HW #7 Solution Due Thursday, November 14, 2002

Transcription

1 HW #7 Solution Due Thursday, November 14, 2002 Question (1): (10-points) Repeat question 1 of HW 2 but this time using Matlab. Attach M-file and output. Write, compile and run a fortran program which generates a table of sine, cosine, and tangent values to 6 decimal places for angles between 0 and 360 degrees in 5 degree increments. Table should be no wider than 80 characters and should have headers explaining what the columns are. How would you change this program if 8 significant digits were required? Include your design of the program, source code and output. The same arguments as used in answering the original question can be used here and the original fortran code can almost directly translated in Matlab. However we can simplify the solution greatly using the array and matrix features of Matlab. The HW6_1.m file contains the full documented Matlab code but the program really only consists of 6 lines of Matlab code. We define an array with the arguments we need and we then generate a matrix with all the table entries. We still have the same problem with tan(90) and tan(270). These could be handled with a for loop and if statement as in Fortan but Matlab can be more elegant. The bad_arg array can be used to address the tab matrix and set the bad argument values to NaN. The remaining lines simply produce the output. arg_deg = [0:5:355]; tab = [ arg_deg', sin(arg_deg'*pi/180), cos(arg_deg'*pi/180), tan(arg_deg'*pi/180) ]'; bad_arg = or(eq(arg_deg,90), eq(arg_deg,270)); tab(4,bad_arg) = NaN; fprintf(1,' Angle Sin Cos Tan \n (deg) \n') fprintf(1,' %4d %9.5f %9.5f %11.5f\n',tab); Output from HW5_1.m HW5_1: Matlab table of sin/cos/tan Angle Sin Cos Tan (deg) NaN

2 Question (2): (15-points) Repeat question 3 of HW 2 using Matlab and the same data files as before. Compare your Matlab results with your Fortran results. (Check some of the functions available in Matlab). For this problem, do not attach the sorted list to your solution. Design and write as program that reads a file containing a single column of numbers, sorts the numbers into ascending order and computes the mean, standard deviation about the mean, median and the 68% probability range of the values. Consider different sort algorithms that could be used. How would this program change if only the mean and standard deviation were needed? In the web page version of the homework, you will find two trial data sets. (I will test your programs with other data sets as well). A quick scan of Matlab s functions shows this is a much easier homework in Matlab than in Fortran mainly because all the functions we need are already available. The function sort will sort the array, median will return the median, mean the mean and std for the standard deviation. The only quanity we need to compute ourselves is the 68% range. Since the input data files are just a string of numbers we can fscanf to read the files. The solution to this problem is fairly easy. In the attached solution file, HW6_2.m, several matlab features have been added. Specifically, a dialog box is used to select the data file to be processed. Several matlab specific issues arise in doing this. Specifically, the list of file names can be reduced using the beginning of the file name (HW02 in this case) but selecting the names based on the end characters (e.g.,.dat) is much more difficult. The output was done graphically. Check the method used for _ in the file names, and that output of % is not easy. 8 STATISTICS HW02_01.dat Mean Median Std Dev % Range

3 10 STATISTICS HW02_02.dat 9 Mean Median Std Dev % Range You selected: HW02_01.dat STATISTICS HW02\_01.dat Mean Median Std Dev % Range Full Range You selected: HW02_02.dat STATISTICS HW02\_02.dat Mean Median Std Dev % Range Full Range Results from Fortran Program: HW2_3: File HW02_3.01.dat has 384 data For 384 data: Statistics are: Mean Median Standard Deviation % confidence *SD caputo[167] hw2_3 HW02_3.02.dat grep -v '^+' HW2_3: File HW02_3.02.dat has 384 data For 384 data: Statistics are: Mean Median Standard Deviation % confidence *SD

4 Question (3): (25-points) Repeat question 1 of HW3 using the original functions in afunc.f. Compare your Matlab results with those from your Fortran solution. Use Matlab to integrate sin(x^10) over the 1 to 1 and 2 to 2 intervals. Compare results with those given in the Homework 3 solution. The solution of this problem is again easier in Matlab because Matlab has a built in numerical integrator called quad, which allows the tolerance on the accuracy of the numerical integration to be set. Quad takes as arguments the name of an M-file that implements the function to be integrated. Design in the Matlab case is mainly determined by how the user will interact with the program. Here we have several choices. The homework asks for 5 different functions to be integrated and we could have 5 separate M-files that implement the functions. The user input would then the name of the M-file to be integrated. This approach has lots of appeal since the help information in the function in the M-files could be designed to report exactly what function the M-file was evaluating. Another option would be to pass an option to a single M-file that indicates which function to integrate. The option could be selected through another M- file, which has the disadvantage that the option selection may not correspond to the correct function if modifications are made to the M-files. In the homework solution, the same M-file is used to select the function as to implement it. In this way the selection code and implementation are in the same file thus reducing the possibility of the two being incompatible. The selection method used is the menu command in Matlab, which has the advantage that only legitimate selections can be made. Commented out in the code is a more terminal based selection process. The solutions to the homework are given in HW6_3.m and afunc.m. Some test results for the sin(x^10) function. This is difficult function to integrate because of the rapid oscillations that develop during the integration. With a specialized integrator, i.e., one that changes step size during the integration, it is not that difficult because the function is well behaved (values always between 1 and 1 with no singularities). The problem here is that a fixed integration step size does not work well. Here are results from several sources: HW3_1 Fortran Matlab Matematica Error Interval 1 to (2e-7) (2e-7) e (1e-4) (2e-7) 1e-3 Interval 2 to (4e-9) (1e-2) e (1e-6) (1e-2) 1e-3 Comments: 1. Matlab generates errors concerning recursion limits being exceeded and results should not be expected to be reliable. 2. Integrating from 0 to 1 and 1 to 2 separately and adding and doubling the results (for the 2 to 0 part of the integral) generates Matlab results an order of magnitude better. Presumably this approach of dividing the interval manually would generate better Matlab results overall.

5 (for the 2 to 0 part of the integral) generates Matlab results an order of magnitude better. Presumably this approach of dividing the interval manually would generate better Matlab results overall. 3. A generic problem with the Mathematica Notebooks is that they really can t be used in systems that have not had the Mathematic fonts installed. 4. Mathematica generates the following solutions: In[1]:= s10 := Sin[x^10] In[9]:= Integrate[s10,{x,-2,2}] In[10]:= N[%,20] Generated the results above but the behavior with the number of digits specified for the output was not predictable i.e., without the number of digits given, only 5 significant digits were generated, and the same number of output digits were given until the 20 was used. The analytic solution was given as: In[11]:= Integrate[s10,x] (x Gamma[--, -I x ]) x Gamma[--, I x ] -I Out[11]= -- ( ) / /10 10 (-I x ) 10 (I x ) The numerical evaluation generates an imaginary component to the integral (magnitude of order 1e-16 for the 1 to 1 integration and 1e-23 for the 2 to 2 integration). The above expression involves complex numbers despite the integral being completely real.

6 Question (4): (50-points) Solve the bouncing ball problem of Exercise 4 HW 4 with realistic floor bouncing. Create a figure that animates the motion of the ball. This basic problem is to solve the equations of motion of a ball moving under the influence of gravity and bouncing on a floor. There are several methods of approaching this problem. The C and C++ versions of this routine used a largely analytic solution but based on the discussion in the first homework where exotic forces were considered a numerical approach would be more flexible. The Matlab routine quad was used in Question 3 but since we need to do a double integral here, another integration technique could be easier to solve. Since we are solving potentially a partial differential equation, a search of the Matlab help files reveals that Matlab has a built in routine for solving first order differential equations. Ours is a second order differential equation but such equations can always be posed as a pair of first order equations. Below we give the characteristics of the different Matlab solvers. Solver Type Accuracy When to Use ode45 Nonstiff Medium Most of the time. This should be the first solver you try. ode23 Nonstiff Low If using crude error tolerances or solving moderately stiff problems. ode113 Nonstiff Low to high If using stringent error tolerances or solving a computationally intensive ODE file. ode15s Stiff Low to medium If ode45 is slow (stiff systems) or there is a mass matrix. ode23s Stiff Low If using crude error tolerances to solve stiff systems or there is a constant mass matrix. ode23t Moderate Low If the problem is only moderately stiff and you need a solution without numerical damping. ode23tb Stiff Low If using crude error tolerances to solve stiff systems or there is a mass matrix. We will use the ode45 solver (we could experiment later with the other ones). We introduce a 4-element vector y where the first two elements are the X and Y positions, and the other 2 elements are the X and Y velocities. The partial differential equations are given in the form y(1)/ t = y(3), y(2)/ t = y(4) and y(3)/ t = horizontal acceleration and y(4)/ t = vertical acceleration. The vertical acceleration always includes a 9.8 m/s 2 term plus other forces that are invoked. The input parameters are obtained by use of the inputdlg Matlab command. The main issue that we need to worry about is the bouncing off the walls. The homework only calls for bouncing off the floor, which is relatively easy. The implementation in the homework bounces off all four walls which somewhat more difficult to implement. The method adopted in the homework solution takes advantage of the output of the odexx routines in Matlab. In these routines a time interval over which to solve the ODE is given and the routines return the intermediate values during the time interval. The homework solve the ODE in 1 second steps. In the bounce module, the end of the step is checked to see if it is past a wall. If it is past a wall, the earliest time at which a wall in encountered is found and the ODEs solved to that time. The velocity component perpendicular to the wall is changed (multiplied by the Wall efficiency WE), and the remainder of the time step is integrated and again checked for wall interaction. For a simple floor bounce there it is unlikely that there will be another bounce but when there are multi-walls there can be

7 solve the ODE in 1 second steps. In the bounce module, the end of the step is checked to see if it is past a wall. If it is past a wall, the earliest time at which a wall in encountered is found and the ODEs solved to that time. The velocity component perpendicular to the wall is changed (multiplied by the Wall efficiency WE), and the remainder of the time step is integrated and again checked for wall interaction. For a simple floor bounce there it is unlikely that there will be another bounce but when there are multi-walls there can be (basically the ball hits near a corner). The solution to the homework is HW6_4.m and it uses M-files acc.m and bounce.m. One test of the program is turn off gravity, make the central object attractive and see if the ball will orbit the central body (the program passes this test). Example output for HW6_4.m for the default settings that include drag, inefficient walls, and a repulsive force at the center of the box. The inputs for the program are by use of dialog box HW6 Question 4 Initial Position Initial Velocity Wall Efficiency Drag Coefficient 1.00e-04 Repulsive Force Redline shows trajectory and blue arrows are the velocity vectors at each of the 1-second steps in the integration. The green square in the middle shows the repulusive element position.

8 Example below shows results when the center force is attractive rather than repulsive HW6 Question 4 Initial Position Initial Velocity Wall Efficiency Drag Coefficient 1.00e-04 Repulsive Force »