Physics 3730/6720 Midterm Test

Rules

The test is open book and notes. You may use the internet to consult references. However, you may not seek or receive assistance from anyone other than the instructor or TA.
There is one answer file. Please submit it just as you have been submitting homework. Use submit p6720 midterm midterm.txt
Please be sure you submit your final version. Replacing files is permitted until the end of the test.

Please put the answers to the following questions in a text file midterm.txt. Identify each answer by beginning the line with the number and letter 1 (a), 1 (b), etc. of the question.

Problem 1. (30 pts)

1 (a) Write a Unix command that lists the contents of your home directory, no matter where your current directory happens to be.

1 (b) You have written a program called addup that adds up a list of numbers. That is, when you type addup at the Unix command prompt and then type a bunch of numbers on the keyboard, followed by Ctrl-d, the program prints the sum of the numbers you typed. You have a file called expense.txt that has two columns, the first gives the name of an item as a single word and the second gives the price. Write the one-line Unix command that uses Unix utilities and addup to get the total of all the prices in the second column of the file.

1 (c) You have a file data that contains five columns of numbers. Write the one-line gnuplot command that makes a two-dimensional plot taking the third column as the horizontal axis and the fifth column as the vertical axis. The plot should have lines connecting the points.

1 (d) Find all the zeros of x - 2*exp(x/4)*cos(x) on the interval [-10,10] to 4 significant figures by any quick method. Explain how you did it (showing all of your work) as well as giving the answer.

1 (e) Suppose you have a file called galaxies with lines with #. You need to remove these lines and create a new file called galaxies.fix. Write a one-line Unix command for doing the job.

Problem 2. (20 pts)

2 (a) In the following code fragment say what values are printed out and why.

   double a = 5;
   double *r = &a;
   double &g = a;
   *r = 2;
   cout << a << "\n";
   cout << g << "\n";

(Be very careful about the data types and the meaning of each of the assignments.)

2 (b) Find the Taylor series expansion of sec(x) about x = 0 up to and including terms in x^6. Show how you did it. For example, if you used Maple, give the Maple command(s).

Problem 3. (30 pts)

You are writing a program to find the roots of a polynomial using the Newton-Raphson method. The main program is shown below, but it is incomplete. For this question you are asked to complete it by writing in your answer file the one line of code needed.

You are using Horner's algorithm to evaluate the polynomial and its derivative, and you put the Horner code in a separate file, which is not shown here. That code defines a subprogram called horner. The subprogram prototype declaration is included below with comments so you can see what arguments it takes. Please pay special attention to the datatypes of the arguments.

// Code for finding roots of a polynomial
// using the Newton-Raphson method

// Physics 6720 8/18/03

// Usage

//   nr

// The user is prompted for the tolerance and the starting point.

#include <iostream>
#include <cmath>
using namespace std;
#define N 100    // Maximum number of iterations

// This is the prototype declaration for the subprogram horner.  It
// evaluates the polynomial at x and returns its value there as pr and its
// derivative there as qp.  The degree of the polynomial is n and the
// coefficients of the polynomial are in the array a.

void horner(int n, double a[], double x, double &pr, double *qp);

int main(){
  double p,pnew;
  double f,dfdx;
  double tol;
  int i,n;

  cout << "Degree of polynomial = ";
  cin >> n;

  double a[n+1];

  cout << "Coefficients, a[0], a[1], ..., a[" << n << "]\n";
  for(i = 0; i <= n; i++)
    cin >> a[i];

  cout << "Enter tolerance: ";
  cin >> tol;
  cout << "Enter starting value x: ";
  cin >> pnew;

  // Main loop
  for(i = 0; i < N; i = i + 1){
    p = pnew;

    // Evaluate the function and its derivative

    // IN YOUR ANSWER FILE WRITE THE ONE LINE STATEMENT
    // THAT BELONGS HERE AND THAT COMPLETES THE CODE.
    // THIS STATEMENT CALLS horner WITH THE APPROPRIATE ARGUMENTS.
    ////////////////////////////////////////////////////


    /////////////////////////////////////////////////////
    // The Newton-Raphson step
    pnew = p - f/dfdx;
    // Check for convergence and quit if done
    if(abs(p-pnew) < tol){
      cout << "Root is " << pnew << " to within " 
	   << tol << "\n";
      return 0;
    }
  }
  // We reach this point only if the iteration failed to converge
  cerr << "Failed to converge after " << N << " iterations.\n";
  return 1;
}

Problem 4. (20 pts)

Answer in a clear, complete sentence.

4 (a) In calculus we learn that a derivative of a continuous function f(x) at x is found by taking the limit as h->0 of

    [f(x+h) - f(x)]/h.

The idea of the limit is that we get steadily closer to the derivative as we make h smaller. Suppose you try taking this limit in a computer program. Note that the numerator will involve a subtraction of two numbers that are nearly equal. Will this quantity get closer and closer to the derivative? Explain.

4 (b) What is the advantage of Horner's algorithm?

Please be sure to save the final version of your answer file and submit them. See instructions at the top.

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Last modified 6 October 2009