MCS 471 --- Computer Problem 1 --- Spring 1997

NonLinear Equations

• General Instructions: Write a single, well documented, program that solves the following finite precision (double precision or 64 bits = 8 bytes is required) problem. You are required to use the following Method, Test Examples, and Output. You may use WATFOR77, regular Fortran, Pascal or C for this problem (else, see Prof. Hanson). However, you must hand in both your program source code and your computer output. Document your own code with a brief description of the problem, an important variables dictionary and headings for important programming steps (use programming language comment lines).

  ---

• General Hybrid Program Method:
  • Hybrid Newton'S Method with BISECTION Back-up, i.e. a hybrid method that STARTS WITH 3 BISECTION ITERATIVES, THEN uses Newton's method STARTING WITH THE LAST BISECTION MIDPOINT until it goes out of the current change of sign interval OR F' IS TOO SMALL, then uses 3 MORE bisection iterations to reduce that interval, AND THEN RETURNS TO NEWTON'S.
    
    
  • Stopping Criterion:
    abs(xnew-xold) + abs(fnew) < tol + 3*eps*(abs(xold)+abs(fold)),
    and
    k < kmax

    with tol = 0.5e-5 for all subroutines, eps is the double precision machine epsilon (see Computer Problem#0), and the iteration limit is kmax = 55.

  • Difficult Test Examples:
    1. F1(x)=exp(-6.1*x) - x*exp(-4.5) on (0.2,1.3);
    2. F2(x)=61. - x*exp(4.6) + exp(3.4*x) on (0.0,1.1);
    3. F3(x)= exp(-10.2*x*x+18.92*x-9.358) - 1.6*exp(-10.6*x*x+18.46*x-9.084) on (0,2).
       {Fortran Caution: Do NOT forget that EXTERNAL statements are needed in Fortran or Watfor77, in all subprograms in which other subprograms are called with FUNCTION arguments; the syntax is like EXTERNAL F1, F2, ..., FN and must come before any executable statements.}
  • Output Requirements:
    1. Name of Your Computer System with Processor if Known
    2. For each example, print a nicely labeled table of neval, xnew, abs(xnew-xold), F(xnew),F'(xold)

       using 5 significant DIGITS, and with the last item only for the Newton's method part of the hybrid method. Be sure to give the starting value for both x and f. The 3rd and 4th items of the table should be printed in e-notation, because errors should almost always be in e-notation.

    3. Print whether the hybrid method converges or not converge after kmax iterations for any of the examples or methods.

  ---

• Maple Symbolic Computation Part:

  Separately, check your answers with Maple or similar symbolic computation language, giving Maple Plots for each examples, and hand in

  1. Your commented Maple Worksheet with Maple Plots, accompanying your program and output requested above.
  2. Explanation for differences between Maple Output and Program Output.
  If you need to change precision, note that Maple works in the background with infinite precision, but only displays 10 digits which may be altered by command,
  evalf(<function>,<n>);
  where <n> is the number of desired digits and <function> is a function or expression.

  For help in editing Maple Worksheet to add comments and plots, click on

  • Editing Maple Worksheets