Fall

10 Iterative Programming for Loop The general syntax and structure are: for index = vector_of_values statements end index: variable that defines iterations. Usually i, j, or k vector_of_values : vector containing either the values to be evaluated in the loop or number of desired iterations statements are the commands to be executed with each iteration EGR Fall

11 Iterative Programming while Loop The general syntax and structure are: Initial while logic-condition statements end Initial is a value with the sole purpose of starting the loop by making the condition true. logic-condition is a logical expression that is either true or false. statements are the commands to be repeated. One should be an update for the value tested in the condition. EGR Fall

12 Root Finding 1. Bisection method (Lecture) 2. Newton-Raphson method (Lecture and Lab 09A) EGR Fall

13 Bisection method Pros: Always converge to a root Cons: Requires two initial guesses. Works best after plotting the function first Fails if you don t have a sign change Newton-Raphson Pros: Only require one initial guess Cons: Can be divergent (you may not find a root) EGR Fall

14 Bisection method algorithm 1. Start with two initial guesses: an upper value, x u and a lower value, x l EGR Fall

15 Bisection method algorithm 1. Start with two initial guesses: an upper value, x u and a lower value, x l 2. Take the difference between them and split it in half. This is the estimate of your root, x r EGR Fall

16 Bisection method algorithm 1. Start with two initial guesses: an upper value, x u and a lower value, x l 2. Take the difference between them and split it in half. This is the estimate of your root, x r 3. Determine if the true root is above or below your root estimate If f(x u )*f(x r )<0 then the true root is above the root estimate. If f(x l )*f(x r )<0 then the true root is below the root estimate EGR Fall

17 Bisection method algorithm 1. Start with two initial guesses: an upper value, x u and a lower value, x l 2. Take the difference between them and split it in half. This is the estimate of your root, x r 3. Determine if the true root is above or below your root estimate 4. Adjust the brackets If the true root is above the estimate, x r becomes the new x l value If the true root is below the estimate, x r becomes the new x u value EGR Fall

18 Bisection method algorithm 1. Start with two initial guesses: an upper value, x u and a lower value, x l 2. Take the difference between them and split it in half. This is the estimate of your root, x r 3. Determine if the true root is above or below your root estimate 4. Adjust the brackets If the true root is above the estimate, x r becomes the new x l value If the true root is below the estimate, x r becomes the new x u value EGR Fall

19 Bisection method algorithm 1. Start with two initial guesses: an upper value, x u and a lower value, x l 2. Take the difference between them and split it in half. This is the estimate of your root, x r 3. Determine if the true root is above or below your root estimate 4. Adjust the brackets If the true root is above the estimate, x r becomes the new x l value If the true root is below the estimate, x r becomes the new x u value 5. Calculate the new estimate, x r EGR Fall

20 Bisection method algorithm 1. Start with two initial guesses: an upper value, x u and a lower value, x l 2. Take the difference between them and split it in half. This is the estimate of your root, x r 3. Determine if the true root is above or below your root estimate 4. Adjust the brackets If the true root is above the estimate, x r becomes the new x l value If the true root is below the estimate, x r becomes the new x u value 5. Calculate the new estimate, x r 6. Repeat steps 2-5 until the root converges EGR Fall

21 What is root converging We determine when to stop iterating by looking at the difference between one root estimate and the next If the current root estimate is very close to the next estimate, we know we are close to the true root! We calculate the percent error, ε t, between the two root estimates: ε t = x r,new x r,old x r,new 100% A stop condition, ε s, must be defined and when ε s >ε t we have successfully found the root of the function EGR Fall

22 Bisection method algorithm Example f x = e x x EGR Fall

23 Bisection method algorithm Example f x = e x x Compare to the Newton- Raphson Method EGR Fall

24 Newton-Raphson method algorithm 1. Start with an initial guess of the root, x i EGR Fall

25 Newton-Raphson method algorithm 1. Start with an initial guess of the root, x i 2. Calculate the derivative of the function, f (x i ), and the function value, f(x i ), at the current value EGR Fall

26 Newton-Raphson method algorithm 1. Start with an initial guess of the root, x i 2. Calculate the derivative of the function, f (x i ), and the function value, f(x i ), at the current value 3. Calculate the next estimate of the root using the function value and the derivative: x i+1 = x i f x i f x i EGR Fall

27 Newton-Raphson Method Algorithm 1. Start with an initial guess of the root, x i 2. Calculate the derivative of the function, f (x i ), and the function value, f(x i ), at the current value 3. Calculate the next estimate of the root using the function value and the derivative: x i+1 = x i f x i f x i 4. i=i+1 5. Repeat step 2 and 3 until the root converges EGR Fall

28 Summary Data Manipulation find() built-in function Regression in MATLAB using ployfit Conditional Programming Iterative Programming Root finding EGR Fall