MCS 471 Practice Problems 1: Nonlinear Equations

Background Reading:

• Gerald and Wheatley, Chapter 1
• Class Maple Web Home Page, introductions and fsolve
  • MSCS Introduction to Maple
  • MCS 471 Beginner's Introduction to Maple in MathLab
  • MSCS Maple Equations (fsolve) Lab
  • MSCS Maple Plot Lab
  • MCS 471 Help for Maple Symbolic Tools

  Topics

  1. Bisection Method
  2. Secant Method
  3. Newton's Method
  4. Fixed Point Interation
  5. Golden Section Search (See Lecture Notes, Not in Text)

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  Homework Problems:

  In problems 1-7, use "EXAM PRECISION": Chop to 4 significant (4C) digits only when you write an intermediate or final answer down and continue calculations with those numbers recorded.
  {Change: Problems 1-4 were changed with 'F(x)=exp(x)-3.5/x' replacing 'F(x)=exp(x)-2.5/x' so that there is a zero in [1,2].
  1. Using the method of BISECTION starting with A(1)=1. and B(1)=2., find the root of F(X)=EXP(X)-3.5/X Record your answer in a table of K,A(K),B(K),F(A(K)),F(B(K)) for each iteration on (A(K),B(K)) for K=1 to 3. Compare your answer to that using fsolve of Maple.
  2. Find the root of F(X)=EXP(X)-3.5/X using the SECANT METHOD for 2 iterations beyond the starting guesses, X(1)=1. and X(2)=2. Record your answers in a table of K,X(K),X(K-1),F(X(K)),F(X(K-1)) for each iteration K. Compare your answer to the Bisection and Maple answers from the first question.
  3. Find the root of F(X)=EXP(X)-3.5/X on [1.,2.] using NEWTON'S METHOD until ABS(X(K)-X(K-1))<.5E-1. Record Results in table of K,X(K),F(X(K)). Use X(1)=1.5 to start. Compare your answer to the Bisection, Secant and Maple answers from the first and second questions.
  4. Numerically solve F(X)=LN(X)-1/X=0 by forming a convergent, fixed point iteration, other than Newton's, starting from X(1)=EXP(1). Record your answers in a table of K, X(K), for K= 1 to 3. (Final Ans.=(4ch) 3,1998). Compare your answer to that using fsolve of Maple. Use the plot function of Maple to plot the problem function G(X).
  5. Find the minimum of G(x)=EXP(x)+7.8/x on (1,2) by the method of GOLDEN SECTION SEARCH for K=1,2,3 iterations. Display your answer in a table of K, AK, BK, XK, UK, GXK, GUX. Use Maple's plot to plot the function G(X) and compare your problem answer to that using minimize of Maple.
     {CAUTION!: The Maple minimize function has a problem with function arguments having reciprocal terms like "1/x". See the example help page for minimize by clicking "http://www.math.uic.edu/~hanson/MAPLE/mcs471minimize.html" (revised 2/04/96).
  6. Find the maximum and its location for G(x)=x*COS(x) on (0.4,1.4) by the method of GOLDEN SECTION SEARCH. Summarize your results by a tabulation of K, AK, BK, XK, UK, GXK, GUK for K=1 to 3. ( Best Final Ans.: (0.8720,0.5609) for (X,G) or (U,G).) {WARNING!: You can Ignore the Maple part of this exercise since "maximize" obviously does not work for this simple trigonometric function. "Maximize" seems to work primarily for polynomial functions over algebraic fields. However, it may work if you approximate "x*cos(x)" by the first few terms of its Taylor series. Use Maple's plot to plot the function G(X) and compare your problem answer to that using maximize of Maple. You can also look for the critical point of the derivative "cos(X)-X*sin(x)" by using fsolve.}
  7. Using Maple, get all the roots, including double and triple roots, of the polynomial
     
     x^5-11x^4+46x^3-90x^2+81x-27.
     
     Also plot the polynomial using the plot function of Maple, on [0,4].
     
     

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