dy = y, dt y(0) = 1, y(t) = e t.

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1 NAME: AME Introduction to Engineering Computing Examination 1 Prof. J. M. Powers 16 October (5) Write a short html script which builds a page that gives a link to the Notre Dame home page: Here is one way to handle this. <html> <a href= to the Notre Dame home page. </html> 2. (5) Write LATEX script which generates the following equations with the given format: = y, y(0) = 1, y(t) = e t. \begin{eqnarray*} { \over } &=& -y, \\ y(0) &=& 1, \\ y(t) &=& e^{-t}. \end{eqnarray*} 3. (10) Give the output of the following Fortran statements: (a) print*, 1. 4 Answer: (b) print*, 2/4*2 Answer: 0 (c) print*, 2/4.*2 Answer:

2 (d) print*,2./4.+2 Answer: (e) print*,2/4+2 Answer: 2 Note there are some slight compiler-dependent variations which also could be acceptable. 4. (5) Write the base 2 (sometimes known as binary) representation of the integer = = }{{}}{{}}{{} (10) Write the UNIX commands for (a) changing the name of a file named program.f90 to new.program.f90, (b) deleting a file named program.f90, (c) listing to the screen the contents of a file named program.f90, (d) copying a file named program.f90 into a file named new.program.f90, (e) creating a directory named newdirectory. (a) mv program.f90 new.program.f90 (b) rm program.f90 (c) cat program.f90 or more program.f90 or less program.f90 (d) cp program.f90 new.program.f90 (e) mkdir newdirectory 6. (5) Which languages require a compiler? (a) Fortran 2003 Answer: needs compiler (b) MATLAB Answer: does not need compiler (c) C Answer: needs compiler (d) C++ Answer: needs compiler (e) Microsoft Excel Answer: does not need compiler 2

3 7. (15) You are given a text file named output.txt containing two columns of numbers. The first column represents discrete values of the variable t, which gives time in units of s. The second column represents corresponding values of the variable y, which gives distance in units of m. Write a short MATLAB program which reads the data and generates a continuous plot of y versus t. Use a plotting format recommended for graphs in this course. clear; load( output.txt ) t = output(:,1); y = output(:,2); plot(t,y),... xlabel( t (s), FontSize,24),... ylabel( y (m), FontSize,24);... set(gca, FontSize,20) 8. (40) Consider the mass-spring-damper problem m d2 y 2 +b +ky = 0, y(0) = y 0, = 0, t [0,t stop ], t=0 where y is the position (m), t is time (s), m is the mass (kg), b is the damping coefficient (Ns/m), k is the spring constant (N/m), y 0 is the initial position (m), and t stop is the final time (s). Compose (a) an input file named input.txt which contains numerical values for m = kg, b = 10 Ns/m, k = 1 N/m, y 0 = 1 m, t stop = s; format the file in an easily understood fashion, (b) a Fortran program named msd.f90 which i. includes at least three useful comment statements, ii. reads the input data from input.txt, iii. reads from the screen the number of time steps n to be employed, iv. uses the forward Euler method to get a numerical estimate of y(t) for t [0,t stop ]. v. writes the output of the estimate for t and y(t) to a file named output.txt. Here is an example input file. 3

4 fall.2014]$ cat input.txt m (kg) b (Ns/m) 10. k (N/m) 1. y0 (m) 1. tstop (s) * * A small amount of analysis is required to put the system in the form for the forward Euler method. Let v =. Then the second order differential equation can be rewritten as m dv +bv +ky = 0. Rearranging this, including the definition of v, recasting the initial conditions leads to dv = v, y(0) = y 0, = b m v k y, v(0) = 0. m Replacing the continuous derivative by a discrete analog, we get y n+1 y n t v n+1 v n t Solving for y n+1 and v n+1, we get = v n, y 1 = y 0, = b m vn k yn, v1 = 0. m Here is an example program. y n+1 = y n + tv n, ( b v n+1 = v n t m vn + k ) m yn. [powers@darrow1-p fall.2014]$ cat msd.f90 program msd implicit none real :: y,ynew,v,vnew,t,,m,b,k,y0,tstop,tnew integer :: n,i open(10,file= input.txt ) open(20,file= output.txt )! read the input data read(10,100) m read(10,100) b read(10,100) k read(10,100) y0 read(10,100) tstop 100 format(10x,f16.10) print*, enter number of time steps, n read*,n! set the initial conditions y = y0 v = 0. t = 0. = tstop/(real(n)) print*, =, print*,t,y,v 4

5 ! main loop in time do i=1,n ynew = y+*v vnew = v-*((b/m)*v+(k/m)*y) tnew = t + y = ynew v = vnew t = tnew write(20,*)t,y,v enddo end program msd 9. (5) Write a short Fortran program which prints to the screen Go Irish! Beat Seminoles! Have the program also print to the screen an estimate of the final score. program ClashmoreMike print*, Go Irish! Beat Seminoles! print*, ND 37, FSU 31 end program ClashmoreMike This actually was my prediction. Had one less flag been dropped, it would have been even more accurate, though not perfect. In our world, which shares many features with non-linear namics, small changes can have large consequences, as demonstrated in the Lorenz system s butterfly effect. 5

Fall 2015 Math 337. Basic MatLab

Fall 2015 Math 337. Basic MatLab Fall 215 Math 337 Basic MatLab MatLab is a powerful software created by MathWorks, which is used extensively in mathematics, engineering, and the sciences. It has powerful numerical and graphic capabilities,