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MCS 471 Practice Problems 1: Nonlinear Equations

Hand Calculator and Maple/MATLAB Calculations

Do NOT Hand In: Practice Problems ONLY!



Background Reading:


Practice Problems 1 Topics

  1. Floating Point Arithmetic (for advances information, see Floating Point Arithmetic References

  2. Bisection Method

  3. Secant Method

  4. Newton's Method

  5. Fixed Point Iteration

  6. Golden Section Search, (See Lecture Notes and or click for Class Notes, but topic is briefly in Heath text)

Practice Problems:

In following computational questions, use 4 Digit Exam Precision: Round to 4 significant decimal digits only when you record an intermediate or final answer in your exam booklet; and continue calculations with these rounded, recorded numbers.

  1. Considering just the field representing the fractional part of a nonzero normalized 32 bit floating point real for the IEEE 754 single precision standard, how many distinct numbers can be represented assuming the exponent and sign are fixed? Record as many digits as your calculator allows. Explain how you got your answer. (I.e., ignore the sign bit and the biased exponent field for this count.)


  2. Using Newton's Method, find a numerical approximation to the Zero of

    on [2.5,2.9] starting with the endpoint (k = 0) having the smallest value of |f|, keeping track of the number of ALL function evaluations kfe, the current change in sign interval [a_k,b_k], and tabulating

    k kfe a_k b_k x_k f_k f_k' x_{k+1} |x_{k+1}-x_k|
    0         
    ...        ...

    until |x_{k+1}-x_k| < 0.5e-2.


  3. Find a numerical approximation to the intersection between two functions

    starting from x_0 = 2.400 at k = 0 by forming a convergent Fixed Point Iteration, other than Newton's method. Demonstrate that the fixed point convergence criterion is satisfied near x_0. Tabulate

    k 0 1 2 3
    x_k        
    for k = 0 to 3 iterations.


  4. Using the method of Golden Section Search, find the Maximum and its Interval of Uncertainty for

    on [3.6,6.0] at k = 1, say. Use the rounded version of the GSS constant (c =(4R) 0.3820). Summarize your results with a table of

    k kfe a_k b_k d_k x_k u_k gx_k gu_k
    1                
    ....               ......

    for k = 1 to 3 iterations with k_{fe} the number of function evaluations. Circle and label your best approximation to the maximum, its location, and state the interval of uncertainty.


      Answer: The tabulated answer is

      k kfe a_k b_k d_k x_k u_k gx_k gu_k
      1 2 3.6 6.0 2.400 4.517 5.083 4.390 4.298
      2 3 3.600 5.083 1.483 4.167 4.517 4.357 4.390
      3 4 4.167 5.083 0.916 4.517 4.733 4.390 4.377

      Hence, the Max[g(x)]=(4R)4.390, ArgMax[g(x)]=(4R)4.517 is the location, and IntervalOfUncertainy=(4R)[4.167,5.083].


In the following older problems prior to Fall 1999, use "CHOPPING EXAM PRECISION": The answers are calculated for chopping to 4 significant (4C) digits since the problems are from a time when chopping was used.

Note: Maple comments are not part of these sample exam problems, but were added afterward in the editing stage to aid in analyzing the problems.

  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. (Corrected (2/14/99) Final Ans.=(4ch) 1.998 using g(x)=exp(1/x)). 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.

  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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