1. CMPSC 200 Fall 2013 Lecture 37 November 18, 2013

2. Numerical Differentiation   Suppose you have a set of measured data, T and P and you want to find the first derivative of the using this data.   First derivative is just the slope so you could calculate the slope (  P/  T) for each pair of points (P 2 – P 1 )/(T 2 - T 1 ), (P 3 – P 2 )/(T 3 – T 2 ), (P 4 – P 3 )/(T 4 – T 3 ), etc.   Use MATLAB diff function: slope = diff(P)./ diff(T)

3. Plotting the slope   If you want to plot the slope as a function of T   Cannot plot slope vs. T because number of points are different   If plot slope vs. T(2: length(T)) or slope vs T(1:length(T)-1) would not position the slope correctly   Plot slope vs. middle of T pairs T(1:length(T) -1) + diff(T)/2 or T(2:length(T)) – diff(T)/2 or T(2:end) – diff(T)/2

4. Questions ???

5. Numerical Integration   Integration is often referred to as the area under a curve and can represent entities that you may want to measure.   If you do not have a function that meets a set of points or you have a function that is not easily integrated, you can use numerical methods and a set of points that you have collected.

7. A = y 1 (X 2 -X 1 )A = 0.5(y 2 +y 1 )(X 2 -X 1 ) A = y 1 (X 2 -X 1 ) + 0.5(y 2 -y 1 )(x 2 -x 1 ) A = y 2 (X 2 -X 1 ) Over the entire range: sum(diff(x).* (y(1:length(y)-1) + diff(y)/2))

8. quad and quadl functions   MATLAB has two built-in functions accept a function or function handle and two points to calculate the area.   The quad function uses Simspon’s quadrature to estimate the area.   The quadl function uses Lobatto’s quadrature to estimate the area.

9. Reference for this portion  Much of this portion of the lecture is based on material from Applied Numerical Methods with MATLAB for Engineers and Scientists by Steven Chapra.

10. Simpson’s 1/3 rule  Use a point midway between the values that you want to integrate. Then fit the three points with a parabola and estimate the area under the parabola.  Example: Let x o be the first point x 2 the second point and x 1 be half way between the two. Then the area would be x 2 - x 0 2 f(x 0 ) + 4f(x 1 ) + f(x 2 ) 3 = (x 2 - x 0 )(f(x 0 ) + 4f(x 1 ) + f(x 2 )) 6 h 3 f(x 0 ) + 4f(x 1 ) + f(x 2 )where h = x 2 - x 0 2 b - a 2 or *

11. Graphical Example a f(a) b f(b)

12. MATLAB Example  Use function from 2x 2 + 3x + 5 from x =1 to x = 3 as a demo  Use ginput if you do not know function.

13. Simpson’s 3/8 rule  Use 2 points equal distance between the values that you want to integrate. Then fit the four points with a parabola and estimate the area under the parabola.  Example: Let x o be the first point and x 3 the second point and x 1 be 1/3of the way between the two and x2 would be 2/3 of the way. Then the area would be (x 3 – x o )(f(x 0 ) + 3f(x 1 ) + 3f(x 2 ) + f(x3)) 8 where h = x 3 - x 0 3 b - a 3 or 3h 8 f(x 0 ) + 3f(x 1 ) + 3f(x 2 ) + f(x 3 )

14. Composites  If you have more than 3 or 4 points you can combine techniques.  Examples  Suppose we knew f(x) for 5 points. Use Simpson’s 1/3 rule for x 0 – x 2 then add to Simpson’s 1/3 rule for x 2 – x 4. (Note x 2 is used twice)  Suppose we knew f(x) for 6 points. Use Simspon’s 1/3 rule for x 0 – x 2 and Simpson’s 3/8 rule for x 2 – x 5.  For 7 points we could use Simpson’s 3/8 rule for x 0 – x 3 and again for x 3 – x 6.

15. Example – know 5 Points (x 0 – x 4 ) Use first 3 points to calculate one area, use last 3 points to calculate second area and sum the two. Note the 3 point is used for both areas. (x 2 -x 0 )(f(x 0 ) + 4f(x 1 ) + f(x 2 ))/6 + (x 4 -x 2 )(f(x 2 ) + 4f(x 3 ) + f(x 4 ))/6

16. Questions

17. Differential Equations  There are a number of functions (solvers) built into MATLAB to solve ordinary differential equations. (See page 512 of your textbook)  Each of these solvers require a function handle to describe the function in terms of 2 variables, a range of interest for first variable, and an initial condition of the second variable.  Example dy/dt = f(t,y)

18. Function Handle  A nickname for a function or reference to a function *.m file  Examples:  my_fun = @(t,y) 4*t could be used specifying dy/dt = 4*t  Or @function name if you refer to *.m function file

19. Using ODE functions  [t,y] = odefunction(func. handle, t-range, init y val)  [t,y] =ode45(my_fun, [-1,1], 2); % calculates 45 points from -1 to 1 for t % calculates 45 points for y solving y = 2t 2 + c % where c is a constant which is 0 for this % example.  [t,y] = ode45(my_fun,[-1,2],2); % calculates 45 points t = -1 to 2 and y = 2t 2 + 0  [t,y] = ode45(my_fun, [-1,1], 3) % calculates 45 points t = -1 to 1 and y = 2t 2 + 1

20. Higher Order Differential Equations  Reduce to a system of first order equations.  See example on pages 529 - 531 in your text book

21. Questions

22. Roots of Equations  Find value(s) of the independent variable when the dependent variable is 0.  Use graphical methods and ginput.  Use “Bracketing Methods”  Need two “guesses” where the values of the of the dependent variable change signs. Therefore you know that a root is inside range  Slow, but usually work  Use “Open Methods”  Need one or more guesses  Fast, do not always work

23. Bracketing Methods  Incremental  Move from first guess to second in a series of steps.  Find positions where result of function changes sign  May find a number of ranges for roots  Bisection  Evaluate function at midpoint between two guesses.  If evaluation at midpoint = 0 (or close), then done  If first guess and midpoint have same sign, then repeat with midpoint and second guess. Otherwise repeat with first guess and midpont.  Continue until reach desired precision

24. Bracketing Methods (cont)  False Position (linear interpolation).  Similar to bisection  Draw a straight line between two guesses.  Use where the line crosses the x-axis as a possibility.  Evaluate function at that possibility.  If evaluation at possibility is 0 or close to 0 stop  Repeat with guess that has opposite sign as possibility.

25. Algorithm for Incremental Search  Create a vector for x from low guess to high guess with number of increments  Calculate values for f(x) using vector  Set number of roots to 0  With a loop that goes from k =1 to k = next to last element.  Compare signs f(xk) to f(xk+1)  If signs are different then add 1 to the number of roots and store values of x that bracket that root.  Sign function – returns -1, 0 or 1

26. Function for Incremental Search  Pass function, low value, high value and number of increments to function.  Create a function handle, then use it in the function call.  myfun = @(x) x.^3 – x.^2 – 4*x +4  incsrch(myfun, -3, 3, 300)  Use the function in the function call  incsrch(@(x) x.^3 – x.^2 – 4*x +4, -3, 3, 300)

27. Questions ???