MCS 471 Practice Problems 4: Numerical Integration

Background Reading:

• Gerald and Wheatley, Chapter 4 (2nd Opinion Text Only)

Practice Problems 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
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. Use an efficient, composite, 5-point Trapezoidal Rule to numerically approximate the integral of a piecewise function given by

   
       0.5                         { (x2+3)/4, if 0.5<x<1.0
      S     f(x) dx,  where f(x) = {                                             
       1.5                         { (x+4)/(x2+4), if 1.0<x<1.5    
   
   

   minimizing the number of floating point operations and function evaluations. Also, tabulate

   i		0	1	2	3	4
   xi
   f i

   for all values used. {Caution: You must treat f(x) as one continuous function.}

   ---

   Answer:
   h=(1.5-0.5)/4=0.25; and the tabulated values are

   i		0	1	2	3	4
   xi		0.5000	0.7500	1.000	1.250	1.500
   f i		0.8125	0.8906	1.000	0.9438	0.8800

   FinalAnswer TR₅ = h*(0.5*(f₀+f₄)+f₁+f₂ +f₃) =(r4) 0.9201 in Efficient Form.

   ---

   (S99a)

2. Use a composite, 1-point Gaussian Quadrature (3×G₁; i.e., 3 points total) to numerically approximate the integral given by

         b            0.75 
        S  f(x) dx = S     (exp(2x)/(x+8)) dx ,
         a            0     
   

   minimizing the number of floating point operations and function evaluations. Also, tabulate

   i		1	2	3
   ti
   xi
   Fi

   for all values used, where F_i=F(t_i).

   ---

   Answer:
   h=(0.75-0)/3=0.25; for i = 1 to 3: { t_i=0 (standardized), w_i=2 (standardized), x_i=x(t_i)=(i-0.5)*h, x=x_i+0.5*h*t, dx=0.5*h*dt, a_i=x_i-0.5*h, b_i=x_i+0.5*h}; F(t)=0.5*h*f(x); and the tabulated values are

   i		1	2	3
   ti		0	0	0
   xi		0.1250	0.3750	0.6250
   Fi		0.01975	0.03159	0.05058

   FinalAnswer 3×G₁ = 2*(F₁+F₂+F₃) =(r4) 0.2038 in Efficient Form.

   ---

   (S99a)

Caution: These old problems may have different answers due to different methods and chopping truncation used, so do not worry if your answers are a few least significant digits different.

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+x2) 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.)