MCS 471 Practice Problems 4: Numerical Integration

Background Reading:

• Gerald and Wheatley, Chapter 4
  
  

  Topics

  • 4.3 Higher-Order Derivatives
  • 4.4 Extrapolation Techniques
  • 4.5 Newton-Cotes Integration Rules
  • 4.6 Trapezoidal Rule
  • 4.7 Simpson's Rule
  • 4.9 Gaussian Quadrature

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  CAUTION: These problems are just for practice and are not to be handed in. Some of these old exam problems and answers may not be relevant to your current course. The capital letter "S" is used for an integral sign in the problems below. Caveat Usor!

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  Practice Problems:

  In computational problems, 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.

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  0. For problems in the Gerald and Wheatley text, try Chapter 4 Exercises (pp. 384-392, 5th ed.)
  
  #1,6,7,11,14,31,32,33,36,37,44,45,51,58,61.
  1. In parts (a) to (d), approximate

                                         1
                                        S  f(x) dx,
                                         0
     

     where

                  {  (1+x)  on  (0,0.5)
           f(x) = {                        ,
                  {  (2-x)   on  (0.5,1.0)
     

     according to the following simple rules:
     
     (a) Trapezoidal Rule on (0,1.). (Ans.: 1.000)
     (b) Simpson's Rule on (0,1.). (Ans.: 1.333)
     (c) Midpoint Rule on (0,1.). (Ans.: 1.500)
     (d) Double Trapezoidal Rule on (0,0.5) + (0.5,1). (Ans.: 1.250)
  2. In parts (a) to (e), approximate

                                         +1
                                        S  ln(1+x^2) dx,
                                         -1
     

     for the following rules:
     
     (a) Simpson's Rule with 2 subintervals (SR3). (Ans.: 0.4620)
     (b) Midpoint Rule with 4 subintervals (MR4). (Ans.: 0.5069 or 0.5068)
     (c) Trapezoidal Rule with 4 subintervals (TR5). (Ans.: 0.5697)
     (d) Simpson's Rule with 8 subintervals (SR9). (Ans.: 0.5278)
     (e) Two-point Gaussian Rule (GR2). (Ans.: 0.5753)
     (f) About how many subintervals (n?) are needed for an n-panel composite Trapezoidal Rule so that the absolute value of the global approximation error is less than 1.e-8, given that abs(f"(x)) <= 3 on the integration interval (-1.,+1.). Round the final answer to the nearest integer. (Ans.: 14142)
  3. Approximate the integral:

                                     3 
                                    S  exp(x)*sqrt(x) dx 
                                     0 
     

     using 7 points and Simpson's composite rule. Minimize the number of function evaluations, multiplications and additions. (Ans.: 27.45 or 27.43 with extra chops)
  4. Estimate how many function evaluations (i.e., points used) are needed for an efficient, composite Simpson's Rule, so that the absolute value of the global approximation error is <= 0.5e-3. The interval of integration is (-3,2), -2^p/p! <= (d/dx)^p*f(x) <= 2^(-p)/p! for all x and p; and the chopping error is assumed to be negligible. (Final Ans.: 15 function evaluations.)
  5. Approximate the integral:

                                          4 
                                    I =  S  (sqrt(x)/exp(x))  dx  
                                          1 
     

     using a 7 point Simpson's rule. Minimize the number of function evaluations and multiplications in calculating the composite rule.

     (Ans.:  
           x =      (1.000  1.500  2.000  2.500  3.000     3.500     4.000)
        f(x) =(4ch) (0.3678 0.2732 0.1913 0.1297 0.8623e-1 0.5645e-1 0.3663e-1),
     I =(4ch) (0.3678+4*(0.2732+0.1297+0.5649e-1)+2*(0.1913+0.8623e-1)+0.3663e-1)/6.
       =(4ch) (0.4661,0.4660,0.4663), depending on the chopping sequence.)
     

  
  
  

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